Anisotropic x16 LOD (LOD)

Percentage Accurate: 76.3% → 79.5%

Time: 29.9s

Alternatives: 13

Speedup: 0.6×

Specification

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0, t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \left|t\_3 \cdot t\_2 - t\_0 \cdot t\_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_5}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_6}{t\_5}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (fmax (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2))))
        (t_5 (sqrt t_4))
        (t_6 (fabs (- (* t_3 t_2) (* t_0 t_1)))))
   (log2
    (if (> (/ t_4 t_6) (floor maxAniso))
      (/ t_5 (floor maxAniso))
      (/ t_6 t_5)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = fmaxf(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	float t_5 = sqrtf(t_4);
	float t_6 = fabsf(((t_3 * t_2) - (t_0 * t_1)));
	float tmp;
	if ((t_4 / t_6) > floorf(maxAniso)) {
		tmp = t_5 / floorf(maxAniso);
	} else {
		tmp = t_6 / t_5;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) != Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))) ? Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) : ((Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) != Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))) ? Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) : max(Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)), Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))))
	t_5 = sqrt(t_4)
	t_6 = abs(Float32(Float32(t_3 * t_2) - Float32(t_0 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_4 / t_6) > floor(maxAniso))
		tmp = Float32(t_5 / floor(maxAniso));
	else
		tmp = Float32(t_6 / t_5);
	end
	return log2(tmp)
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) * dX_46_u;
	t_4 = max(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	t_5 = sqrt(t_4);
	t_6 = abs(((t_3 * t_2) - (t_0 * t_1)));
	tmp = single(0.0);
	if ((t_4 / t_6) > floor(maxAniso))
		tmp = t_5 / floor(maxAniso);
	else
		tmp = t_6 / t_5;
	end
	tmp_2 = log2(tmp);
end

Initial Program: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0, t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \left|t\_3 \cdot t\_2 - t\_0 \cdot t\_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_5}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_6}{t\_5}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (fmax (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2))))
        (t_5 (sqrt t_4))
        (t_6 (fabs (- (* t_3 t_2) (* t_0 t_1)))))
   (log2
    (if (> (/ t_4 t_6) (floor maxAniso))
      (/ t_5 (floor maxAniso))
      (/ t_6 t_5)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = fmaxf(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	float t_5 = sqrtf(t_4);
	float t_6 = fabsf(((t_3 * t_2) - (t_0 * t_1)));
	float tmp;
	if ((t_4 / t_6) > floorf(maxAniso)) {
		tmp = t_5 / floorf(maxAniso);
	} else {
		tmp = t_6 / t_5;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) != Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))) ? Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) : ((Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) != Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))) ? Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) : max(Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)), Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))))
	t_5 = sqrt(t_4)
	t_6 = abs(Float32(Float32(t_3 * t_2) - Float32(t_0 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_4 / t_6) > floor(maxAniso))
		tmp = Float32(t_5 / floor(maxAniso));
	else
		tmp = Float32(t_6 / t_5);
	end
	return log2(tmp)
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) * dX_46_u;
	t_4 = max(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	t_5 = sqrt(t_4);
	t_6 = abs(((t_3 * t_2) - (t_0 * t_1)));
	tmp = single(0.0);
	if ((t_4 / t_6) > floor(maxAniso))
		tmp = t_5 / floor(maxAniso);
	else
		tmp = t_6 / t_5;
	end
	tmp_2 = log2(tmp);
end

Alternative 1: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := t\_2 \cdot t\_2\\ t_4 := \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ t_5 := t\_1 \cdot dX.v\\ t_6 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_7 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_8 := t\_7 \cdot t\_7 + t\_6 \cdot t\_6\\ t_9 := \mathsf{max}\left(\mathsf{fma}\left(t\_5, dX.v, \left(t\_0 \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left(t\_1 \cdot dY.v, dY.v, \left(t\_0 \cdot dY.u\right) \cdot dY.u\right)\right)\\ t_10 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_11 := \mathsf{max}\left(t\_10 \cdot t\_10 + t\_3, t\_8\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \left|t\_6 \cdot t\_10 - t\_7 \cdot t\_2\right|\\ t_14 := \frac{t\_11}{t\_13} > \left\lfloor maxAniso\right\rfloor \\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;t\_14:\\ \;\;\;\;\frac{t\_12}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_12}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;t\_14:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot t\_1 + t\_3, t\_8\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{\sqrt{\mathsf{max}\left(t\_5 \cdot dX.v, t\_8\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_9}{t\_4} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_9}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{t\_9}} \cdot t\_4\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* dX.u (floor w)))
        (t_3 (* t_2 t_2))
        (t_4
         (fabs (* (* (- (* dY.u dX.v) (* dY.v dX.u)) (floor w)) (floor h))))
        (t_5 (* t_1 dX.v))
        (t_6 (* dY.u (floor w)))
        (t_7 (* dY.v (floor h)))
        (t_8 (+ (* t_7 t_7) (* t_6 t_6)))
        (t_9
         (fmax
          (fma t_5 dX.v (* (* t_0 dX.u) dX.u))
          (fma (* t_1 dY.v) dY.v (* (* t_0 dY.u) dY.u))))
        (t_10 (* dX.v (floor h)))
        (t_11 (fmax (+ (* t_10 t_10) t_3) t_8))
        (t_12 (sqrt t_11))
        (t_13 (fabs (- (* t_6 t_10) (* t_7 t_2))))
        (t_14 (> (/ t_11 t_13) (floor maxAniso))))
   (if (<=
        (if t_14 (/ t_12 (floor maxAniso)) (/ t_13 t_12))
        1999999968613499000.0)
     (log2
      (if t_14
        (/ (sqrt (fmax (+ (* (* dX.v dX.v) t_1) t_3) t_8)) (floor maxAniso))
        (/ t_13 (sqrt (fmax (* t_5 dX.v) t_8)))))
     (log2
      (if (> (/ t_9 t_4) (floor maxAniso))
        (/ (sqrt t_9) (floor maxAniso))
        (* (sqrt (/ 1.0 t_9)) t_4))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = dX_46_u * floorf(w);
	float t_3 = t_2 * t_2;
	float t_4 = fabsf(((((dY_46_u * dX_46_v) - (dY_46_v * dX_46_u)) * floorf(w)) * floorf(h)));
	float t_5 = t_1 * dX_46_v;
	float t_6 = dY_46_u * floorf(w);
	float t_7 = dY_46_v * floorf(h);
	float t_8 = (t_7 * t_7) + (t_6 * t_6);
	float t_9 = fmaxf(fmaf(t_5, dX_46_v, ((t_0 * dX_46_u) * dX_46_u)), fmaf((t_1 * dY_46_v), dY_46_v, ((t_0 * dY_46_u) * dY_46_u)));
	float t_10 = dX_46_v * floorf(h);
	float t_11 = fmaxf(((t_10 * t_10) + t_3), t_8);
	float t_12 = sqrtf(t_11);
	float t_13 = fabsf(((t_6 * t_10) - (t_7 * t_2)));
	int t_14 = (t_11 / t_13) > floorf(maxAniso);
	float tmp;
	if (t_14) {
		tmp = t_12 / floorf(maxAniso);
	} else {
		tmp = t_13 / t_12;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if (t_14) {
			tmp_3 = sqrtf(fmaxf((((dX_46_v * dX_46_v) * t_1) + t_3), t_8)) / floorf(maxAniso);
		} else {
			tmp_3 = t_13 / sqrtf(fmaxf((t_5 * dX_46_v), t_8));
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_9 / t_4) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_9) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / t_9)) * t_4;
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(t_2 * t_2)
	t_4 = abs(Float32(Float32(Float32(Float32(dY_46_u * dX_46_v) - Float32(dY_46_v * dX_46_u)) * floor(w)) * floor(h)))
	t_5 = Float32(t_1 * dX_46_v)
	t_6 = Float32(dY_46_u * floor(w))
	t_7 = Float32(dY_46_v * floor(h))
	t_8 = Float32(Float32(t_7 * t_7) + Float32(t_6 * t_6))
	t_9 = (fma(t_5, dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u)) != fma(t_5, dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u))) ? fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u)) : ((fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u)) != fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u))) ? fma(t_5, dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u)) : max(fma(t_5, dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u)), fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u))))
	t_10 = Float32(dX_46_v * floor(h))
	t_11 = (Float32(Float32(t_10 * t_10) + t_3) != Float32(Float32(t_10 * t_10) + t_3)) ? t_8 : ((t_8 != t_8) ? Float32(Float32(t_10 * t_10) + t_3) : max(Float32(Float32(t_10 * t_10) + t_3), t_8))
	t_12 = sqrt(t_11)
	t_13 = abs(Float32(Float32(t_6 * t_10) - Float32(t_7 * t_2)))
	t_14 = Float32(t_11 / t_13) > floor(maxAniso)
	tmp = Float32(0.0)
	if (t_14)
		tmp = Float32(t_12 / floor(maxAniso));
	else
		tmp = Float32(t_13 / t_12);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (t_14)
			tmp_3 = Float32(sqrt(((Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_3) != Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_3)) ? t_8 : ((t_8 != t_8) ? Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_3) : max(Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_3), t_8)))) / floor(maxAniso));
		else
			tmp_3 = Float32(t_13 / sqrt(((Float32(t_5 * dX_46_v) != Float32(t_5 * dX_46_v)) ? t_8 : ((t_8 != t_8) ? Float32(t_5 * dX_46_v) : max(Float32(t_5 * dX_46_v), t_8)))));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_9 / t_4) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_9) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / t_9)) * t_4);
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)} - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. lower-floor.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\color{blue}{\left\lfloor w\right\rfloor } \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   5. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. pow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. unpow-prod-down N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. pow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      8. lower-*.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   7. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   8. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_2 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_3 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_4 := t\_3 \cdot t\_3 + t\_2 \cdot t\_2\\ t_5 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_6 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_7 := \mathsf{max}\left(t\_6 \cdot t\_6 + t\_1 \cdot t\_1, t\_4\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := \frac{t\_8}{\left\lfloor maxAniso\right\rfloor }\\ t_10 := \left|t\_2 \cdot t\_6 - t\_3 \cdot t\_1\right|\\ t_11 := \frac{t\_7}{t\_10} > \left\lfloor maxAniso\right\rfloor \\ t_12 := t\_0 \cdot dX.v\\ t_13 := \mathsf{max}\left(\mathsf{fma}\left(t\_12, dX.v, \left(t\_5 \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left(t\_0 \cdot dY.v, dY.v, \left(t\_5 \cdot dY.u\right) \cdot dY.u\right)\right)\\ t_14 := \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;t\_11:\\ \;\;\;\;t\_9\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_10}{t\_8}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;t\_11:\\ \;\;\;\;t\_9\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_10}{\sqrt{\mathsf{max}\left(t\_12 \cdot dX.v, t\_4\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_13}{t\_14} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_13}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{t\_13}} \cdot t\_14\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* dX.u (floor w)))
        (t_2 (* dY.u (floor w)))
        (t_3 (* dY.v (floor h)))
        (t_4 (+ (* t_3 t_3) (* t_2 t_2)))
        (t_5 (pow (floor w) 2.0))
        (t_6 (* dX.v (floor h)))
        (t_7 (fmax (+ (* t_6 t_6) (* t_1 t_1)) t_4))
        (t_8 (sqrt t_7))
        (t_9 (/ t_8 (floor maxAniso)))
        (t_10 (fabs (- (* t_2 t_6) (* t_3 t_1))))
        (t_11 (> (/ t_7 t_10) (floor maxAniso)))
        (t_12 (* t_0 dX.v))
        (t_13
         (fmax
          (fma t_12 dX.v (* (* t_5 dX.u) dX.u))
          (fma (* t_0 dY.v) dY.v (* (* t_5 dY.u) dY.u))))
        (t_14
         (fabs (* (* (- (* dY.u dX.v) (* dY.v dX.u)) (floor w)) (floor h)))))
   (if (<= (if t_11 t_9 (/ t_10 t_8)) 1999999968613499000.0)
     (log2 (if t_11 t_9 (/ t_10 (sqrt (fmax (* t_12 dX.v) t_4)))))
     (log2
      (if (> (/ t_13 t_14) (floor maxAniso))
        (/ (sqrt t_13) (floor maxAniso))
        (* (sqrt (/ 1.0 t_13)) t_14))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = dX_46_u * floorf(w);
	float t_2 = dY_46_u * floorf(w);
	float t_3 = dY_46_v * floorf(h);
	float t_4 = (t_3 * t_3) + (t_2 * t_2);
	float t_5 = powf(floorf(w), 2.0f);
	float t_6 = dX_46_v * floorf(h);
	float t_7 = fmaxf(((t_6 * t_6) + (t_1 * t_1)), t_4);
	float t_8 = sqrtf(t_7);
	float t_9 = t_8 / floorf(maxAniso);
	float t_10 = fabsf(((t_2 * t_6) - (t_3 * t_1)));
	int t_11 = (t_7 / t_10) > floorf(maxAniso);
	float t_12 = t_0 * dX_46_v;
	float t_13 = fmaxf(fmaf(t_12, dX_46_v, ((t_5 * dX_46_u) * dX_46_u)), fmaf((t_0 * dY_46_v), dY_46_v, ((t_5 * dY_46_u) * dY_46_u)));
	float t_14 = fabsf(((((dY_46_u * dX_46_v) - (dY_46_v * dX_46_u)) * floorf(w)) * floorf(h)));
	float tmp;
	if (t_11) {
		tmp = t_9;
	} else {
		tmp = t_10 / t_8;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if (t_11) {
			tmp_3 = t_9;
		} else {
			tmp_3 = t_10 / sqrtf(fmaxf((t_12 * dX_46_v), t_4));
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_13 / t_14) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_13) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / t_13)) * t_14;
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(dX_46_u * floor(w))
	t_2 = Float32(dY_46_u * floor(w))
	t_3 = Float32(dY_46_v * floor(h))
	t_4 = Float32(Float32(t_3 * t_3) + Float32(t_2 * t_2))
	t_5 = floor(w) ^ Float32(2.0)
	t_6 = Float32(dX_46_v * floor(h))
	t_7 = (Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1)) != Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1))) ? t_4 : ((t_4 != t_4) ? Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1)) : max(Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1)), t_4))
	t_8 = sqrt(t_7)
	t_9 = Float32(t_8 / floor(maxAniso))
	t_10 = abs(Float32(Float32(t_2 * t_6) - Float32(t_3 * t_1)))
	t_11 = Float32(t_7 / t_10) > floor(maxAniso)
	t_12 = Float32(t_0 * dX_46_v)
	t_13 = (fma(t_12, dX_46_v, Float32(Float32(t_5 * dX_46_u) * dX_46_u)) != fma(t_12, dX_46_v, Float32(Float32(t_5 * dX_46_u) * dX_46_u))) ? fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(t_5 * dY_46_u) * dY_46_u)) : ((fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(t_5 * dY_46_u) * dY_46_u)) != fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(t_5 * dY_46_u) * dY_46_u))) ? fma(t_12, dX_46_v, Float32(Float32(t_5 * dX_46_u) * dX_46_u)) : max(fma(t_12, dX_46_v, Float32(Float32(t_5 * dX_46_u) * dX_46_u)), fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(t_5 * dY_46_u) * dY_46_u))))
	t_14 = abs(Float32(Float32(Float32(Float32(dY_46_u * dX_46_v) - Float32(dY_46_v * dX_46_u)) * floor(w)) * floor(h)))
	tmp = Float32(0.0)
	if (t_11)
		tmp = t_9;
	else
		tmp = Float32(t_10 / t_8);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (t_11)
			tmp_3 = t_9;
		else
			tmp_3 = Float32(t_10 / sqrt(((Float32(t_12 * dX_46_v) != Float32(t_12 * dX_46_v)) ? t_4 : ((t_4 != t_4) ? Float32(t_12 * dX_46_v) : max(Float32(t_12 * dX_46_v), t_4)))));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_13 / t_14) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_13) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / t_13)) * t_14);
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)} - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. lower-floor.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\color{blue}{\left\lfloor w\right\rfloor } \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   5. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   8. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_4 := \mathsf{max}\left(\mathsf{fma}\left(t\_3 \cdot dX.v, dX.v, \left(t\_0 \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left(t\_3 \cdot dY.v, dY.v, \left(t\_0 \cdot dY.u\right) \cdot dY.u\right)\right)\\ t_5 := \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ t_6 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_7 := \left|\left(dY.v \cdot t\_2 - t\_6 \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right|\\ t_8 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_9 := \mathsf{max}\left(t\_8 \cdot t\_8 + t\_2 \cdot t\_2, t\_1 \cdot t\_1 + t\_6 \cdot t\_6\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{max}\left({t\_2}^{2} + {t\_8}^{2}, {t\_6}^{2} + {t\_1}^{2}\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \left|t\_6 \cdot t\_8 - t\_1 \cdot t\_2\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_9}{t\_13} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_10}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_10}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_11}{t\_7} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_12}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_7}{t\_12}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{t\_5} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_4}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{t\_4}} \cdot t\_5\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dY.v (floor h)))
        (t_2 (* dX.u (floor w)))
        (t_3 (pow (floor h) 2.0))
        (t_4
         (fmax
          (fma (* t_3 dX.v) dX.v (* (* t_0 dX.u) dX.u))
          (fma (* t_3 dY.v) dY.v (* (* t_0 dY.u) dY.u))))
        (t_5
         (fabs (* (* (- (* dY.u dX.v) (* dY.v dX.u)) (floor w)) (floor h))))
        (t_6 (* dY.u (floor w)))
        (t_7 (fabs (* (- (* dY.v t_2) (* t_6 dX.v)) (floor h))))
        (t_8 (* dX.v (floor h)))
        (t_9 (fmax (+ (* t_8 t_8) (* t_2 t_2)) (+ (* t_1 t_1) (* t_6 t_6))))
        (t_10 (sqrt t_9))
        (t_11
         (fmax
          (+ (pow t_2 2.0) (pow t_8 2.0))
          (+ (pow t_6 2.0) (pow t_1 2.0))))
        (t_12 (sqrt t_11))
        (t_13 (fabs (- (* t_6 t_8) (* t_1 t_2)))))
   (if (<=
        (if (> (/ t_9 t_13) (floor maxAniso))
          (/ t_10 (floor maxAniso))
          (/ t_13 t_10))
        1999999968613499000.0)
     (log2
      (if (> (/ t_11 t_7) (floor maxAniso))
        (/ t_12 (floor maxAniso))
        (/ t_7 t_12)))
     (log2
      (if (> (/ t_4 t_5) (floor maxAniso))
        (/ (sqrt t_4) (floor maxAniso))
        (* (sqrt (/ 1.0 t_4)) t_5))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = dY_46_v * floorf(h);
	float t_2 = dX_46_u * floorf(w);
	float t_3 = powf(floorf(h), 2.0f);
	float t_4 = fmaxf(fmaf((t_3 * dX_46_v), dX_46_v, ((t_0 * dX_46_u) * dX_46_u)), fmaf((t_3 * dY_46_v), dY_46_v, ((t_0 * dY_46_u) * dY_46_u)));
	float t_5 = fabsf(((((dY_46_u * dX_46_v) - (dY_46_v * dX_46_u)) * floorf(w)) * floorf(h)));
	float t_6 = dY_46_u * floorf(w);
	float t_7 = fabsf((((dY_46_v * t_2) - (t_6 * dX_46_v)) * floorf(h)));
	float t_8 = dX_46_v * floorf(h);
	float t_9 = fmaxf(((t_8 * t_8) + (t_2 * t_2)), ((t_1 * t_1) + (t_6 * t_6)));
	float t_10 = sqrtf(t_9);
	float t_11 = fmaxf((powf(t_2, 2.0f) + powf(t_8, 2.0f)), (powf(t_6, 2.0f) + powf(t_1, 2.0f)));
	float t_12 = sqrtf(t_11);
	float t_13 = fabsf(((t_6 * t_8) - (t_1 * t_2)));
	float tmp;
	if ((t_9 / t_13) > floorf(maxAniso)) {
		tmp = t_10 / floorf(maxAniso);
	} else {
		tmp = t_13 / t_10;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_11 / t_7) > floorf(maxAniso)) {
			tmp_3 = t_12 / floorf(maxAniso);
		} else {
			tmp_3 = t_7 / t_12;
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_4 / t_5) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_4) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / t_4)) * t_5;
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dY_46_v * floor(h))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = floor(h) ^ Float32(2.0)
	t_4 = (fma(Float32(t_3 * dX_46_v), dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u)) != fma(Float32(t_3 * dX_46_v), dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u))) ? fma(Float32(t_3 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u)) : ((fma(Float32(t_3 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u)) != fma(Float32(t_3 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u))) ? fma(Float32(t_3 * dX_46_v), dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u)) : max(fma(Float32(t_3 * dX_46_v), dX_46_v, Float32(Float32(t_0 * dX_46_u) * dX_46_u)), fma(Float32(t_3 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u))))
	t_5 = abs(Float32(Float32(Float32(Float32(dY_46_u * dX_46_v) - Float32(dY_46_v * dX_46_u)) * floor(w)) * floor(h)))
	t_6 = Float32(dY_46_u * floor(w))
	t_7 = abs(Float32(Float32(Float32(dY_46_v * t_2) - Float32(t_6 * dX_46_v)) * floor(h)))
	t_8 = Float32(dX_46_v * floor(h))
	t_9 = (Float32(Float32(t_8 * t_8) + Float32(t_2 * t_2)) != Float32(Float32(t_8 * t_8) + Float32(t_2 * t_2))) ? Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6)) : ((Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6)) != Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6))) ? Float32(Float32(t_8 * t_8) + Float32(t_2 * t_2)) : max(Float32(Float32(t_8 * t_8) + Float32(t_2 * t_2)), Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6))))
	t_10 = sqrt(t_9)
	t_11 = (Float32((t_2 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) != Float32((t_2 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))) ? Float32((t_6 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) : ((Float32((t_6 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) != Float32((t_6 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) ? Float32((t_2 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) : max(Float32((t_2 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))), Float32((t_6 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))))
	t_12 = sqrt(t_11)
	t_13 = abs(Float32(Float32(t_6 * t_8) - Float32(t_1 * t_2)))
	tmp = Float32(0.0)
	if (Float32(t_9 / t_13) > floor(maxAniso))
		tmp = Float32(t_10 / floor(maxAniso));
	else
		tmp = Float32(t_13 / t_10);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_11 / t_7) > floor(maxAniso))
			tmp_3 = Float32(t_12 / floor(maxAniso));
		else
			tmp_3 = Float32(t_7 / t_12);
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_4 / t_5) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_4) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / t_4)) * t_5);
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   8. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right|}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := t\_3 \cdot t\_3\\ t_5 := t\_1 \cdot dX.v\\ t_6 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_7 := t\_2 \cdot t\_2 + t\_6 \cdot t\_6\\ t_8 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_9 := \mathsf{max}\left(\mathsf{fma}\left(t\_5, dX.v, \left(t\_8 \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left(t\_1 \cdot dY.v, dY.v, \left(t\_8 \cdot dY.u\right) \cdot dY.u\right)\right)\\ t_10 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_11 := \mathsf{max}\left(t\_10 \cdot t\_10 + t\_4, t\_7\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \left|t\_6 \cdot t\_10 - t\_2 \cdot t\_3\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_11}{t\_13} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_12}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_12}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_11}{\left|\left(\left\lfloor h\right\rfloor \cdot dX.u\right) \cdot \left(dY.v \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot t\_1 + t\_4, t\_7\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\frac{dY.v \cdot dX.u}{dY.u} - dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(t\_5 \cdot dX.v, t\_7\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_9}{t\_0} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_9}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{t\_9}} \cdot t\_0\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (fabs (* (* (- (* dY.u dX.v) (* dY.v dX.u)) (floor w)) (floor h))))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* dY.v (floor h)))
        (t_3 (* dX.u (floor w)))
        (t_4 (* t_3 t_3))
        (t_5 (* t_1 dX.v))
        (t_6 (* dY.u (floor w)))
        (t_7 (+ (* t_2 t_2) (* t_6 t_6)))
        (t_8 (pow (floor w) 2.0))
        (t_9
         (fmax
          (fma t_5 dX.v (* (* t_8 dX.u) dX.u))
          (fma (* t_1 dY.v) dY.v (* (* t_8 dY.u) dY.u))))
        (t_10 (* dX.v (floor h)))
        (t_11 (fmax (+ (* t_10 t_10) t_4) t_7))
        (t_12 (sqrt t_11))
        (t_13 (fabs (- (* t_6 t_10) (* t_2 t_3)))))
   (if (<=
        (if (> (/ t_11 t_13) (floor maxAniso))
          (/ t_12 (floor maxAniso))
          (/ t_13 t_12))
        1999999968613499000.0)
     (log2
      (if (>
           (/ t_11 (fabs (* (* (floor h) dX.u) (* dY.v (floor w)))))
           (floor maxAniso))
        (/ (sqrt (fmax (+ (* (* dX.v dX.v) t_1) t_4) t_7)) (floor maxAniso))
        (/
         (fabs
          (* (* (- (/ (* dY.v dX.u) dY.u) dX.v) (* (floor h) (floor w))) dY.u))
         (sqrt (fmax (* t_5 dX.v) t_7)))))
     (log2
      (if (> (/ t_9 t_0) (floor maxAniso))
        (/ (sqrt t_9) (floor maxAniso))
        (* (sqrt (/ 1.0 t_9)) t_0))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = fabsf(((((dY_46_u * dX_46_v) - (dY_46_v * dX_46_u)) * floorf(w)) * floorf(h)));
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = dY_46_v * floorf(h);
	float t_3 = dX_46_u * floorf(w);
	float t_4 = t_3 * t_3;
	float t_5 = t_1 * dX_46_v;
	float t_6 = dY_46_u * floorf(w);
	float t_7 = (t_2 * t_2) + (t_6 * t_6);
	float t_8 = powf(floorf(w), 2.0f);
	float t_9 = fmaxf(fmaf(t_5, dX_46_v, ((t_8 * dX_46_u) * dX_46_u)), fmaf((t_1 * dY_46_v), dY_46_v, ((t_8 * dY_46_u) * dY_46_u)));
	float t_10 = dX_46_v * floorf(h);
	float t_11 = fmaxf(((t_10 * t_10) + t_4), t_7);
	float t_12 = sqrtf(t_11);
	float t_13 = fabsf(((t_6 * t_10) - (t_2 * t_3)));
	float tmp;
	if ((t_11 / t_13) > floorf(maxAniso)) {
		tmp = t_12 / floorf(maxAniso);
	} else {
		tmp = t_13 / t_12;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_11 / fabsf(((floorf(h) * dX_46_u) * (dY_46_v * floorf(w))))) > floorf(maxAniso)) {
			tmp_3 = sqrtf(fmaxf((((dX_46_v * dX_46_v) * t_1) + t_4), t_7)) / floorf(maxAniso);
		} else {
			tmp_3 = fabsf((((((dY_46_v * dX_46_u) / dY_46_u) - dX_46_v) * (floorf(h) * floorf(w))) * dY_46_u)) / sqrtf(fmaxf((t_5 * dX_46_v), t_7));
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_9 / t_0) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_9) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / t_9)) * t_0;
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = abs(Float32(Float32(Float32(Float32(dY_46_u * dX_46_v) - Float32(dY_46_v * dX_46_u)) * floor(w)) * floor(h)))
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(dY_46_v * floor(h))
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(t_3 * t_3)
	t_5 = Float32(t_1 * dX_46_v)
	t_6 = Float32(dY_46_u * floor(w))
	t_7 = Float32(Float32(t_2 * t_2) + Float32(t_6 * t_6))
	t_8 = floor(w) ^ Float32(2.0)
	t_9 = (fma(t_5, dX_46_v, Float32(Float32(t_8 * dX_46_u) * dX_46_u)) != fma(t_5, dX_46_v, Float32(Float32(t_8 * dX_46_u) * dX_46_u))) ? fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_8 * dY_46_u) * dY_46_u)) : ((fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_8 * dY_46_u) * dY_46_u)) != fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_8 * dY_46_u) * dY_46_u))) ? fma(t_5, dX_46_v, Float32(Float32(t_8 * dX_46_u) * dX_46_u)) : max(fma(t_5, dX_46_v, Float32(Float32(t_8 * dX_46_u) * dX_46_u)), fma(Float32(t_1 * dY_46_v), dY_46_v, Float32(Float32(t_8 * dY_46_u) * dY_46_u))))
	t_10 = Float32(dX_46_v * floor(h))
	t_11 = (Float32(Float32(t_10 * t_10) + t_4) != Float32(Float32(t_10 * t_10) + t_4)) ? t_7 : ((t_7 != t_7) ? Float32(Float32(t_10 * t_10) + t_4) : max(Float32(Float32(t_10 * t_10) + t_4), t_7))
	t_12 = sqrt(t_11)
	t_13 = abs(Float32(Float32(t_6 * t_10) - Float32(t_2 * t_3)))
	tmp = Float32(0.0)
	if (Float32(t_11 / t_13) > floor(maxAniso))
		tmp = Float32(t_12 / floor(maxAniso));
	else
		tmp = Float32(t_13 / t_12);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_11 / abs(Float32(Float32(floor(h) * dX_46_u) * Float32(dY_46_v * floor(w))))) > floor(maxAniso))
			tmp_3 = Float32(sqrt(((Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_4) != Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_4)) ? t_7 : ((t_7 != t_7) ? Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_4) : max(Float32(Float32(Float32(dX_46_v * dX_46_v) * t_1) + t_4), t_7)))) / floor(maxAniso));
		else
			tmp_3 = Float32(abs(Float32(Float32(Float32(Float32(Float32(dY_46_v * dX_46_u) / dY_46_u) - dX_46_v) * Float32(floor(h) * floor(w))) * dY_46_u)) / sqrt(((Float32(t_5 * dX_46_v) != Float32(t_5 * dX_46_v)) ? t_7 : ((t_7 != t_7) ? Float32(t_5 * dX_46_v) : max(Float32(t_5 * dX_46_v), t_7)))));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_9 / t_0) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_9) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / t_9)) * t_0);
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)} - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. lower-floor.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\color{blue}{\left\lfloor w\right\rfloor } \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   5. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. pow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. unpow-prod-down N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. pow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      8. lower-*.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   7. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   8. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|dX.u \cdot \color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.v\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dX.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.v}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(\left(dX.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)} \cdot dY.v\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dX.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.v\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dY.v \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      8. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dY.v \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      9. lower-floor.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      10. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \color{blue}{\left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      11. lower-floor.f32 99.4

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   10. Applied rewrites 99.4%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|dY.u \cdot \left(\frac{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}{dY.u} - dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\frac{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}{dY.u} - dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\frac{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}{dY.u} - dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   13. Applied rewrites 99.4%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(\frac{dY.v \cdot dX.u}{dY.u} - dX.v\right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   8. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot dX.u\right) \cdot \left(dY.v \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\frac{dY.v \cdot dX.u}{dY.u} - dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\left(\left(dY.u \cdot dX.v - dY.v \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_1 := \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot t\_0\right|\\ t_2 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := t\_3 \cdot t\_3\\ t_5 := t\_2 \cdot dX.v\\ t_6 := t\_5 \cdot dX.v\\ t_7 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_8 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_9 := \left(t\_8 \cdot dX.u\right) \cdot dX.u\\ t_10 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_11 := t\_7 \cdot t\_7 + t\_10 \cdot t\_10\\ t_12 := \mathsf{fma}\left(t\_2 \cdot dY.v, dY.v, \left(t\_8 \cdot dY.u\right) \cdot dY.u\right)\\ t_13 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_14 := \mathsf{max}\left(t\_13 \cdot t\_13 + t\_4, t\_11\right)\\ t_15 := \sqrt{t\_14}\\ t_16 := \left|t\_10 \cdot t\_13 - t\_7 \cdot t\_3\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_14}{t\_16} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_15}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_16}{t\_15}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_14}{\left|\left(\left\lfloor h\right\rfloor \cdot dX.u\right) \cdot \left(dY.v \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot t\_2 + t\_4, t\_11\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\frac{dY.v \cdot dX.u}{dY.u} - dX.v\right) \cdot t\_0\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(t\_6, t\_11\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_9, t\_12\right)}{t\_1} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_6, \mathsf{fma}\left(\left\lfloor w\right\rfloor , t\_10 \cdot dY.u, {t\_7}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_5, dX.v, t\_9\right), t\_12\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) (floor w)))
        (t_1 (fabs (* (fma (- dX.u) dY.v (* dY.u dX.v)) t_0)))
        (t_2 (pow (floor h) 2.0))
        (t_3 (* dX.u (floor w)))
        (t_4 (* t_3 t_3))
        (t_5 (* t_2 dX.v))
        (t_6 (* t_5 dX.v))
        (t_7 (* dY.v (floor h)))
        (t_8 (pow (floor w) 2.0))
        (t_9 (* (* t_8 dX.u) dX.u))
        (t_10 (* dY.u (floor w)))
        (t_11 (+ (* t_7 t_7) (* t_10 t_10)))
        (t_12 (fma (* t_2 dY.v) dY.v (* (* t_8 dY.u) dY.u)))
        (t_13 (* dX.v (floor h)))
        (t_14 (fmax (+ (* t_13 t_13) t_4) t_11))
        (t_15 (sqrt t_14))
        (t_16 (fabs (- (* t_10 t_13) (* t_7 t_3)))))
   (if (<=
        (if (> (/ t_14 t_16) (floor maxAniso))
          (/ t_15 (floor maxAniso))
          (/ t_16 t_15))
        1999999968613499000.0)
     (log2
      (if (>
           (/ t_14 (fabs (* (* (floor h) dX.u) (* dY.v (floor w)))))
           (floor maxAniso))
        (/ (sqrt (fmax (+ (* (* dX.v dX.v) t_2) t_4) t_11)) (floor maxAniso))
        (/
         (fabs (* (* (- (/ (* dY.v dX.u) dY.u) dX.v) t_0) dY.u))
         (sqrt (fmax t_6 t_11)))))
     (log2
      (if (> (/ (fmax t_9 t_12) t_1) (floor maxAniso))
        (/
         (sqrt (fmax t_6 (fma (floor w) (* t_10 dY.u) (pow t_7 2.0))))
         (floor maxAniso))
        (* t_1 (sqrt (/ 1.0 (fmax (fma t_5 dX.v t_9) t_12)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * floorf(w);
	float t_1 = fabsf((fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v)) * t_0));
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = dX_46_u * floorf(w);
	float t_4 = t_3 * t_3;
	float t_5 = t_2 * dX_46_v;
	float t_6 = t_5 * dX_46_v;
	float t_7 = dY_46_v * floorf(h);
	float t_8 = powf(floorf(w), 2.0f);
	float t_9 = (t_8 * dX_46_u) * dX_46_u;
	float t_10 = dY_46_u * floorf(w);
	float t_11 = (t_7 * t_7) + (t_10 * t_10);
	float t_12 = fmaf((t_2 * dY_46_v), dY_46_v, ((t_8 * dY_46_u) * dY_46_u));
	float t_13 = dX_46_v * floorf(h);
	float t_14 = fmaxf(((t_13 * t_13) + t_4), t_11);
	float t_15 = sqrtf(t_14);
	float t_16 = fabsf(((t_10 * t_13) - (t_7 * t_3)));
	float tmp;
	if ((t_14 / t_16) > floorf(maxAniso)) {
		tmp = t_15 / floorf(maxAniso);
	} else {
		tmp = t_16 / t_15;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_14 / fabsf(((floorf(h) * dX_46_u) * (dY_46_v * floorf(w))))) > floorf(maxAniso)) {
			tmp_3 = sqrtf(fmaxf((((dX_46_v * dX_46_v) * t_2) + t_4), t_11)) / floorf(maxAniso);
		} else {
			tmp_3 = fabsf((((((dY_46_v * dX_46_u) / dY_46_u) - dX_46_v) * t_0) * dY_46_u)) / sqrtf(fmaxf(t_6, t_11));
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((fmaxf(t_9, t_12) / t_1) > floorf(maxAniso)) {
			tmp_4 = sqrtf(fmaxf(t_6, fmaf(floorf(w), (t_10 * dY_46_u), powf(t_7, 2.0f)))) / floorf(maxAniso);
		} else {
			tmp_4 = t_1 * sqrtf((1.0f / fmaxf(fmaf(t_5, dX_46_v, t_9), t_12)));
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * floor(w))
	t_1 = abs(Float32(fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v)) * t_0))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(t_3 * t_3)
	t_5 = Float32(t_2 * dX_46_v)
	t_6 = Float32(t_5 * dX_46_v)
	t_7 = Float32(dY_46_v * floor(h))
	t_8 = floor(w) ^ Float32(2.0)
	t_9 = Float32(Float32(t_8 * dX_46_u) * dX_46_u)
	t_10 = Float32(dY_46_u * floor(w))
	t_11 = Float32(Float32(t_7 * t_7) + Float32(t_10 * t_10))
	t_12 = fma(Float32(t_2 * dY_46_v), dY_46_v, Float32(Float32(t_8 * dY_46_u) * dY_46_u))
	t_13 = Float32(dX_46_v * floor(h))
	t_14 = (Float32(Float32(t_13 * t_13) + t_4) != Float32(Float32(t_13 * t_13) + t_4)) ? t_11 : ((t_11 != t_11) ? Float32(Float32(t_13 * t_13) + t_4) : max(Float32(Float32(t_13 * t_13) + t_4), t_11))
	t_15 = sqrt(t_14)
	t_16 = abs(Float32(Float32(t_10 * t_13) - Float32(t_7 * t_3)))
	tmp = Float32(0.0)
	if (Float32(t_14 / t_16) > floor(maxAniso))
		tmp = Float32(t_15 / floor(maxAniso));
	else
		tmp = Float32(t_16 / t_15);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_14 / abs(Float32(Float32(floor(h) * dX_46_u) * Float32(dY_46_v * floor(w))))) > floor(maxAniso))
			tmp_3 = Float32(sqrt(((Float32(Float32(Float32(dX_46_v * dX_46_v) * t_2) + t_4) != Float32(Float32(Float32(dX_46_v * dX_46_v) * t_2) + t_4)) ? t_11 : ((t_11 != t_11) ? Float32(Float32(Float32(dX_46_v * dX_46_v) * t_2) + t_4) : max(Float32(Float32(Float32(dX_46_v * dX_46_v) * t_2) + t_4), t_11)))) / floor(maxAniso));
		else
			tmp_3 = Float32(abs(Float32(Float32(Float32(Float32(Float32(dY_46_v * dX_46_u) / dY_46_u) - dX_46_v) * t_0) * dY_46_u)) / sqrt(((t_6 != t_6) ? t_11 : ((t_11 != t_11) ? t_6 : max(t_6, t_11)))));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(((t_9 != t_9) ? t_12 : ((t_12 != t_12) ? t_9 : max(t_9, t_12))) / t_1) > floor(maxAniso))
			tmp_4 = Float32(sqrt(((t_6 != t_6) ? fma(floor(w), Float32(t_10 * dY_46_u), (t_7 ^ Float32(2.0))) : ((fma(floor(w), Float32(t_10 * dY_46_u), (t_7 ^ Float32(2.0))) != fma(floor(w), Float32(t_10 * dY_46_u), (t_7 ^ Float32(2.0)))) ? t_6 : max(t_6, fma(floor(w), Float32(t_10 * dY_46_u), (t_7 ^ Float32(2.0))))))) / floor(maxAniso));
		else
			tmp_4 = Float32(t_1 * sqrt(Float32(Float32(1.0) / ((fma(t_5, dX_46_v, t_9) != fma(t_5, dX_46_v, t_9)) ? t_12 : ((t_12 != t_12) ? fma(t_5, dX_46_v, t_9) : max(fma(t_5, dX_46_v, t_9), t_12))))));
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)} - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. lower-floor.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\color{blue}{\left\lfloor w\right\rfloor } \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   5. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)}\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. pow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. unpow-prod-down N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. pow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      8. lower-*.f32 100.0

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   7. Applied rewrites 100.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   8. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|dX.u \cdot \color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.v\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      2. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dX.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.v}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(\left(dX.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)} \cdot dY.v\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      4. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dX.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.v\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dY.v \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      8. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dY.v \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      9. lower-floor.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      10. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \color{blue}{\left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

      11. lower-floor.f32 99.4

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   10. Applied rewrites 99.4%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|dY.u \cdot \left(\frac{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}{dY.u} - dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\frac{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}{dY.u} - dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\frac{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}{dY.u} - dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   13. Applied rewrites 99.4%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(\frac{dY.v \cdot dX.u}{dY.u} - dX.v\right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.4%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   9. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.5%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 17.9%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot dX.u\right) \cdot \left(dY.v \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\frac{dY.v \cdot dX.u}{dY.u} - dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) \cdot dY.u\right|}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_2 := \left(t\_0 \cdot dX.u\right) \cdot dX.u\\ t_3 := \mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\\ t_4 := \left(\left|t\_3\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \\ t_5 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_6 := {t\_5}^{2}\\ t_7 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_8 := t\_7 \cdot dX.v\\ t_9 := \mathsf{fma}\left(t\_7 \cdot dY.v, dY.v, \left(t\_0 \cdot dY.u\right) \cdot dY.u\right)\\ t_10 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_11 := {t\_10}^{2} + t\_6\\ t_12 := \left|t\_3 \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ t_13 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_14 := {t\_13}^{2}\\ t_15 := \mathsf{max}\left(t\_13 \cdot t\_13 + t\_1 \cdot t\_1, t\_5 \cdot t\_5 + t\_10 \cdot t\_10\right)\\ t_16 := \sqrt{t\_15}\\ t_17 := \mathsf{max}\left({t\_1}^{2} + t\_14, t\_11\right)\\ t_18 := \left|t\_10 \cdot t\_13 - t\_5 \cdot t\_1\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_15}{t\_18} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_16}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_18}{t\_16}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_17}{t\_4} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_17}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{max}\left(t\_14, t\_11\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_2, t\_9\right)}{t\_12} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_8 \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , t\_10 \cdot dY.u, t\_6\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_12 \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_8, dX.v, t\_2\right), t\_9\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dX.u (floor w)))
        (t_2 (* (* t_0 dX.u) dX.u))
        (t_3 (fma (- dX.u) dY.v (* dY.u dX.v)))
        (t_4 (* (* (fabs t_3) (floor w)) (floor h)))
        (t_5 (* dY.v (floor h)))
        (t_6 (pow t_5 2.0))
        (t_7 (pow (floor h) 2.0))
        (t_8 (* t_7 dX.v))
        (t_9 (fma (* t_7 dY.v) dY.v (* (* t_0 dY.u) dY.u)))
        (t_10 (* dY.u (floor w)))
        (t_11 (+ (pow t_10 2.0) t_6))
        (t_12 (fabs (* t_3 (* (floor h) (floor w)))))
        (t_13 (* dX.v (floor h)))
        (t_14 (pow t_13 2.0))
        (t_15
         (fmax (+ (* t_13 t_13) (* t_1 t_1)) (+ (* t_5 t_5) (* t_10 t_10))))
        (t_16 (sqrt t_15))
        (t_17 (fmax (+ (pow t_1 2.0) t_14) t_11))
        (t_18 (fabs (- (* t_10 t_13) (* t_5 t_1)))))
   (if (<=
        (if (> (/ t_15 t_18) (floor maxAniso))
          (/ t_16 (floor maxAniso))
          (/ t_18 t_16))
        1999999968613499000.0)
     (log2
      (if (> (/ t_17 t_4) (floor maxAniso))
        (/ (sqrt t_17) (floor maxAniso))
        (/ t_4 (sqrt (fmax t_14 t_11)))))
     (log2
      (if (> (/ (fmax t_2 t_9) t_12) (floor maxAniso))
        (/
         (sqrt (fmax (* t_8 dX.v) (fma (floor w) (* t_10 dY.u) t_6)))
         (floor maxAniso))
        (* t_12 (sqrt (/ 1.0 (fmax (fma t_8 dX.v t_2) t_9)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = dX_46_u * floorf(w);
	float t_2 = (t_0 * dX_46_u) * dX_46_u;
	float t_3 = fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v));
	float t_4 = (fabsf(t_3) * floorf(w)) * floorf(h);
	float t_5 = dY_46_v * floorf(h);
	float t_6 = powf(t_5, 2.0f);
	float t_7 = powf(floorf(h), 2.0f);
	float t_8 = t_7 * dX_46_v;
	float t_9 = fmaf((t_7 * dY_46_v), dY_46_v, ((t_0 * dY_46_u) * dY_46_u));
	float t_10 = dY_46_u * floorf(w);
	float t_11 = powf(t_10, 2.0f) + t_6;
	float t_12 = fabsf((t_3 * (floorf(h) * floorf(w))));
	float t_13 = dX_46_v * floorf(h);
	float t_14 = powf(t_13, 2.0f);
	float t_15 = fmaxf(((t_13 * t_13) + (t_1 * t_1)), ((t_5 * t_5) + (t_10 * t_10)));
	float t_16 = sqrtf(t_15);
	float t_17 = fmaxf((powf(t_1, 2.0f) + t_14), t_11);
	float t_18 = fabsf(((t_10 * t_13) - (t_5 * t_1)));
	float tmp;
	if ((t_15 / t_18) > floorf(maxAniso)) {
		tmp = t_16 / floorf(maxAniso);
	} else {
		tmp = t_18 / t_16;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_17 / t_4) > floorf(maxAniso)) {
			tmp_3 = sqrtf(t_17) / floorf(maxAniso);
		} else {
			tmp_3 = t_4 / sqrtf(fmaxf(t_14, t_11));
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((fmaxf(t_2, t_9) / t_12) > floorf(maxAniso)) {
			tmp_4 = sqrtf(fmaxf((t_8 * dX_46_v), fmaf(floorf(w), (t_10 * dY_46_u), t_6))) / floorf(maxAniso);
		} else {
			tmp_4 = t_12 * sqrtf((1.0f / fmaxf(fmaf(t_8, dX_46_v, t_2), t_9)));
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dX_46_u * floor(w))
	t_2 = Float32(Float32(t_0 * dX_46_u) * dX_46_u)
	t_3 = fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v))
	t_4 = Float32(Float32(abs(t_3) * floor(w)) * floor(h))
	t_5 = Float32(dY_46_v * floor(h))
	t_6 = t_5 ^ Float32(2.0)
	t_7 = floor(h) ^ Float32(2.0)
	t_8 = Float32(t_7 * dX_46_v)
	t_9 = fma(Float32(t_7 * dY_46_v), dY_46_v, Float32(Float32(t_0 * dY_46_u) * dY_46_u))
	t_10 = Float32(dY_46_u * floor(w))
	t_11 = Float32((t_10 ^ Float32(2.0)) + t_6)
	t_12 = abs(Float32(t_3 * Float32(floor(h) * floor(w))))
	t_13 = Float32(dX_46_v * floor(h))
	t_14 = t_13 ^ Float32(2.0)
	t_15 = (Float32(Float32(t_13 * t_13) + Float32(t_1 * t_1)) != Float32(Float32(t_13 * t_13) + Float32(t_1 * t_1))) ? Float32(Float32(t_5 * t_5) + Float32(t_10 * t_10)) : ((Float32(Float32(t_5 * t_5) + Float32(t_10 * t_10)) != Float32(Float32(t_5 * t_5) + Float32(t_10 * t_10))) ? Float32(Float32(t_13 * t_13) + Float32(t_1 * t_1)) : max(Float32(Float32(t_13 * t_13) + Float32(t_1 * t_1)), Float32(Float32(t_5 * t_5) + Float32(t_10 * t_10))))
	t_16 = sqrt(t_15)
	t_17 = (Float32((t_1 ^ Float32(2.0)) + t_14) != Float32((t_1 ^ Float32(2.0)) + t_14)) ? t_11 : ((t_11 != t_11) ? Float32((t_1 ^ Float32(2.0)) + t_14) : max(Float32((t_1 ^ Float32(2.0)) + t_14), t_11))
	t_18 = abs(Float32(Float32(t_10 * t_13) - Float32(t_5 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_15 / t_18) > floor(maxAniso))
		tmp = Float32(t_16 / floor(maxAniso));
	else
		tmp = Float32(t_18 / t_16);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_17 / t_4) > floor(maxAniso))
			tmp_3 = Float32(sqrt(t_17) / floor(maxAniso));
		else
			tmp_3 = Float32(t_4 / sqrt(((t_14 != t_14) ? t_11 : ((t_11 != t_11) ? t_14 : max(t_14, t_11)))));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(((t_2 != t_2) ? t_9 : ((t_9 != t_9) ? t_2 : max(t_2, t_9))) / t_12) > floor(maxAniso))
			tmp_4 = Float32(sqrt(((Float32(t_8 * dX_46_v) != Float32(t_8 * dX_46_v)) ? fma(floor(w), Float32(t_10 * dY_46_u), t_6) : ((fma(floor(w), Float32(t_10 * dY_46_u), t_6) != fma(floor(w), Float32(t_10 * dY_46_u), t_6)) ? Float32(t_8 * dX_46_v) : max(Float32(t_8 * dX_46_v), fma(floor(w), Float32(t_10 * dY_46_u), t_6))))) / floor(maxAniso));
		else
			tmp_4 = Float32(t_12 * sqrt(Float32(Float32(1.0) / ((fma(t_8, dX_46_v, t_2) != fma(t_8, dX_46_v, t_2)) ? t_9 : ((t_9 != t_9) ? fma(t_8, dX_46_v, t_2) : max(fma(t_8, dX_46_v, t_2), t_9))))));
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   10. Applied rewrites 75.8%

       \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 20.5%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.4%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   9. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.9%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 17.8%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := t\_0 \cdot dX.v\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := {t\_3}^{2}\\ t_5 := \mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\\ t_6 := \left(\left|t\_5\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \\ t_7 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_8 := \mathsf{fma}\left(t\_0 \cdot dY.v, dY.v, \left(t\_7 \cdot dY.u\right) \cdot dY.u\right)\\ t_9 := \left(t\_7 \cdot dX.u\right) \cdot dX.u\\ t_10 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_11 := {t\_10}^{2}\\ t_12 := {t\_1}^{2} + t\_11\\ t_13 := \left|t\_5 \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ t_14 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_15 := \sqrt{\mathsf{max}\left(t\_4 + {t\_14}^{2}, t\_12\right)}\\ t_16 := \mathsf{max}\left(t\_14 \cdot t\_14 + t\_3 \cdot t\_3, t\_10 \cdot t\_10 + t\_1 \cdot t\_1\right)\\ t_17 := \sqrt{t\_16}\\ t_18 := \left|t\_1 \cdot t\_14 - t\_10 \cdot t\_3\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_16}{t\_18} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_17}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_18}{t\_17}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_4, t\_12\right)}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_15}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_6}{t\_15}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_9, t\_8\right)}{t\_13} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_2 \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , t\_1 \cdot dY.u, t\_11\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_13 \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_2, dX.v, t\_9\right), t\_8\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* dY.u (floor w)))
        (t_2 (* t_0 dX.v))
        (t_3 (* dX.u (floor w)))
        (t_4 (pow t_3 2.0))
        (t_5 (fma (- dX.u) dY.v (* dY.u dX.v)))
        (t_6 (* (* (fabs t_5) (floor w)) (floor h)))
        (t_7 (pow (floor w) 2.0))
        (t_8 (fma (* t_0 dY.v) dY.v (* (* t_7 dY.u) dY.u)))
        (t_9 (* (* t_7 dX.u) dX.u))
        (t_10 (* dY.v (floor h)))
        (t_11 (pow t_10 2.0))
        (t_12 (+ (pow t_1 2.0) t_11))
        (t_13 (fabs (* t_5 (* (floor h) (floor w)))))
        (t_14 (* dX.v (floor h)))
        (t_15 (sqrt (fmax (+ t_4 (pow t_14 2.0)) t_12)))
        (t_16
         (fmax (+ (* t_14 t_14) (* t_3 t_3)) (+ (* t_10 t_10) (* t_1 t_1))))
        (t_17 (sqrt t_16))
        (t_18 (fabs (- (* t_1 t_14) (* t_10 t_3)))))
   (if (<=
        (if (> (/ t_16 t_18) (floor maxAniso))
          (/ t_17 (floor maxAniso))
          (/ t_18 t_17))
        1999999968613499000.0)
     (log2
      (if (> (/ (fmax t_4 t_12) t_6) (floor maxAniso))
        (/ t_15 (floor maxAniso))
        (/ t_6 t_15)))
     (log2
      (if (> (/ (fmax t_9 t_8) t_13) (floor maxAniso))
        (/
         (sqrt (fmax (* t_2 dX.v) (fma (floor w) (* t_1 dY.u) t_11)))
         (floor maxAniso))
        (* t_13 (sqrt (/ 1.0 (fmax (fma t_2 dX.v t_9) t_8)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = dY_46_u * floorf(w);
	float t_2 = t_0 * dX_46_v;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = powf(t_3, 2.0f);
	float t_5 = fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v));
	float t_6 = (fabsf(t_5) * floorf(w)) * floorf(h);
	float t_7 = powf(floorf(w), 2.0f);
	float t_8 = fmaf((t_0 * dY_46_v), dY_46_v, ((t_7 * dY_46_u) * dY_46_u));
	float t_9 = (t_7 * dX_46_u) * dX_46_u;
	float t_10 = dY_46_v * floorf(h);
	float t_11 = powf(t_10, 2.0f);
	float t_12 = powf(t_1, 2.0f) + t_11;
	float t_13 = fabsf((t_5 * (floorf(h) * floorf(w))));
	float t_14 = dX_46_v * floorf(h);
	float t_15 = sqrtf(fmaxf((t_4 + powf(t_14, 2.0f)), t_12));
	float t_16 = fmaxf(((t_14 * t_14) + (t_3 * t_3)), ((t_10 * t_10) + (t_1 * t_1)));
	float t_17 = sqrtf(t_16);
	float t_18 = fabsf(((t_1 * t_14) - (t_10 * t_3)));
	float tmp;
	if ((t_16 / t_18) > floorf(maxAniso)) {
		tmp = t_17 / floorf(maxAniso);
	} else {
		tmp = t_18 / t_17;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((fmaxf(t_4, t_12) / t_6) > floorf(maxAniso)) {
			tmp_3 = t_15 / floorf(maxAniso);
		} else {
			tmp_3 = t_6 / t_15;
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((fmaxf(t_9, t_8) / t_13) > floorf(maxAniso)) {
			tmp_4 = sqrtf(fmaxf((t_2 * dX_46_v), fmaf(floorf(w), (t_1 * dY_46_u), t_11))) / floorf(maxAniso);
		} else {
			tmp_4 = t_13 * sqrtf((1.0f / fmaxf(fmaf(t_2, dX_46_v, t_9), t_8)));
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(t_0 * dX_46_v)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = t_3 ^ Float32(2.0)
	t_5 = fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v))
	t_6 = Float32(Float32(abs(t_5) * floor(w)) * floor(h))
	t_7 = floor(w) ^ Float32(2.0)
	t_8 = fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(t_7 * dY_46_u) * dY_46_u))
	t_9 = Float32(Float32(t_7 * dX_46_u) * dX_46_u)
	t_10 = Float32(dY_46_v * floor(h))
	t_11 = t_10 ^ Float32(2.0)
	t_12 = Float32((t_1 ^ Float32(2.0)) + t_11)
	t_13 = abs(Float32(t_5 * Float32(floor(h) * floor(w))))
	t_14 = Float32(dX_46_v * floor(h))
	t_15 = sqrt(((Float32(t_4 + (t_14 ^ Float32(2.0))) != Float32(t_4 + (t_14 ^ Float32(2.0)))) ? t_12 : ((t_12 != t_12) ? Float32(t_4 + (t_14 ^ Float32(2.0))) : max(Float32(t_4 + (t_14 ^ Float32(2.0))), t_12))))
	t_16 = (Float32(Float32(t_14 * t_14) + Float32(t_3 * t_3)) != Float32(Float32(t_14 * t_14) + Float32(t_3 * t_3))) ? Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1)) : ((Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1)) != Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1))) ? Float32(Float32(t_14 * t_14) + Float32(t_3 * t_3)) : max(Float32(Float32(t_14 * t_14) + Float32(t_3 * t_3)), Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1))))
	t_17 = sqrt(t_16)
	t_18 = abs(Float32(Float32(t_1 * t_14) - Float32(t_10 * t_3)))
	tmp = Float32(0.0)
	if (Float32(t_16 / t_18) > floor(maxAniso))
		tmp = Float32(t_17 / floor(maxAniso));
	else
		tmp = Float32(t_18 / t_17);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(((t_4 != t_4) ? t_12 : ((t_12 != t_12) ? t_4 : max(t_4, t_12))) / t_6) > floor(maxAniso))
			tmp_3 = Float32(t_15 / floor(maxAniso));
		else
			tmp_3 = Float32(t_6 / t_15);
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(((t_9 != t_9) ? t_8 : ((t_8 != t_8) ? t_9 : max(t_9, t_8))) / t_13) > floor(maxAniso))
			tmp_4 = Float32(sqrt(((Float32(t_2 * dX_46_v) != Float32(t_2 * dX_46_v)) ? fma(floor(w), Float32(t_1 * dY_46_u), t_11) : ((fma(floor(w), Float32(t_1 * dY_46_u), t_11) != fma(floor(w), Float32(t_1 * dY_46_u), t_11)) ? Float32(t_2 * dX_46_v) : max(Float32(t_2 * dX_46_v), fma(floor(w), Float32(t_1 * dY_46_u), t_11))))) / floor(maxAniso));
		else
			tmp_4 = Float32(t_13 * sqrt(Float32(Float32(1.0) / ((fma(t_2, dX_46_v, t_9) != fma(t_2, dX_46_v, t_9)) ? t_8 : ((t_8 != t_8) ? fma(t_2, dX_46_v, t_9) : max(fma(t_2, dX_46_v, t_9), t_8))))));
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 16.5%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.6%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   10. Applied rewrites 75.2%

       \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 20.4%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   9. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.6%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 17.8%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := {t\_1}^{2}\\ t_3 := t\_0 \cdot dX.v\\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_5 := {t\_4}^{2}\\ t_6 := \mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\\ t_7 := \left(\left|t\_6\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \\ t_8 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_9 := \mathsf{fma}\left(t\_0 \cdot dY.v, dY.v, \left(t\_8 \cdot dY.u\right) \cdot dY.u\right)\\ t_10 := \left(t\_8 \cdot dX.u\right) \cdot dX.u\\ t_11 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_12 := t\_2 + {t\_11}^{2}\\ t_13 := \left|t\_6 \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ t_14 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_15 := \mathsf{max}\left(t\_14 \cdot t\_14 + t\_4 \cdot t\_4, t\_11 \cdot t\_11 + t\_1 \cdot t\_1\right)\\ t_16 := \sqrt{t\_15}\\ t_17 := \sqrt{\mathsf{max}\left(t\_5 + {t\_14}^{2}, t\_12\right)}\\ t_18 := \left|t\_1 \cdot t\_14 - t\_11 \cdot t\_4\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_15}{t\_18} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_16}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_18}{t\_16}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_5, t\_12\right)}{t\_7} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_17}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_7}{t\_17}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_10, \mathsf{fma}\left(t\_0, dY.v \cdot dY.v, t\_2\right)\right)}{t\_13} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_3 \cdot dX.v, t\_9\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_13 \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_3, dX.v, t\_10\right), t\_9\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* dY.u (floor w)))
        (t_2 (pow t_1 2.0))
        (t_3 (* t_0 dX.v))
        (t_4 (* dX.u (floor w)))
        (t_5 (pow t_4 2.0))
        (t_6 (fma (- dX.u) dY.v (* dY.u dX.v)))
        (t_7 (* (* (fabs t_6) (floor w)) (floor h)))
        (t_8 (pow (floor w) 2.0))
        (t_9 (fma (* t_0 dY.v) dY.v (* (* t_8 dY.u) dY.u)))
        (t_10 (* (* t_8 dX.u) dX.u))
        (t_11 (* dY.v (floor h)))
        (t_12 (+ t_2 (pow t_11 2.0)))
        (t_13 (fabs (* t_6 (* (floor h) (floor w)))))
        (t_14 (* dX.v (floor h)))
        (t_15
         (fmax (+ (* t_14 t_14) (* t_4 t_4)) (+ (* t_11 t_11) (* t_1 t_1))))
        (t_16 (sqrt t_15))
        (t_17 (sqrt (fmax (+ t_5 (pow t_14 2.0)) t_12)))
        (t_18 (fabs (- (* t_1 t_14) (* t_11 t_4)))))
   (if (<=
        (if (> (/ t_15 t_18) (floor maxAniso))
          (/ t_16 (floor maxAniso))
          (/ t_18 t_16))
        1999999968613499000.0)
     (log2
      (if (> (/ (fmax t_5 t_12) t_7) (floor maxAniso))
        (/ t_17 (floor maxAniso))
        (/ t_7 t_17)))
     (log2
      (if (> (/ (fmax t_10 (fma t_0 (* dY.v dY.v) t_2)) t_13) (floor maxAniso))
        (/ (sqrt (fmax (* t_3 dX.v) t_9)) (floor maxAniso))
        (* t_13 (sqrt (/ 1.0 (fmax (fma t_3 dX.v t_10) t_9)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = dY_46_u * floorf(w);
	float t_2 = powf(t_1, 2.0f);
	float t_3 = t_0 * dX_46_v;
	float t_4 = dX_46_u * floorf(w);
	float t_5 = powf(t_4, 2.0f);
	float t_6 = fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v));
	float t_7 = (fabsf(t_6) * floorf(w)) * floorf(h);
	float t_8 = powf(floorf(w), 2.0f);
	float t_9 = fmaf((t_0 * dY_46_v), dY_46_v, ((t_8 * dY_46_u) * dY_46_u));
	float t_10 = (t_8 * dX_46_u) * dX_46_u;
	float t_11 = dY_46_v * floorf(h);
	float t_12 = t_2 + powf(t_11, 2.0f);
	float t_13 = fabsf((t_6 * (floorf(h) * floorf(w))));
	float t_14 = dX_46_v * floorf(h);
	float t_15 = fmaxf(((t_14 * t_14) + (t_4 * t_4)), ((t_11 * t_11) + (t_1 * t_1)));
	float t_16 = sqrtf(t_15);
	float t_17 = sqrtf(fmaxf((t_5 + powf(t_14, 2.0f)), t_12));
	float t_18 = fabsf(((t_1 * t_14) - (t_11 * t_4)));
	float tmp;
	if ((t_15 / t_18) > floorf(maxAniso)) {
		tmp = t_16 / floorf(maxAniso);
	} else {
		tmp = t_18 / t_16;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((fmaxf(t_5, t_12) / t_7) > floorf(maxAniso)) {
			tmp_3 = t_17 / floorf(maxAniso);
		} else {
			tmp_3 = t_7 / t_17;
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((fmaxf(t_10, fmaf(t_0, (dY_46_v * dY_46_v), t_2)) / t_13) > floorf(maxAniso)) {
			tmp_4 = sqrtf(fmaxf((t_3 * dX_46_v), t_9)) / floorf(maxAniso);
		} else {
			tmp_4 = t_13 * sqrtf((1.0f / fmaxf(fmaf(t_3, dX_46_v, t_10), t_9)));
		}
		tmp_2 = log2f(tmp_4);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32(t_0 * dX_46_v)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = t_4 ^ Float32(2.0)
	t_6 = fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v))
	t_7 = Float32(Float32(abs(t_6) * floor(w)) * floor(h))
	t_8 = floor(w) ^ Float32(2.0)
	t_9 = fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(t_8 * dY_46_u) * dY_46_u))
	t_10 = Float32(Float32(t_8 * dX_46_u) * dX_46_u)
	t_11 = Float32(dY_46_v * floor(h))
	t_12 = Float32(t_2 + (t_11 ^ Float32(2.0)))
	t_13 = abs(Float32(t_6 * Float32(floor(h) * floor(w))))
	t_14 = Float32(dX_46_v * floor(h))
	t_15 = (Float32(Float32(t_14 * t_14) + Float32(t_4 * t_4)) != Float32(Float32(t_14 * t_14) + Float32(t_4 * t_4))) ? Float32(Float32(t_11 * t_11) + Float32(t_1 * t_1)) : ((Float32(Float32(t_11 * t_11) + Float32(t_1 * t_1)) != Float32(Float32(t_11 * t_11) + Float32(t_1 * t_1))) ? Float32(Float32(t_14 * t_14) + Float32(t_4 * t_4)) : max(Float32(Float32(t_14 * t_14) + Float32(t_4 * t_4)), Float32(Float32(t_11 * t_11) + Float32(t_1 * t_1))))
	t_16 = sqrt(t_15)
	t_17 = sqrt(((Float32(t_5 + (t_14 ^ Float32(2.0))) != Float32(t_5 + (t_14 ^ Float32(2.0)))) ? t_12 : ((t_12 != t_12) ? Float32(t_5 + (t_14 ^ Float32(2.0))) : max(Float32(t_5 + (t_14 ^ Float32(2.0))), t_12))))
	t_18 = abs(Float32(Float32(t_1 * t_14) - Float32(t_11 * t_4)))
	tmp = Float32(0.0)
	if (Float32(t_15 / t_18) > floor(maxAniso))
		tmp = Float32(t_16 / floor(maxAniso));
	else
		tmp = Float32(t_18 / t_16);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(((t_5 != t_5) ? t_12 : ((t_12 != t_12) ? t_5 : max(t_5, t_12))) / t_7) > floor(maxAniso))
			tmp_3 = Float32(t_17 / floor(maxAniso));
		else
			tmp_3 = Float32(t_7 / t_17);
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(((t_10 != t_10) ? fma(t_0, Float32(dY_46_v * dY_46_v), t_2) : ((fma(t_0, Float32(dY_46_v * dY_46_v), t_2) != fma(t_0, Float32(dY_46_v * dY_46_v), t_2)) ? t_10 : max(t_10, fma(t_0, Float32(dY_46_v * dY_46_v), t_2)))) / t_13) > floor(maxAniso))
			tmp_4 = Float32(sqrt(((Float32(t_3 * dX_46_v) != Float32(t_3 * dX_46_v)) ? t_9 : ((t_9 != t_9) ? Float32(t_3 * dX_46_v) : max(Float32(t_3 * dX_46_v), t_9)))) / floor(maxAniso));
		else
			tmp_4 = Float32(t_13 * sqrt(Float32(Float32(1.0) / ((fma(t_3, dX_46_v, t_10) != fma(t_3, dX_46_v, t_10)) ? t_9 : ((t_9 != t_9) ? fma(t_3, dX_46_v, t_10) : max(fma(t_3, dX_46_v, t_10), t_9))))));
		end
		tmp_2 = log2(tmp_4);
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))))) < 1.99999997e18

   1. Initial program 100.0%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   10. Applied rewrites 75.3%

       \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   if 1.99999997e18 < (if (>.f32 (/.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u))))) (floor.f32 maxAniso)) (/.f32 (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))))) (floor.f32 maxAniso)) (/.f32 (fabs.f32 (-.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 h) dY.v)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 w) dY.u)))) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))))

   1. Initial program 5.6%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

   5. Applied rewrites 20.6%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.0%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

   9. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.4%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 17.1%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.v \cdot dY.v, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) - \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right|}{\sqrt{\mathsf{max}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right), \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.v \cdot dY.v, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := \left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \\ t_2 := {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := \sqrt{\mathsf{max}\left(t\_2 + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, t\_0\right)}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_2, t\_0\right)}{t\_1} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_3}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 2.0)))
        (t_1
         (* (* (fabs (fma (- dX.u) dY.v (* dY.u dX.v))) (floor w)) (floor h)))
        (t_2 (pow (* dX.u (floor w)) 2.0))
        (t_3 (sqrt (fmax (+ t_2 (pow (* dX.v (floor h)) 2.0)) t_0))))
   (log2
    (if (> (/ (fmax t_2 t_0) t_1) (floor maxAniso))
      (/ t_3 (floor maxAniso))
      (/ t_1 t_3)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 2.0f);
	float t_1 = (fabsf(fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v))) * floorf(w)) * floorf(h);
	float t_2 = powf((dX_46_u * floorf(w)), 2.0f);
	float t_3 = sqrtf(fmaxf((t_2 + powf((dX_46_v * floorf(h)), 2.0f)), t_0));
	float tmp;
	if ((fmaxf(t_2, t_0) / t_1) > floorf(maxAniso)) {
		tmp = t_3 / floorf(maxAniso);
	} else {
		tmp = t_1 / t_3;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))
	t_1 = Float32(Float32(abs(fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v))) * floor(w)) * floor(h))
	t_2 = Float32(dX_46_u * floor(w)) ^ Float32(2.0)
	t_3 = sqrt(((Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) != Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) ? t_0 : ((t_0 != t_0) ? Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) : max(Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), t_0))))
	tmp = Float32(0.0)
	if (Float32(((t_2 != t_2) ? t_0 : ((t_0 != t_0) ? t_2 : max(t_2, t_0))) / t_1) > floor(maxAniso))
		tmp = Float32(t_3 / floor(maxAniso));
	else
		tmp = Float32(t_1 / t_3);
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.0%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in dY.v around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

5. Applied rewrites 18.2%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

6. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 30.5%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

10. Applied rewrites 57.4%

    \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

11. Final simplification 56.2%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]
12. Add Preprocessing

Alternative 10: 50.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right|\\ t_2 := {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := \sqrt{\mathsf{max}\left(t\_2 + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + t\_0\right)}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\frac{\mathsf{max}\left(t\_2, t\_0\right)}{t\_1}}{\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_3}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (* dY.v (floor h)) 2.0))
        (t_1 (fabs (fma (- dX.u) dY.v (* dY.u dX.v))))
        (t_2 (pow (* dX.u (floor w)) 2.0))
        (t_3
         (sqrt
          (fmax
           (+ t_2 (pow (* dX.v (floor h)) 2.0))
           (+ (pow (* dY.u (floor w)) 2.0) t_0)))))
   (log2
    (if (> (/ (/ (fmax t_2 t_0) t_1) (* (floor h) (floor w))) (floor maxAniso))
      (/ t_3 (floor maxAniso))
      (/ (* (* t_1 (floor w)) (floor h)) t_3)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf((dY_46_v * floorf(h)), 2.0f);
	float t_1 = fabsf(fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v)));
	float t_2 = powf((dX_46_u * floorf(w)), 2.0f);
	float t_3 = sqrtf(fmaxf((t_2 + powf((dX_46_v * floorf(h)), 2.0f)), (powf((dY_46_u * floorf(w)), 2.0f) + t_0)));
	float tmp;
	if (((fmaxf(t_2, t_0) / t_1) / (floorf(h) * floorf(w))) > floorf(maxAniso)) {
		tmp = t_3 / floorf(maxAniso);
	} else {
		tmp = ((t_1 * floorf(w)) * floorf(h)) / t_3;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	t_1 = abs(fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v)))
	t_2 = Float32(dX_46_u * floor(w)) ^ Float32(2.0)
	t_3 = sqrt(((Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) != Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) ? Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + t_0) : ((Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + t_0) != Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + t_0)) ? Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) : max(Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + t_0)))))
	tmp = Float32(0.0)
	if (Float32(Float32(((t_2 != t_2) ? t_0 : ((t_0 != t_0) ? t_2 : max(t_2, t_0))) / t_1) / Float32(floor(h) * floor(w))) > floor(maxAniso))
		tmp = Float32(t_3 / floor(maxAniso));
	else
		tmp = Float32(Float32(Float32(t_1 * floor(w)) * floor(h)) / t_3);
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.0%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in dY.v around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

5. Applied rewrites 17.5%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

6. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 30.0%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

9. Taylor expanded in dY.v around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 29.2%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

13. Applied rewrites 49.2%

    \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right|}}{\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

14. Final simplification 49.2%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right|}}{\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor } > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right)\right| \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]
15. Add Preprocessing

Alternative 11: 16.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := t\_0 \cdot dY.v\\ t_2 := t\_1 \cdot dY.v\\ t_3 := \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ t_4 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_5 := \left(t\_4 \cdot dX.u\right) \cdot dX.u\\ t_6 := \mathsf{fma}\left(t\_0 \cdot dX.v, dX.v, t\_5\right)\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_5, t\_2\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_6, \mathsf{fma}\left(t\_1, dY.v, \left(t\_4 \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(t\_6, t\_2\right)}} \cdot t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* t_0 dY.v))
        (t_2 (* t_1 dY.v))
        (t_3
         (fabs (* (fma (- dX.u) dY.v (* dY.u dX.v)) (* (floor h) (floor w)))))
        (t_4 (pow (floor w) 2.0))
        (t_5 (* (* t_4 dX.u) dX.u))
        (t_6 (fma (* t_0 dX.v) dX.v t_5)))
   (log2
    (if (> (/ (fmax t_5 t_2) t_3) (floor maxAniso))
      (/
       (sqrt (fmax t_6 (fma t_1 dY.v (* (* t_4 dY.u) dY.u))))
       (floor maxAniso))
      (* (sqrt (/ 1.0 (fmax t_6 t_2))) t_3)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = t_0 * dY_46_v;
	float t_2 = t_1 * dY_46_v;
	float t_3 = fabsf((fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v)) * (floorf(h) * floorf(w))));
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = (t_4 * dX_46_u) * dX_46_u;
	float t_6 = fmaf((t_0 * dX_46_v), dX_46_v, t_5);
	float tmp;
	if ((fmaxf(t_5, t_2) / t_3) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(t_6, fmaf(t_1, dY_46_v, ((t_4 * dY_46_u) * dY_46_u)))) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf(t_6, t_2))) * t_3;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_v)
	t_2 = Float32(t_1 * dY_46_v)
	t_3 = abs(Float32(fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v)) * Float32(floor(h) * floor(w))))
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = Float32(Float32(t_4 * dX_46_u) * dX_46_u)
	t_6 = fma(Float32(t_0 * dX_46_v), dX_46_v, t_5)
	tmp = Float32(0.0)
	if (Float32(((t_5 != t_5) ? t_2 : ((t_2 != t_2) ? t_5 : max(t_5, t_2))) / t_3) > floor(maxAniso))
		tmp = Float32(sqrt(((t_6 != t_6) ? fma(t_1, dY_46_v, Float32(Float32(t_4 * dY_46_u) * dY_46_u)) : ((fma(t_1, dY_46_v, Float32(Float32(t_4 * dY_46_u) * dY_46_u)) != fma(t_1, dY_46_v, Float32(Float32(t_4 * dY_46_u) * dY_46_u))) ? t_6 : max(t_6, fma(t_1, dY_46_v, Float32(Float32(t_4 * dY_46_u) * dY_46_u)))))) / floor(maxAniso));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / ((t_6 != t_6) ? t_2 : ((t_2 != t_2) ? t_6 : max(t_6, t_2))))) * t_3);
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.0%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in dY.v around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

6. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 29.6%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

9. Taylor expanded in dY.v around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 28.9%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

12. Taylor expanded in dY.v around inf

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
13. Step-by-step derivation

14. Applied rewrites 29.5%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

15. Final simplification 29.2%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ \end{array} \]
16. Add Preprocessing

Alternative 12: 16.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := t\_0 \cdot dY.v\\ t_2 := \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := \mathsf{fma}\left(t\_1, dY.v, \left(t\_3 \cdot dY.u\right) \cdot dY.u\right)\\ t_5 := \left(t\_3 \cdot dX.u\right) \cdot dX.u\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_5, t\_1 \cdot dY.v\right)}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.v, dX.v, t\_5\right), t\_4\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(t\_5, t\_4\right)}} \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* t_0 dY.v))
        (t_2
         (fabs (* (fma (- dX.u) dY.v (* dY.u dX.v)) (* (floor h) (floor w)))))
        (t_3 (pow (floor w) 2.0))
        (t_4 (fma t_1 dY.v (* (* t_3 dY.u) dY.u)))
        (t_5 (* (* t_3 dX.u) dX.u)))
   (log2
    (if (> (/ (fmax t_5 (* t_1 dY.v)) t_2) (floor maxAniso))
      (/ (sqrt (fmax (fma (* t_0 dX.v) dX.v t_5) t_4)) (floor maxAniso))
      (* (sqrt (/ 1.0 (fmax t_5 t_4))) t_2)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = t_0 * dY_46_v;
	float t_2 = fabsf((fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v)) * (floorf(h) * floorf(w))));
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = fmaf(t_1, dY_46_v, ((t_3 * dY_46_u) * dY_46_u));
	float t_5 = (t_3 * dX_46_u) * dX_46_u;
	float tmp;
	if ((fmaxf(t_5, (t_1 * dY_46_v)) / t_2) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf((t_0 * dX_46_v), dX_46_v, t_5), t_4)) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf(t_5, t_4))) * t_2;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_v)
	t_2 = abs(Float32(fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v)) * Float32(floor(h) * floor(w))))
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = fma(t_1, dY_46_v, Float32(Float32(t_3 * dY_46_u) * dY_46_u))
	t_5 = Float32(Float32(t_3 * dX_46_u) * dX_46_u)
	tmp = Float32(0.0)
	if (Float32(((t_5 != t_5) ? Float32(t_1 * dY_46_v) : ((Float32(t_1 * dY_46_v) != Float32(t_1 * dY_46_v)) ? t_5 : max(t_5, Float32(t_1 * dY_46_v)))) / t_2) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(Float32(t_0 * dX_46_v), dX_46_v, t_5) != fma(Float32(t_0 * dX_46_v), dX_46_v, t_5)) ? t_4 : ((t_4 != t_4) ? fma(Float32(t_0 * dX_46_v), dX_46_v, t_5) : max(fma(Float32(t_0 * dX_46_v), dX_46_v, t_5), t_4)))) / floor(maxAniso));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / ((t_5 != t_5) ? t_4 : ((t_4 != t_4) ? t_5 : max(t_5, t_4))))) * t_2);
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.0%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in dY.v around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

5. Applied rewrites 18.2%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

6. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 30.3%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

9. Taylor expanded in dY.v around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 28.9%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

12. Taylor expanded in dX.u around inf

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
13. Step-by-step derivation

14. Applied rewrites 28.8%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

15. Final simplification 28.8%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ \end{array} \]
16. Add Preprocessing

Alternative 13: 15.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := t\_0 \cdot dY.v\\ t_2 := t\_0 \cdot dX.v\\ t_3 := \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ t_4 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_5 := \mathsf{fma}\left(t\_1, dY.v, \left(t\_4 \cdot dY.u\right) \cdot dY.u\right)\\ t_6 := \left(t\_4 \cdot dX.u\right) \cdot dX.u\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_6, t\_1 \cdot dY.v\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2, dX.v, t\_6\right), t\_5\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(t\_2 \cdot dX.v, t\_5\right)}} \cdot t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* t_0 dY.v))
        (t_2 (* t_0 dX.v))
        (t_3
         (fabs (* (fma (- dX.u) dY.v (* dY.u dX.v)) (* (floor h) (floor w)))))
        (t_4 (pow (floor w) 2.0))
        (t_5 (fma t_1 dY.v (* (* t_4 dY.u) dY.u)))
        (t_6 (* (* t_4 dX.u) dX.u)))
   (log2
    (if (> (/ (fmax t_6 (* t_1 dY.v)) t_3) (floor maxAniso))
      (/ (sqrt (fmax (fma t_2 dX.v t_6) t_5)) (floor maxAniso))
      (* (sqrt (/ 1.0 (fmax (* t_2 dX.v) t_5))) t_3)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = t_0 * dY_46_v;
	float t_2 = t_0 * dX_46_v;
	float t_3 = fabsf((fmaf(-dX_46_u, dY_46_v, (dY_46_u * dX_46_v)) * (floorf(h) * floorf(w))));
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = fmaf(t_1, dY_46_v, ((t_4 * dY_46_u) * dY_46_u));
	float t_6 = (t_4 * dX_46_u) * dX_46_u;
	float tmp;
	if ((fmaxf(t_6, (t_1 * dY_46_v)) / t_3) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(t_2, dX_46_v, t_6), t_5)) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf((t_2 * dX_46_v), t_5))) * t_3;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_v)
	t_2 = Float32(t_0 * dX_46_v)
	t_3 = abs(Float32(fma(Float32(-dX_46_u), dY_46_v, Float32(dY_46_u * dX_46_v)) * Float32(floor(h) * floor(w))))
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = fma(t_1, dY_46_v, Float32(Float32(t_4 * dY_46_u) * dY_46_u))
	t_6 = Float32(Float32(t_4 * dX_46_u) * dX_46_u)
	tmp = Float32(0.0)
	if (Float32(((t_6 != t_6) ? Float32(t_1 * dY_46_v) : ((Float32(t_1 * dY_46_v) != Float32(t_1 * dY_46_v)) ? t_6 : max(t_6, Float32(t_1 * dY_46_v)))) / t_3) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(t_2, dX_46_v, t_6) != fma(t_2, dX_46_v, t_6)) ? t_5 : ((t_5 != t_5) ? fma(t_2, dX_46_v, t_6) : max(fma(t_2, dX_46_v, t_6), t_5)))) / floor(maxAniso));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / ((Float32(t_2 * dX_46_v) != Float32(t_2 * dX_46_v)) ? t_5 : ((t_5 != t_5) ? Float32(t_2 * dX_46_v) : max(Float32(t_2 * dX_46_v), t_5))))) * t_3);
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.0%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in dY.v around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array}} \]

5. Applied rewrites 17.1%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]

6. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 29.8%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

9. Taylor expanded in dY.v around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 28.7%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

12. Taylor expanded in dX.u around 0

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]
13. Step-by-step derivation

14. Applied rewrites 28.1%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array} \]

15. Final simplification 28.0%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}{\left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}} \cdot \left|\mathsf{fma}\left(-dX.u, dY.v, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right|\\ \end{array} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024272 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (LOD)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (log2 (if (> (/ (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (fabs (- (* (* (floor w) dX.u) (* (floor h) dY.v)) (* (* (floor h) dX.v) (* (floor w) dY.u))))) (floor maxAniso)) (/ (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v))))) (floor maxAniso)) (/ (fabs (- (* (* (floor w) dX.u) (* (floor h) dY.v)) (* (* (floor h) dX.v) (* (floor w) dY.u)))) (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))))))