Anisotropic x16 LOD (LOD)

Percentage Accurate: 75.2% → 78.1%

Time: 24.6s

Alternatives: 10

Speedup: 0.5×

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: 75.2% 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: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\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 := {t\_2}^{2}\\ t_4 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_5 := \mathsf{fma}\left(t\_4 \cdot dY.u, dY.u, \left(t\_0 \cdot dY.v\right) \cdot dY.v\right)\\ t_6 := \mathsf{max}\left(\mathsf{fma}\left(t\_4 \cdot dX.u, dX.u, \left(t\_0 \cdot dX.v\right) \cdot dX.v\right), t\_5\right)\\ t_7 := \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ t_8 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_9 := \left|\left(dY.v \cdot t\_2 - t\_8 \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right|\\ t_10 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_11 := \mathsf{max}\left(t\_10 \cdot t\_10 + t\_2 \cdot t\_2, t\_1 \cdot t\_1 + t\_8 \cdot t\_8\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \mathsf{max}\left({t\_10}^{2} + t\_3, {t\_1}^{2} + {t\_8}^{2}\right)\\ t_14 := \sqrt{t\_13}\\ t_15 := \left|t\_8 \cdot t\_10 - t\_1 \cdot t\_2\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_11}{t\_15} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_12}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_15}{t\_12}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_13}{t\_9} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_14}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_9}{t\_14}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_6}{t\_7} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_6}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_10 \cdot dX.v, \left\lfloor h\right\rfloor , t\_3\right), t\_5\right)}} \cdot t\_7\\ \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.v (floor h)))
        (t_2 (* dX.u (floor w)))
        (t_3 (pow t_2 2.0))
        (t_4 (pow (floor w) 2.0))
        (t_5 (fma (* t_4 dY.u) dY.u (* (* t_0 dY.v) dY.v)))
        (t_6 (fmax (fma (* t_4 dX.u) dX.u (* (* t_0 dX.v) dX.v)) t_5))
        (t_7
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_8 (* dY.u (floor w)))
        (t_9 (fabs (* (- (* dY.v t_2) (* t_8 dX.v)) (floor h))))
        (t_10 (* dX.v (floor h)))
        (t_11 (fmax (+ (* t_10 t_10) (* t_2 t_2)) (+ (* t_1 t_1) (* t_8 t_8))))
        (t_12 (sqrt t_11))
        (t_13 (fmax (+ (pow t_10 2.0) t_3) (+ (pow t_1 2.0) (pow t_8 2.0))))
        (t_14 (sqrt t_13))
        (t_15 (fabs (- (* t_8 t_10) (* t_1 t_2)))))
   (if (<=
        (if (> (/ t_11 t_15) (floor maxAniso))
          (/ t_12 (floor maxAniso))
          (/ t_15 t_12))
        1999999968613499000.0)
     (log2
      (if (> (/ t_13 t_9) (floor maxAniso))
        (/ t_14 (floor maxAniso))
        (/ t_9 t_14)))
     (log2
      (if (> (/ t_6 t_7) (floor maxAniso))
        (/ (sqrt t_6) (floor maxAniso))
        (*
         (sqrt (/ 1.0 (fmax (fma (* t_10 dX.v) (floor h) t_3) t_5)))
         t_7))))))

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_v * floorf(h);
	float t_2 = dX_46_u * floorf(w);
	float t_3 = powf(t_2, 2.0f);
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = fmaf((t_4 * dY_46_u), dY_46_u, ((t_0 * dY_46_v) * dY_46_v));
	float t_6 = fmaxf(fmaf((t_4 * dX_46_u), dX_46_u, ((t_0 * dX_46_v) * dX_46_v)), t_5);
	float t_7 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_8 = dY_46_u * floorf(w);
	float t_9 = fabsf((((dY_46_v * t_2) - (t_8 * dX_46_v)) * floorf(h)));
	float t_10 = dX_46_v * floorf(h);
	float t_11 = fmaxf(((t_10 * t_10) + (t_2 * t_2)), ((t_1 * t_1) + (t_8 * t_8)));
	float t_12 = sqrtf(t_11);
	float t_13 = fmaxf((powf(t_10, 2.0f) + t_3), (powf(t_1, 2.0f) + powf(t_8, 2.0f)));
	float t_14 = sqrtf(t_13);
	float t_15 = fabsf(((t_8 * t_10) - (t_1 * t_2)));
	float tmp;
	if ((t_11 / t_15) > floorf(maxAniso)) {
		tmp = t_12 / floorf(maxAniso);
	} else {
		tmp = t_15 / t_12;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_13 / t_9) > floorf(maxAniso)) {
			tmp_3 = t_14 / floorf(maxAniso);
		} else {
			tmp_3 = t_9 / t_14;
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_6 / t_7) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_6) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / fmaxf(fmaf((t_10 * dX_46_v), floorf(h), t_3), t_5))) * t_7;
		}
		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_v * floor(h))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = t_2 ^ Float32(2.0)
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = fma(Float32(t_4 * dY_46_u), dY_46_u, Float32(Float32(t_0 * dY_46_v) * dY_46_v))
	t_6 = (fma(Float32(t_4 * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_v) * dX_46_v)) != fma(Float32(t_4 * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_v) * dX_46_v))) ? t_5 : ((t_5 != t_5) ? fma(Float32(t_4 * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_v) * dX_46_v)) : max(fma(Float32(t_4 * dX_46_u), dX_46_u, Float32(Float32(t_0 * dX_46_v) * dX_46_v)), t_5))
	t_7 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_8 = Float32(dY_46_u * floor(w))
	t_9 = abs(Float32(Float32(Float32(dY_46_v * t_2) - Float32(t_8 * dX_46_v)) * floor(h)))
	t_10 = Float32(dX_46_v * floor(h))
	t_11 = (Float32(Float32(t_10 * t_10) + Float32(t_2 * t_2)) != Float32(Float32(t_10 * t_10) + Float32(t_2 * t_2))) ? Float32(Float32(t_1 * t_1) + Float32(t_8 * t_8)) : ((Float32(Float32(t_1 * t_1) + Float32(t_8 * t_8)) != Float32(Float32(t_1 * t_1) + Float32(t_8 * t_8))) ? Float32(Float32(t_10 * t_10) + Float32(t_2 * t_2)) : max(Float32(Float32(t_10 * t_10) + Float32(t_2 * t_2)), Float32(Float32(t_1 * t_1) + Float32(t_8 * t_8))))
	t_12 = sqrt(t_11)
	t_13 = (Float32((t_10 ^ Float32(2.0)) + t_3) != Float32((t_10 ^ Float32(2.0)) + t_3)) ? Float32((t_1 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) : ((Float32((t_1 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) != Float32((t_1 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))) ? Float32((t_10 ^ Float32(2.0)) + t_3) : max(Float32((t_10 ^ Float32(2.0)) + t_3), Float32((t_1 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))))
	t_14 = sqrt(t_13)
	t_15 = abs(Float32(Float32(t_8 * t_10) - Float32(t_1 * t_2)))
	tmp = Float32(0.0)
	if (Float32(t_11 / t_15) > floor(maxAniso))
		tmp = Float32(t_12 / floor(maxAniso));
	else
		tmp = Float32(t_15 / t_12);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_13 / t_9) > floor(maxAniso))
			tmp_3 = Float32(t_14 / floor(maxAniso));
		else
			tmp_3 = Float32(t_9 / t_14);
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_6 / t_7) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_6) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / ((fma(Float32(t_10 * dX_46_v), floor(h), t_3) != fma(Float32(t_10 * dX_46_v), floor(h), t_3)) ? t_5 : ((t_5 != t_5) ? fma(Float32(t_10 * dX_46_v), floor(h), t_3) : max(fma(Float32(t_10 * dX_46_v), floor(h), t_3), t_5))))) * t_7);
		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 99.9%

      \[\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 99.9%

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

   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 5.8%

      \[\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 w 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 18.4%

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

   7. Applied rewrites 16.2%

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

4. Final simplification 77.0%

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

Alternative 2: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ t_1 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_2 := {t\_1}^{2}\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_5 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_6 := \mathsf{fma}\left(t\_3 \cdot dY.u, dY.u, \left(t\_5 \cdot dY.v\right) \cdot dY.v\right)\\ t_7 := \mathsf{max}\left(\mathsf{fma}\left(t\_3 \cdot dX.u, dX.u, \left(t\_5 \cdot dX.v\right) \cdot dX.v\right), t\_6\right)\\ t_8 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_9 := \left|\left(dY.v \cdot t\_1 - t\_8 \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right|\\ t_10 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_11 := \mathsf{max}\left(t\_10 \cdot t\_10 + t\_1 \cdot t\_1, t\_4 \cdot t\_4 + t\_8 \cdot t\_8\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \mathsf{max}\left({t\_10}^{2} + t\_2, {t\_4}^{2} + {t\_8}^{2}\right)\\ t_14 := \left|t\_8 \cdot t\_10 - t\_4 \cdot t\_1\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_11}{t\_14} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_12}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_14}{t\_12}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_13}{t\_9} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{e^{0.5 \cdot \log t\_13}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_9}{\sqrt{t\_13}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_7}{t\_0} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_7}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_10 \cdot dX.v, \left\lfloor h\right\rfloor , t\_2\right), t\_6\right)}} \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 (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_1 (* dX.u (floor w)))
        (t_2 (pow t_1 2.0))
        (t_3 (pow (floor w) 2.0))
        (t_4 (* dY.v (floor h)))
        (t_5 (pow (floor h) 2.0))
        (t_6 (fma (* t_3 dY.u) dY.u (* (* t_5 dY.v) dY.v)))
        (t_7 (fmax (fma (* t_3 dX.u) dX.u (* (* t_5 dX.v) dX.v)) t_6))
        (t_8 (* dY.u (floor w)))
        (t_9 (fabs (* (- (* dY.v t_1) (* t_8 dX.v)) (floor h))))
        (t_10 (* dX.v (floor h)))
        (t_11 (fmax (+ (* t_10 t_10) (* t_1 t_1)) (+ (* t_4 t_4) (* t_8 t_8))))
        (t_12 (sqrt t_11))
        (t_13 (fmax (+ (pow t_10 2.0) t_2) (+ (pow t_4 2.0) (pow t_8 2.0))))
        (t_14 (fabs (- (* t_8 t_10) (* t_4 t_1)))))
   (if (<=
        (if (> (/ t_11 t_14) (floor maxAniso))
          (/ t_12 (floor maxAniso))
          (/ t_14 t_12))
        1999999968613499000.0)
     (log2
      (if (> (/ t_13 t_9) (floor maxAniso))
        (/ (exp (* 0.5 (log t_13))) (floor maxAniso))
        (/ t_9 (sqrt t_13))))
     (log2
      (if (> (/ t_7 t_0) (floor maxAniso))
        (/ (sqrt t_7) (floor maxAniso))
        (*
         (sqrt (/ 1.0 (fmax (fma (* t_10 dX.v) (floor h) t_2) t_6)))
         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(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_1 = dX_46_u * floorf(w);
	float t_2 = powf(t_1, 2.0f);
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = dY_46_v * floorf(h);
	float t_5 = powf(floorf(h), 2.0f);
	float t_6 = fmaf((t_3 * dY_46_u), dY_46_u, ((t_5 * dY_46_v) * dY_46_v));
	float t_7 = fmaxf(fmaf((t_3 * dX_46_u), dX_46_u, ((t_5 * dX_46_v) * dX_46_v)), t_6);
	float t_8 = dY_46_u * floorf(w);
	float t_9 = fabsf((((dY_46_v * t_1) - (t_8 * dX_46_v)) * floorf(h)));
	float t_10 = dX_46_v * floorf(h);
	float t_11 = fmaxf(((t_10 * t_10) + (t_1 * t_1)), ((t_4 * t_4) + (t_8 * t_8)));
	float t_12 = sqrtf(t_11);
	float t_13 = fmaxf((powf(t_10, 2.0f) + t_2), (powf(t_4, 2.0f) + powf(t_8, 2.0f)));
	float t_14 = fabsf(((t_8 * t_10) - (t_4 * t_1)));
	float tmp;
	if ((t_11 / t_14) > floorf(maxAniso)) {
		tmp = t_12 / floorf(maxAniso);
	} else {
		tmp = t_14 / t_12;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_13 / t_9) > floorf(maxAniso)) {
			tmp_3 = expf((0.5f * logf(t_13))) / floorf(maxAniso);
		} else {
			tmp_3 = t_9 / sqrtf(t_13);
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_7 / t_0) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_7) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / fmaxf(fmaf((t_10 * dX_46_v), floorf(h), t_2), t_6))) * 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(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_1 = Float32(dX_46_u * floor(w))
	t_2 = t_1 ^ Float32(2.0)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = Float32(dY_46_v * floor(h))
	t_5 = floor(h) ^ Float32(2.0)
	t_6 = fma(Float32(t_3 * dY_46_u), dY_46_u, Float32(Float32(t_5 * dY_46_v) * dY_46_v))
	t_7 = (fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_5 * dX_46_v) * dX_46_v)) != fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_5 * dX_46_v) * dX_46_v))) ? t_6 : ((t_6 != t_6) ? fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_5 * dX_46_v) * dX_46_v)) : max(fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_5 * dX_46_v) * dX_46_v)), t_6))
	t_8 = Float32(dY_46_u * floor(w))
	t_9 = abs(Float32(Float32(Float32(dY_46_v * t_1) - Float32(t_8 * dX_46_v)) * floor(h)))
	t_10 = Float32(dX_46_v * floor(h))
	t_11 = (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_4 * t_4) + Float32(t_8 * t_8)) : ((Float32(Float32(t_4 * t_4) + Float32(t_8 * t_8)) != Float32(Float32(t_4 * t_4) + Float32(t_8 * t_8))) ? Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1)) : max(Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1)), Float32(Float32(t_4 * t_4) + Float32(t_8 * t_8))))
	t_12 = sqrt(t_11)
	t_13 = (Float32((t_10 ^ Float32(2.0)) + t_2) != Float32((t_10 ^ Float32(2.0)) + t_2)) ? Float32((t_4 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) : ((Float32((t_4 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) != Float32((t_4 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))) ? Float32((t_10 ^ Float32(2.0)) + t_2) : max(Float32((t_10 ^ Float32(2.0)) + t_2), Float32((t_4 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))))
	t_14 = abs(Float32(Float32(t_8 * t_10) - Float32(t_4 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_11 / t_14) > floor(maxAniso))
		tmp = Float32(t_12 / floor(maxAniso));
	else
		tmp = Float32(t_14 / t_12);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_13 / t_9) > floor(maxAniso))
			tmp_3 = Float32(exp(Float32(Float32(0.5) * log(t_13))) / floor(maxAniso));
		else
			tmp_3 = Float32(t_9 / sqrt(t_13));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_7 / t_0) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_7) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / ((fma(Float32(t_10 * dX_46_v), floor(h), t_2) != fma(Float32(t_10 * dX_46_v), floor(h), t_2)) ? t_6 : ((t_6 != t_6) ? fma(Float32(t_10 * dX_46_v), floor(h), t_2) : max(fma(Float32(t_10 * dX_46_v), floor(h), t_2), t_6))))) * 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 99.9%

      \[\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 99.9%

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

   6. Applied rewrites 99.1%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\color{blue}{e^{\log \left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right) \cdot 0.5}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   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 5.8%

      \[\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 w 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 18.0%

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

   7. Applied rewrites 16.0%

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

4. Final simplification 76.1%

   \[\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)}^{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(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{e^{0.5 \cdot \log \left(\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)\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.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}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ t_1 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_2 := {t\_1}^{2}\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_5 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_6 := \mathsf{fma}\left(t\_3 \cdot dY.u, dY.u, \left(t\_4 \cdot dY.v\right) \cdot dY.v\right)\\ t_7 := \mathsf{max}\left(\mathsf{fma}\left(t\_3 \cdot dX.u, dX.u, \left(t\_4 \cdot dX.v\right) \cdot dX.v\right), t\_6\right)\\ t_8 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_9 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_10 := \mathsf{max}\left(t\_9 \cdot t\_9 + t\_1 \cdot t\_1, t\_5 \cdot t\_5 + t\_8 \cdot t\_8\right)\\ t_11 := \sqrt{t\_10}\\ t_12 := \mathsf{max}\left({t\_9}^{2} + t\_2, {t\_5}^{2} + {t\_8}^{2}\right)\\ t_13 := \sqrt{t\_12}\\ t_14 := \left|t\_8 \cdot t\_9 - t\_5 \cdot t\_1\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_10}{t\_14} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_11}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_14}{t\_11}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_12}{\left|\left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_13}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(dY.v \cdot t\_1 - t\_8 \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right|}{t\_13}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_7}{t\_0} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_7}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_9 \cdot dX.v, \left\lfloor h\right\rfloor , t\_2\right), t\_6\right)}} \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 (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_1 (* dX.u (floor w)))
        (t_2 (pow t_1 2.0))
        (t_3 (pow (floor w) 2.0))
        (t_4 (pow (floor h) 2.0))
        (t_5 (* dY.v (floor h)))
        (t_6 (fma (* t_3 dY.u) dY.u (* (* t_4 dY.v) dY.v)))
        (t_7 (fmax (fma (* t_3 dX.u) dX.u (* (* t_4 dX.v) dX.v)) t_6))
        (t_8 (* dY.u (floor w)))
        (t_9 (* dX.v (floor h)))
        (t_10 (fmax (+ (* t_9 t_9) (* t_1 t_1)) (+ (* t_5 t_5) (* t_8 t_8))))
        (t_11 (sqrt t_10))
        (t_12 (fmax (+ (pow t_9 2.0) t_2) (+ (pow t_5 2.0) (pow t_8 2.0))))
        (t_13 (sqrt t_12))
        (t_14 (fabs (- (* t_8 t_9) (* t_5 t_1)))))
   (if (<=
        (if (> (/ t_10 t_14) (floor maxAniso))
          (/ t_11 (floor maxAniso))
          (/ t_14 t_11))
        1999999968613499000.0)
     (log2
      (if (>
           (/ t_12 (fabs (* (* (* dY.v (floor w)) dX.u) (floor h))))
           (floor maxAniso))
        (/ t_13 (floor maxAniso))
        (/ (fabs (* (- (* dY.v t_1) (* t_8 dX.v)) (floor h))) t_13)))
     (log2
      (if (> (/ t_7 t_0) (floor maxAniso))
        (/ (sqrt t_7) (floor maxAniso))
        (* (sqrt (/ 1.0 (fmax (fma (* t_9 dX.v) (floor h) t_2) t_6))) 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(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_1 = dX_46_u * floorf(w);
	float t_2 = powf(t_1, 2.0f);
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = powf(floorf(h), 2.0f);
	float t_5 = dY_46_v * floorf(h);
	float t_6 = fmaf((t_3 * dY_46_u), dY_46_u, ((t_4 * dY_46_v) * dY_46_v));
	float t_7 = fmaxf(fmaf((t_3 * dX_46_u), dX_46_u, ((t_4 * dX_46_v) * dX_46_v)), t_6);
	float t_8 = dY_46_u * floorf(w);
	float t_9 = dX_46_v * floorf(h);
	float t_10 = fmaxf(((t_9 * t_9) + (t_1 * t_1)), ((t_5 * t_5) + (t_8 * t_8)));
	float t_11 = sqrtf(t_10);
	float t_12 = fmaxf((powf(t_9, 2.0f) + t_2), (powf(t_5, 2.0f) + powf(t_8, 2.0f)));
	float t_13 = sqrtf(t_12);
	float t_14 = fabsf(((t_8 * t_9) - (t_5 * t_1)));
	float tmp;
	if ((t_10 / t_14) > floorf(maxAniso)) {
		tmp = t_11 / floorf(maxAniso);
	} else {
		tmp = t_14 / t_11;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_12 / fabsf((((dY_46_v * floorf(w)) * dX_46_u) * floorf(h)))) > floorf(maxAniso)) {
			tmp_3 = t_13 / floorf(maxAniso);
		} else {
			tmp_3 = fabsf((((dY_46_v * t_1) - (t_8 * dX_46_v)) * floorf(h))) / t_13;
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_7 / t_0) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_7) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / fmaxf(fmaf((t_9 * dX_46_v), floorf(h), t_2), t_6))) * 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(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_1 = Float32(dX_46_u * floor(w))
	t_2 = t_1 ^ Float32(2.0)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = floor(h) ^ Float32(2.0)
	t_5 = Float32(dY_46_v * floor(h))
	t_6 = fma(Float32(t_3 * dY_46_u), dY_46_u, Float32(Float32(t_4 * dY_46_v) * dY_46_v))
	t_7 = (fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)) != fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v))) ? t_6 : ((t_6 != t_6) ? fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)) : max(fma(Float32(t_3 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)), t_6))
	t_8 = Float32(dY_46_u * floor(w))
	t_9 = Float32(dX_46_v * floor(h))
	t_10 = (Float32(Float32(t_9 * t_9) + Float32(t_1 * t_1)) != Float32(Float32(t_9 * t_9) + Float32(t_1 * t_1))) ? Float32(Float32(t_5 * t_5) + Float32(t_8 * t_8)) : ((Float32(Float32(t_5 * t_5) + Float32(t_8 * t_8)) != Float32(Float32(t_5 * t_5) + Float32(t_8 * t_8))) ? Float32(Float32(t_9 * t_9) + Float32(t_1 * t_1)) : max(Float32(Float32(t_9 * t_9) + Float32(t_1 * t_1)), Float32(Float32(t_5 * t_5) + Float32(t_8 * t_8))))
	t_11 = sqrt(t_10)
	t_12 = (Float32((t_9 ^ Float32(2.0)) + t_2) != Float32((t_9 ^ Float32(2.0)) + t_2)) ? Float32((t_5 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) : ((Float32((t_5 ^ Float32(2.0)) + (t_8 ^ Float32(2.0))) != Float32((t_5 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))) ? Float32((t_9 ^ Float32(2.0)) + t_2) : max(Float32((t_9 ^ Float32(2.0)) + t_2), Float32((t_5 ^ Float32(2.0)) + (t_8 ^ Float32(2.0)))))
	t_13 = sqrt(t_12)
	t_14 = abs(Float32(Float32(t_8 * t_9) - Float32(t_5 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_10 / t_14) > floor(maxAniso))
		tmp = Float32(t_11 / floor(maxAniso));
	else
		tmp = Float32(t_14 / t_11);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_12 / abs(Float32(Float32(Float32(dY_46_v * floor(w)) * dX_46_u) * floor(h)))) > floor(maxAniso))
			tmp_3 = Float32(t_13 / floor(maxAniso));
		else
			tmp_3 = Float32(abs(Float32(Float32(Float32(dY_46_v * t_1) - Float32(t_8 * dX_46_v)) * floor(h))) / t_13);
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_7 / t_0) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_7) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / ((fma(Float32(t_9 * dX_46_v), floor(h), t_2) != fma(Float32(t_9 * dX_46_v), floor(h), t_2)) ? t_6 : ((t_6 != t_6) ? fma(Float32(t_9 * dX_46_v), floor(h), t_2) : max(fma(Float32(t_9 * dX_46_v), floor(h), t_2), t_6))))) * 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 99.9%

      \[\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 99.9%

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

   5. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-floor.f32 99.0

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

   7. Applied rewrites 99.0%

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

   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 5.8%

      \[\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 w 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 18.0%

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

   7. Applied rewrites 15.6%

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

4. Final simplification 76.3%

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

Alternative 4: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := {t\_0}^{2}\\ t_2 := \left|\left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor h\right\rfloor \right|\\ t_3 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_4 := \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ t_5 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_6 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_7 := \mathsf{fma}\left(t\_5 \cdot dY.u, dY.u, \left(t\_3 \cdot dY.v\right) \cdot dY.v\right)\\ t_8 := \mathsf{max}\left(\mathsf{fma}\left(t\_5 \cdot dX.u, dX.u, \left(t\_3 \cdot dX.v\right) \cdot dX.v\right), t\_7\right)\\ t_9 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_10 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_11 := \mathsf{max}\left(t\_10 \cdot t\_10 + t\_0 \cdot t\_0, t\_6 \cdot t\_6 + t\_9 \cdot t\_9\right)\\ t_12 := \sqrt{t\_11}\\ t_13 := \mathsf{max}\left({t\_10}^{2} + t\_1, {t\_6}^{2} + {t\_9}^{2}\right)\\ t_14 := \left|t\_9 \cdot t\_10 - t\_6 \cdot t\_0\right|\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;\frac{t\_11}{t\_14} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_12}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_14}{t\_12}\\ \end{array} \leq 1999999968613499000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_13}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({t\_13}^{0.25}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_13}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_8}{t\_4} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_8}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_10 \cdot dX.v, \left\lfloor h\right\rfloor , t\_1\right), t\_7\right)}} \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 (* dX.u (floor w)))
        (t_1 (pow t_0 2.0))
        (t_2 (fabs (* (* (* dY.v (floor w)) dX.u) (floor h))))
        (t_3 (pow (floor h) 2.0))
        (t_4
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_5 (pow (floor w) 2.0))
        (t_6 (* dY.v (floor h)))
        (t_7 (fma (* t_5 dY.u) dY.u (* (* t_3 dY.v) dY.v)))
        (t_8 (fmax (fma (* t_5 dX.u) dX.u (* (* t_3 dX.v) dX.v)) t_7))
        (t_9 (* dY.u (floor w)))
        (t_10 (* dX.v (floor h)))
        (t_11 (fmax (+ (* t_10 t_10) (* t_0 t_0)) (+ (* t_6 t_6) (* t_9 t_9))))
        (t_12 (sqrt t_11))
        (t_13 (fmax (+ (pow t_10 2.0) t_1) (+ (pow t_6 2.0) (pow t_9 2.0))))
        (t_14 (fabs (- (* t_9 t_10) (* t_6 t_0)))))
   (if (<=
        (if (> (/ t_11 t_14) (floor maxAniso))
          (/ t_12 (floor maxAniso))
          (/ t_14 t_12))
        1999999968613499000.0)
     (log2
      (if (> (/ t_13 t_2) (floor maxAniso))
        (/ (pow (pow t_13 0.25) 2.0) (floor maxAniso))
        (/ t_2 (sqrt t_13))))
     (log2
      (if (> (/ t_8 t_4) (floor maxAniso))
        (/ (sqrt t_8) (floor maxAniso))
        (*
         (sqrt (/ 1.0 (fmax (fma (* t_10 dX.v) (floor h) t_1) t_7)))
         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 = dX_46_u * floorf(w);
	float t_1 = powf(t_0, 2.0f);
	float t_2 = fabsf((((dY_46_v * floorf(w)) * dX_46_u) * floorf(h)));
	float t_3 = powf(floorf(h), 2.0f);
	float t_4 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_5 = powf(floorf(w), 2.0f);
	float t_6 = dY_46_v * floorf(h);
	float t_7 = fmaf((t_5 * dY_46_u), dY_46_u, ((t_3 * dY_46_v) * dY_46_v));
	float t_8 = fmaxf(fmaf((t_5 * dX_46_u), dX_46_u, ((t_3 * dX_46_v) * dX_46_v)), t_7);
	float t_9 = dY_46_u * floorf(w);
	float t_10 = dX_46_v * floorf(h);
	float t_11 = fmaxf(((t_10 * t_10) + (t_0 * t_0)), ((t_6 * t_6) + (t_9 * t_9)));
	float t_12 = sqrtf(t_11);
	float t_13 = fmaxf((powf(t_10, 2.0f) + t_1), (powf(t_6, 2.0f) + powf(t_9, 2.0f)));
	float t_14 = fabsf(((t_9 * t_10) - (t_6 * t_0)));
	float tmp;
	if ((t_11 / t_14) > floorf(maxAniso)) {
		tmp = t_12 / floorf(maxAniso);
	} else {
		tmp = t_14 / t_12;
	}
	float tmp_2;
	if (tmp <= 1999999968613499000.0f) {
		float tmp_3;
		if ((t_13 / t_2) > floorf(maxAniso)) {
			tmp_3 = powf(powf(t_13, 0.25f), 2.0f) / floorf(maxAniso);
		} else {
			tmp_3 = t_2 / sqrtf(t_13);
		}
		tmp_2 = log2f(tmp_3);
	} else {
		float tmp_4;
		if ((t_8 / t_4) > floorf(maxAniso)) {
			tmp_4 = sqrtf(t_8) / floorf(maxAniso);
		} else {
			tmp_4 = sqrtf((1.0f / fmaxf(fmaf((t_10 * dX_46_v), floorf(h), t_1), t_7))) * 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 = Float32(dX_46_u * floor(w))
	t_1 = t_0 ^ Float32(2.0)
	t_2 = abs(Float32(Float32(Float32(dY_46_v * floor(w)) * dX_46_u) * floor(h)))
	t_3 = floor(h) ^ Float32(2.0)
	t_4 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_5 = floor(w) ^ Float32(2.0)
	t_6 = Float32(dY_46_v * floor(h))
	t_7 = fma(Float32(t_5 * dY_46_u), dY_46_u, Float32(Float32(t_3 * dY_46_v) * dY_46_v))
	t_8 = (fma(Float32(t_5 * dX_46_u), dX_46_u, Float32(Float32(t_3 * dX_46_v) * dX_46_v)) != fma(Float32(t_5 * dX_46_u), dX_46_u, Float32(Float32(t_3 * dX_46_v) * dX_46_v))) ? t_7 : ((t_7 != t_7) ? fma(Float32(t_5 * dX_46_u), dX_46_u, Float32(Float32(t_3 * dX_46_v) * dX_46_v)) : max(fma(Float32(t_5 * dX_46_u), dX_46_u, Float32(Float32(t_3 * dX_46_v) * dX_46_v)), t_7))
	t_9 = Float32(dY_46_u * floor(w))
	t_10 = Float32(dX_46_v * floor(h))
	t_11 = (Float32(Float32(t_10 * t_10) + Float32(t_0 * t_0)) != Float32(Float32(t_10 * t_10) + Float32(t_0 * t_0))) ? Float32(Float32(t_6 * t_6) + Float32(t_9 * t_9)) : ((Float32(Float32(t_6 * t_6) + Float32(t_9 * t_9)) != Float32(Float32(t_6 * t_6) + Float32(t_9 * t_9))) ? Float32(Float32(t_10 * t_10) + Float32(t_0 * t_0)) : max(Float32(Float32(t_10 * t_10) + Float32(t_0 * t_0)), Float32(Float32(t_6 * t_6) + Float32(t_9 * t_9))))
	t_12 = sqrt(t_11)
	t_13 = (Float32((t_10 ^ Float32(2.0)) + t_1) != Float32((t_10 ^ Float32(2.0)) + t_1)) ? Float32((t_6 ^ Float32(2.0)) + (t_9 ^ Float32(2.0))) : ((Float32((t_6 ^ Float32(2.0)) + (t_9 ^ Float32(2.0))) != Float32((t_6 ^ Float32(2.0)) + (t_9 ^ Float32(2.0)))) ? Float32((t_10 ^ Float32(2.0)) + t_1) : max(Float32((t_10 ^ Float32(2.0)) + t_1), Float32((t_6 ^ Float32(2.0)) + (t_9 ^ Float32(2.0)))))
	t_14 = abs(Float32(Float32(t_9 * t_10) - Float32(t_6 * t_0)))
	tmp = Float32(0.0)
	if (Float32(t_11 / t_14) > floor(maxAniso))
		tmp = Float32(t_12 / floor(maxAniso));
	else
		tmp = Float32(t_14 / t_12);
	end
	tmp_2 = Float32(0.0)
	if (tmp <= Float32(1999999968613499000.0))
		tmp_3 = Float32(0.0)
		if (Float32(t_13 / t_2) > floor(maxAniso))
			tmp_3 = Float32(((t_13 ^ Float32(0.25)) ^ Float32(2.0)) / floor(maxAniso));
		else
			tmp_3 = Float32(t_2 / sqrt(t_13));
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = Float32(0.0)
		if (Float32(t_8 / t_4) > floor(maxAniso))
			tmp_4 = Float32(sqrt(t_8) / floor(maxAniso));
		else
			tmp_4 = Float32(sqrt(Float32(Float32(1.0) / ((fma(Float32(t_10 * dX_46_v), floor(h), t_1) != fma(Float32(t_10 * dX_46_v), floor(h), t_1)) ? t_7 : ((t_7 != t_7) ? fma(Float32(t_10 * dX_46_v), floor(h), t_1) : max(fma(Float32(t_10 * dX_46_v), floor(h), t_1), t_7))))) * 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 99.9%

      \[\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 99.9%

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

   6. Applied rewrites 99.9%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\color{blue}{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   7. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \color{blue}{\left(dX.u \cdot \left(dY.v \cdot \left\lfloor w\right\rfloor \right)\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \color{blue}{\left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \color{blue}{\left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.v\right)} \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.v\right)} \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

      5. lower-floor.f32 98.9

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\color{blue}{\left\lfloor w\right\rfloor } \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   9. Applied rewrites 98.9%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) - \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.v\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   10. Taylor expanded in dX.u around inf

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(dX.u \cdot \left(dY.v \cdot \left\lfloor w\right\rfloor \right)\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

       3. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

       4. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{4}}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

       5. lower-floor.f32 98.9

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   12. Applied rewrites 98.9%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor h\right\rfloor \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.v\right) \cdot dX.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   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 5.8%

      \[\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 w 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 18.0%

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

   7. Applied rewrites 16.8%

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

4. Final simplification 76.1%

   \[\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)}^{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(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor h\right\rfloor \right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left({\left(\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)\right)}^{0.25}\right)}^{2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(dY.v \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor h\right\rfloor \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}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := t\_1 \cdot dY.u\\ t_3 := \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ t_4 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_5 := \mathsf{fma}\left(t\_2, dY.u, \left(t\_4 \cdot dY.v\right) \cdot dY.v\right)\\ t_6 := \mathsf{max}\left(t\_0, t\_5\right)\\ t_7 := \mathsf{fma}\left(t\_1 \cdot dX.u, dX.u, \left(t\_4 \cdot dX.v\right) \cdot dX.v\right)\\ t_8 := \sqrt{\frac{1}{\mathsf{max}\left(t\_7, t\_5\right)}} \cdot t\_3\\ \mathbf{if}\;dX.v \leq 0.5199999809265137:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_0, t\_2 \cdot dY.u\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_7, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_6}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_6}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0)))
        (t_1 (pow (floor w) 2.0))
        (t_2 (* t_1 dY.u))
        (t_3
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_4 (pow (floor h) 2.0))
        (t_5 (fma t_2 dY.u (* (* t_4 dY.v) dY.v)))
        (t_6 (fmax t_0 t_5))
        (t_7 (fma (* t_1 dX.u) dX.u (* (* t_4 dX.v) dX.v)))
        (t_8 (* (sqrt (/ 1.0 (fmax t_7 t_5))) t_3)))
   (if (<= dX.v 0.5199999809265137)
     (log2
      (if (> (/ (fmax t_0 (* t_2 dY.u)) t_3) (floor maxAniso))
        (/
         (sqrt
          (fmax
           t_7
           (+ (pow (* dY.v (floor h)) 2.0) (pow (* dY.u (floor w)) 2.0))))
         (floor maxAniso))
        t_8))
     (log2
      (if (> (/ t_6 t_3) (floor maxAniso))
        (/ (sqrt t_6) (floor maxAniso))
        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((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float t_2 = t_1 * dY_46_u;
	float t_3 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_4 = powf(floorf(h), 2.0f);
	float t_5 = fmaf(t_2, dY_46_u, ((t_4 * dY_46_v) * dY_46_v));
	float t_6 = fmaxf(t_0, t_5);
	float t_7 = fmaf((t_1 * dX_46_u), dX_46_u, ((t_4 * dX_46_v) * dX_46_v));
	float t_8 = sqrtf((1.0f / fmaxf(t_7, t_5))) * t_3;
	float tmp_1;
	if (dX_46_v <= 0.5199999809265137f) {
		float tmp_2;
		if ((fmaxf(t_0, (t_2 * dY_46_u)) / t_3) > floorf(maxAniso)) {
			tmp_2 = sqrtf(fmaxf(t_7, (powf((dY_46_v * floorf(h)), 2.0f) + powf((dY_46_u * floorf(w)), 2.0f)))) / floorf(maxAniso);
		} else {
			tmp_2 = t_8;
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if ((t_6 / t_3) > floorf(maxAniso)) {
			tmp_3 = sqrtf(t_6) / floorf(maxAniso);
		} else {
			tmp_3 = t_8;
		}
		tmp_1 = log2f(tmp_3);
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))
	t_1 = floor(w) ^ Float32(2.0)
	t_2 = Float32(t_1 * dY_46_u)
	t_3 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_4 = floor(h) ^ Float32(2.0)
	t_5 = fma(t_2, dY_46_u, Float32(Float32(t_4 * dY_46_v) * dY_46_v))
	t_6 = (t_0 != t_0) ? t_5 : ((t_5 != t_5) ? t_0 : max(t_0, t_5))
	t_7 = fma(Float32(t_1 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v))
	t_8 = Float32(sqrt(Float32(Float32(1.0) / ((t_7 != t_7) ? t_5 : ((t_5 != t_5) ? t_7 : max(t_7, t_5))))) * t_3)
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(0.5199999809265137))
		tmp_2 = Float32(0.0)
		if (Float32(((t_0 != t_0) ? Float32(t_2 * dY_46_u) : ((Float32(t_2 * dY_46_u) != Float32(t_2 * dY_46_u)) ? t_0 : max(t_0, Float32(t_2 * dY_46_u)))) / t_3) > floor(maxAniso))
			tmp_2 = Float32(sqrt(((t_7 != t_7) ? Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? t_7 : max(t_7, Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))) / floor(maxAniso));
		else
			tmp_2 = t_8;
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (Float32(t_6 / t_3) > floor(maxAniso))
			tmp_3 = Float32(sqrt(t_6) / floor(maxAniso));
		else
			tmp_3 = t_8;
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.519999981

   1. Initial program 71.1%

      \[\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 w 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 19.2%

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

   7. Applied rewrites 22.4%

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

   8. Taylor expanded in dY.u around inf

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

   10. Applied rewrites 14.0%

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

   12. Applied rewrites 43.3%

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

   if 0.519999981 < dX.v

   1. Initial program 80.6%

      \[\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 w 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 16.6%

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

   7. Applied rewrites 43.2%

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

   9. Applied rewrites 52.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.5199999809265137:\\ \;\;\;\;\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({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), {\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}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\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}, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \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}, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := t\_1 \cdot dY.u\\ t_3 := \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ t_4 := \frac{\mathsf{max}\left(t\_0, t\_2 \cdot dY.u\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor \\ t_5 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_6 := \mathsf{fma}\left(t\_2, dY.u, \left(t\_5 \cdot dY.v\right) \cdot dY.v\right)\\ t_7 := \mathsf{fma}\left(t\_1 \cdot dX.u, dX.u, \left(t\_5 \cdot dX.v\right) \cdot dX.v\right)\\ t_8 := \sqrt{\frac{1}{\mathsf{max}\left(t\_7, t\_6\right)}} \cdot t\_3\\ \mathbf{if}\;dX.v \leq 0.5199999809265137:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;t\_4:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_7, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;t\_4:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_0, t\_6\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0)))
        (t_1 (pow (floor w) 2.0))
        (t_2 (* t_1 dY.u))
        (t_3
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_4 (> (/ (fmax t_0 (* t_2 dY.u)) t_3) (floor maxAniso)))
        (t_5 (pow (floor h) 2.0))
        (t_6 (fma t_2 dY.u (* (* t_5 dY.v) dY.v)))
        (t_7 (fma (* t_1 dX.u) dX.u (* (* t_5 dX.v) dX.v)))
        (t_8 (* (sqrt (/ 1.0 (fmax t_7 t_6))) t_3)))
   (if (<= dX.v 0.5199999809265137)
     (log2
      (if t_4
        (/
         (sqrt
          (fmax
           t_7
           (+ (pow (* dY.v (floor h)) 2.0) (pow (* dY.u (floor w)) 2.0))))
         (floor maxAniso))
        t_8))
     (log2 (if t_4 (/ (sqrt (fmax t_0 t_6)) (floor maxAniso)) 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((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float t_2 = t_1 * dY_46_u;
	float t_3 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	int t_4 = (fmaxf(t_0, (t_2 * dY_46_u)) / t_3) > floorf(maxAniso);
	float t_5 = powf(floorf(h), 2.0f);
	float t_6 = fmaf(t_2, dY_46_u, ((t_5 * dY_46_v) * dY_46_v));
	float t_7 = fmaf((t_1 * dX_46_u), dX_46_u, ((t_5 * dX_46_v) * dX_46_v));
	float t_8 = sqrtf((1.0f / fmaxf(t_7, t_6))) * t_3;
	float tmp_1;
	if (dX_46_v <= 0.5199999809265137f) {
		float tmp_2;
		if (t_4) {
			tmp_2 = sqrtf(fmaxf(t_7, (powf((dY_46_v * floorf(h)), 2.0f) + powf((dY_46_u * floorf(w)), 2.0f)))) / floorf(maxAniso);
		} else {
			tmp_2 = t_8;
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if (t_4) {
			tmp_3 = sqrtf(fmaxf(t_0, t_6)) / floorf(maxAniso);
		} else {
			tmp_3 = t_8;
		}
		tmp_1 = log2f(tmp_3);
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))
	t_1 = floor(w) ^ Float32(2.0)
	t_2 = Float32(t_1 * dY_46_u)
	t_3 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_4 = Float32(((t_0 != t_0) ? Float32(t_2 * dY_46_u) : ((Float32(t_2 * dY_46_u) != Float32(t_2 * dY_46_u)) ? t_0 : max(t_0, Float32(t_2 * dY_46_u)))) / t_3) > floor(maxAniso)
	t_5 = floor(h) ^ Float32(2.0)
	t_6 = fma(t_2, dY_46_u, Float32(Float32(t_5 * dY_46_v) * dY_46_v))
	t_7 = fma(Float32(t_1 * dX_46_u), dX_46_u, Float32(Float32(t_5 * dX_46_v) * dX_46_v))
	t_8 = Float32(sqrt(Float32(Float32(1.0) / ((t_7 != t_7) ? t_6 : ((t_6 != t_6) ? t_7 : max(t_7, t_6))))) * t_3)
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(0.5199999809265137))
		tmp_2 = Float32(0.0)
		if (t_4)
			tmp_2 = Float32(sqrt(((t_7 != t_7) ? Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? t_7 : max(t_7, Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))) / floor(maxAniso));
		else
			tmp_2 = t_8;
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (t_4)
			tmp_3 = Float32(sqrt(((t_0 != t_0) ? t_6 : ((t_6 != t_6) ? t_0 : max(t_0, t_6)))) / floor(maxAniso));
		else
			tmp_3 = t_8;
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.519999981

   1. Initial program 71.1%

      \[\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 w 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 18.4%

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in dY.u around inf

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

   10. Applied rewrites 14.0%

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

   12. Applied rewrites 43.3%

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

   if 0.519999981 < dX.v

   1. Initial program 80.6%

      \[\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 w 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 16.6%

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

   7. Applied rewrites 44.7%

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

   8. Taylor expanded in dY.u around inf

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

   10. Applied rewrites 12.8%

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

   12. Applied rewrites 49.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.5199999809265137:\\ \;\;\;\;\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({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), {\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}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\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({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}{\left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \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}, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right)\right|\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.8% accurate, 1.3× speedup

Math

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

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 (* t_0 dY.u))
        (t_2 (pow (floor h) 2.0))
        (t_3 (fma t_1 dY.u (* (* t_2 dY.v) dY.v)))
        (t_4 (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0)))
        (t_5
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v))))))
   (log2
    (if (> (/ (fmax t_4 (* t_1 dY.u)) t_5) (floor maxAniso))
      (/ (sqrt (fmax t_4 t_3)) (floor maxAniso))
      (*
       (sqrt (/ 1.0 (fmax (fma (* t_0 dX.u) dX.u (* (* t_2 dX.v) dX.v)) t_3)))
       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 = t_0 * dY_46_u;
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = fmaf(t_1, dY_46_u, ((t_2 * dY_46_v) * dY_46_v));
	float t_4 = powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f);
	float t_5 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float tmp;
	if ((fmaxf(t_4, (t_1 * dY_46_u)) / t_5) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(t_4, t_3)) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf(fmaf((t_0 * dX_46_u), dX_46_u, ((t_2 * dX_46_v) * dX_46_v)), t_3))) * 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 = floor(w) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_u)
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = fma(t_1, dY_46_u, Float32(Float32(t_2 * dY_46_v) * dY_46_v))
	t_4 = Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))
	t_5 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	tmp = Float32(0.0)
	if (Float32(((t_4 != t_4) ? Float32(t_1 * dY_46_u) : ((Float32(t_1 * dY_46_u) != Float32(t_1 * dY_46_u)) ? t_4 : max(t_4, Float32(t_1 * dY_46_u)))) / t_5) > floor(maxAniso))
		tmp = Float32(sqrt(((t_4 != t_4) ? t_3 : ((t_3 != t_3) ? t_4 : max(t_4, t_3)))) / floor(maxAniso));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / ((fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_2 * dX_46_v) * dX_46_v)) != fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_2 * dX_46_v) * dX_46_v))) ? t_3 : ((t_3 != t_3) ? fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_2 * dX_46_v) * dX_46_v)) : max(fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_2 * dX_46_v) * dX_46_v)), t_3))))) * t_5);
	end
	return log2(tmp)
end

Derivation

1. Initial program 73.5%

   \[\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 w 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 19.2%

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

7. Applied rewrites 27.7%

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

8. Taylor expanded in dY.u around inf

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

10. Applied rewrites 14.2%

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

12. Applied rewrites 35.7%

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

13. Final simplification 35.5%

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

Alternative 8: 25.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_1 (pow (floor h) 2.0))
        (t_2 (pow (floor w) 2.0))
        (t_3 (* t_2 dY.u))
        (t_4 (* t_2 dX.u))
        (t_5 (fma t_3 dY.u (* (* t_1 dY.v) dY.v))))
   (log2
    (if (>
         (/
          (fmax
           (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0))
           (* t_3 dY.u))
          t_0)
         (floor maxAniso))
      (/ (sqrt (fmax (* t_4 dX.u) t_5)) (floor maxAniso))
      (*
       (sqrt (/ 1.0 (fmax (fma t_4 dX.u (* (* t_1 dX.v) dX.v)) t_5)))
       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(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = powf(floorf(w), 2.0f);
	float t_3 = t_2 * dY_46_u;
	float t_4 = t_2 * dX_46_u;
	float t_5 = fmaf(t_3, dY_46_u, ((t_1 * dY_46_v) * dY_46_v));
	float tmp;
	if ((fmaxf((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)), (t_3 * dY_46_u)) / t_0) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf((t_4 * dX_46_u), t_5)) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf(fmaf(t_4, dX_46_u, ((t_1 * dX_46_v) * dX_46_v)), t_5))) * t_0;
	}
	return log2f(tmp);
}

Julia

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

Derivation

1. Initial program 73.5%

   \[\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 w 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 18.3%

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

7. Applied rewrites 27.8%

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

8. Taylor expanded in dY.u around inf

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

10. Applied rewrites 14.0%

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

11. Taylor expanded in dX.u around inf

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

13. Applied rewrites 25.4%

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

14. Final simplification 25.4%

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

Alternative 9: 15.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* t_0 dY.u))
        (t_2 (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0)))
        (t_3
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_4 (pow (floor h) 2.0))
        (t_5 (fma t_1 dY.u (* (* t_4 dY.v) dY.v))))
   (log2
    (if (> (/ (fmax t_2 (* t_1 dY.u)) t_3) (floor maxAniso))
      (/
       (sqrt (fmax (fma (* t_0 dX.u) dX.u (* (* t_4 dX.v) dX.v)) t_5))
       (floor maxAniso))
      (* (sqrt (/ 1.0 (fmax t_2 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(w), 2.0f);
	float t_1 = t_0 * dY_46_u;
	float t_2 = powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f);
	float t_3 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_4 = powf(floorf(h), 2.0f);
	float t_5 = fmaf(t_1, dY_46_u, ((t_4 * dY_46_v) * dY_46_v));
	float tmp;
	if ((fmaxf(t_2, (t_1 * dY_46_u)) / t_3) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf((t_0 * dX_46_u), dX_46_u, ((t_4 * dX_46_v) * dX_46_v)), t_5)) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf(t_2, 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(w) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_u)
	t_2 = Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))
	t_3 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_4 = floor(h) ^ Float32(2.0)
	t_5 = fma(t_1, dY_46_u, Float32(Float32(t_4 * dY_46_v) * dY_46_v))
	tmp = Float32(0.0)
	if (Float32(((t_2 != t_2) ? Float32(t_1 * dY_46_u) : ((Float32(t_1 * dY_46_u) != Float32(t_1 * dY_46_u)) ? t_2 : max(t_2, Float32(t_1 * dY_46_u)))) / t_3) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)) != fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v))) ? t_5 : ((t_5 != t_5) ? fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)) : max(fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)), t_5)))) / floor(maxAniso));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / ((t_2 != t_2) ? t_5 : ((t_5 != t_5) ? t_2 : max(t_2, t_5))))) * t_3);
	end
	return log2(tmp)
end

Derivation

1. Initial program 73.5%

   \[\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 w 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 18.7%

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

7. Applied rewrites 27.5%

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

8. Taylor expanded in dY.u around inf

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

10. Applied rewrites 13.7%

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

12. Applied rewrites 15.9%

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

13. Final simplification 15.9%

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

Alternative 10: 14.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* t_0 dY.u))
        (t_2 (* t_0 dX.u))
        (t_3
         (fabs (* (* (floor h) (floor w)) (fma (- dY.v) dX.u (* dY.u dX.v)))))
        (t_4 (pow (floor h) 2.0))
        (t_5 (fma t_1 dY.u (* (* t_4 dY.v) dY.v))))
   (log2
    (if (>
         (/
          (fmax
           (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0))
           (* t_1 dY.u))
          t_3)
         (floor maxAniso))
      (/
       (sqrt (fmax (fma t_2 dX.u (* (* t_4 dX.v) dX.v)) t_5))
       (floor maxAniso))
      (* (sqrt (/ 1.0 (fmax (* t_2 dX.u) 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(w), 2.0f);
	float t_1 = t_0 * dY_46_u;
	float t_2 = t_0 * dX_46_u;
	float t_3 = fabsf(((floorf(h) * floorf(w)) * fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v))));
	float t_4 = powf(floorf(h), 2.0f);
	float t_5 = fmaf(t_1, dY_46_u, ((t_4 * dY_46_v) * dY_46_v));
	float tmp;
	if ((fmaxf((powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f)), (t_1 * dY_46_u)) / t_3) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(t_2, dX_46_u, ((t_4 * dX_46_v) * dX_46_v)), t_5)) / floorf(maxAniso);
	} else {
		tmp = sqrtf((1.0f / fmaxf((t_2 * dX_46_u), 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(w) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_u)
	t_2 = Float32(t_0 * dX_46_u)
	t_3 = abs(Float32(Float32(floor(h) * floor(w)) * fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v))))
	t_4 = floor(h) ^ Float32(2.0)
	t_5 = fma(t_1, dY_46_u, Float32(Float32(t_4 * dY_46_v) * dY_46_v))
	tmp = Float32(0.0)
	if (Float32(((Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) != Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) ? Float32(t_1 * dY_46_u) : ((Float32(t_1 * dY_46_u) != Float32(t_1 * dY_46_u)) ? Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) : max(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))), Float32(t_1 * dY_46_u)))) / t_3) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(t_2, dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)) != fma(t_2, dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v))) ? t_5 : ((t_5 != t_5) ? fma(t_2, dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)) : max(fma(t_2, dX_46_u, Float32(Float32(t_4 * dX_46_v) * dX_46_v)), t_5)))) / floor(maxAniso));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / ((Float32(t_2 * dX_46_u) != Float32(t_2 * dX_46_u)) ? t_5 : ((t_5 != t_5) ? Float32(t_2 * dX_46_u) : max(Float32(t_2 * dX_46_u), t_5))))) * t_3);
	end
	return log2(tmp)
end

Derivation

1. Initial program 73.5%

   \[\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 w 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%

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

7. Applied rewrites 27.3%

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

8. Taylor expanded in dY.u around inf

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

10. Applied rewrites 13.8%

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

11. Taylor expanded in dX.u around inf

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

13. Applied rewrites 15.4%

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

14. Final simplification 15.3%

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

Reproduce

herbie shell --seed 2024304 
(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)))))))))