Anisotropic x16 LOD (LOD)

Percentage Accurate: 76.2% → 76.2%

Time: 24.4s

Alternatives: 7

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 76.2% 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: 76.2% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = (Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) != Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))) ? Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) : ((Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) != Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) ? Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) : max(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (t_0 ^ Float32(2.0))), Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))))
	t_3 = sqrt(t_2)
	t_4 = abs(Float32(floor(h) * Float32(Float32(t_1 * dX_46_v) - Float32(dY_46_v * t_0))))
	tmp = Float32(0.0)
	if (Float32(t_2 / t_4) > floor(maxAniso))
		tmp = Float32(t_3 / floor(maxAniso));
	else
		tmp = Float32(t_4 / t_3);
	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 = dX_46_u * floor(w);
	t_1 = dY_46_u * floor(w);
	t_2 = max((((dX_46_v * floor(h)) ^ single(2.0)) + (t_0 ^ single(2.0))), (((dY_46_v * floor(h)) ^ single(2.0)) + (t_1 ^ single(2.0))));
	t_3 = sqrt(t_2);
	t_4 = abs((floor(h) * ((t_1 * dX_46_v) - (dY_46_v * t_0))));
	tmp = single(0.0);
	if ((t_2 / t_4) > floor(maxAniso))
		tmp = t_3 / floor(maxAniso);
	else
		tmp = t_4 / t_3;
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 80.8%

   \[\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

4. Applied rewrites 80.8%

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

5. Final simplification 80.8%

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

Alternative 2: 75.3% accurate, 1.2× speedup

Math

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

FPCore

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

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = (Float32((Float32(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((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)))) ? 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((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))))
	t_1 = sqrt(t_0)
	t_2 = abs(Float32(floor(h) * Float32(Float32(floor(w) * dY_46_v) * dX_46_u)))
	tmp = Float32(0.0)
	if (Float32(t_0 / t_2) > floor(maxAniso))
		tmp = Float32(t_1 / floor(maxAniso));
	else
		tmp = Float32(t_2 / t_1);
	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 = max((((dX_46_v * floor(h)) ^ single(2.0)) + ((dX_46_u * floor(w)) ^ single(2.0))), (((dY_46_v * floor(h)) ^ single(2.0)) + ((dY_46_u * floor(w)) ^ single(2.0))));
	t_1 = sqrt(t_0);
	t_2 = abs((floor(h) * ((floor(w) * dY_46_v) * dX_46_u)));
	tmp = single(0.0);
	if ((t_0 / t_2) > floor(maxAniso))
		tmp = t_1 / floor(maxAniso);
	else
		tmp = t_2 / t_1;
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 80.8%

   \[\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

4. Applied rewrites 80.8%

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

5. Taylor expanded in dX.u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-floor.f32 80.2

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

7. Applied rewrites 80.2%

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

8. Taylor expanded in dX.u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-floor.f32 80.5

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

10. Applied rewrites 80.5%

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

Alternative 3: 49.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\\ t_2 := t\_0 \cdot dY.u\\ t_3 := \left|\mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ t_4 := \frac{\mathsf{max}\left(t\_1, t\_2 \cdot dY.u\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor \\ t_5 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_6 := \mathsf{fma}\left(t\_2, dY.u, \left(t\_5 \cdot dY.v\right) \cdot dY.v\right)\\ t_7 := \mathsf{fma}\left(t\_0 \cdot dX.u, dX.u, \left(t\_5 \cdot dX.v\right) \cdot dX.v\right)\\ t_8 := \sqrt{\frac{1}{\mathsf{max}\left(t\_7, t\_6\right)}} \cdot t\_3\\ \mathbf{if}\;dY.u \leq -9.999999747378752 \cdot 10^{-5} \lor \neg \left(dY.u \leq 3500\right):\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;t\_4:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_7, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;t\_4:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_1, t\_6\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (+ (pow (* (floor w) dX.u) 2.0) (pow (* (floor h) dX.v) 2.0)))
        (t_2 (* t_0 dY.u))
        (t_3
         (fabs (* (fma (- dY.v) dX.u (* dY.u dX.v)) (* (floor w) (floor h)))))
        (t_4 (> (/ (fmax t_1 (* 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_0 dX.u) dX.u (* (* t_5 dX.v) dX.v)))
        (t_8 (* (sqrt (/ 1.0 (fmax t_7 t_6))) t_3)))
   (if (or (<= dY.u -9.999999747378752e-5) (not (<= dY.u 3500.0)))
     (log2
      (if t_4
        (/
         (sqrt
          (fmax
           t_7
           (+ (pow (* (floor h) dY.v) 2.0) (pow (* dY.u (floor w)) 2.0))))
         (floor maxAniso))
        t_8))
     (log2 (if t_4 (/ (sqrt (fmax t_1 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(floorf(w), 2.0f);
	float t_1 = powf((floorf(w) * dX_46_u), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f);
	float t_2 = t_0 * dY_46_u;
	float t_3 = fabsf((fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v)) * (floorf(w) * floorf(h))));
	int t_4 = (fmaxf(t_1, (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_0 * 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 ((dY_46_u <= -9.999999747378752e-5f) || !(dY_46_u <= 3500.0f)) {
		float tmp_2;
		if (t_4) {
			tmp_2 = sqrtf(fmaxf(t_7, (powf((floorf(h) * dY_46_v), 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_1, 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 = floor(w) ^ Float32(2.0)
	t_1 = Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))
	t_2 = Float32(t_0 * dY_46_u)
	t_3 = abs(Float32(fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v)) * Float32(floor(w) * floor(h))))
	t_4 = Float32(((t_1 != t_1) ? Float32(t_2 * dY_46_u) : ((Float32(t_2 * dY_46_u) != Float32(t_2 * dY_46_u)) ? t_1 : max(t_1, 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_0 * 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 ((dY_46_u <= Float32(-9.999999747378752e-5)) || !(dY_46_u <= Float32(3500.0)))
		tmp_2 = Float32(0.0)
		if (t_4)
			tmp_2 = Float32(sqrt(((t_7 != t_7) ? Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? t_7 : max(t_7, Float32((Float32(floor(h) * dY_46_v) ^ 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_1 != t_1) ? t_6 : ((t_6 != t_6) ? t_1 : max(t_1, 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 dY.u < -9.99999975e-5 or 3500 < dY.u

   1. Initial program 77.3%

      \[\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 18.8%

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

   7. Applied rewrites 23.7%

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

   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 12.6%

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

   12. Applied rewrites 53.0%

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

   if -9.99999975e-5 < dY.u < 3500

   1. Initial program 83.9%

      \[\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

   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 16.1%

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

   7. Applied rewrites 31.1%

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

   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 14.4%

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

   12. Applied rewrites 43.9%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq -9.999999747378752 \cdot 10^{-5} \lor \neg \left(dY.u \leq 3500\right):\\ \;\;\;\;\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(\left\lfloor h\right\rfloor \cdot dY.v\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}\\ \mathbf{else}:\\ \;\;\;\;\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}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.9% accurate, 1.3× speedup

Math

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

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 (* (floor w) dX.u) 2.0) (pow (* (floor h) dX.v) 2.0)))
        (t_5
         (fabs (* (fma (- dY.v) dX.u (* dY.u dX.v)) (* (floor w) (floor h))))))
   (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((floorf(w) * dX_46_u), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f);
	float t_5 = fabsf((fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v)) * (floorf(w) * floorf(h))));
	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(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))
	t_5 = abs(Float32(fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v)) * Float32(floor(w) * floor(h))))
	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 80.8%

   \[\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 17.0%

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

7. Applied rewrites 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 13.4%

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

12. Applied rewrites 36.0%

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

Alternative 5: 23.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 80.8%

   \[\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 17.2%

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

7. Applied rewrites 28.2%

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

8. 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 13.5%

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

11. Taylor expanded in dX.u around inf

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

13. Applied rewrites 14.8%

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

Alternative 6: 24.3% accurate, 1.3× speedup

Math

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

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 (* (floor w) dX.u) 2.0) (pow (* (floor h) dX.v) 2.0)))
        (t_3
         (fabs (* (fma (- dY.v) dX.u (* dY.u dX.v)) (* (floor w) (floor h)))))
        (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((floorf(w) * dX_46_u), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f);
	float t_3 = fabsf((fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v)) * (floorf(w) * floorf(h))));
	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(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))
	t_3 = abs(Float32(fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v)) * Float32(floor(w) * floor(h))))
	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 80.8%

   \[\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 17.4%

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

7. Applied rewrites 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 13.4%

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

12. Applied rewrites 15.3%

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

Alternative 7: 24.0% accurate, 1.4× speedup

Math

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

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 (* (fma (- dY.v) dX.u (* dY.u dX.v)) (* (floor w) (floor h)))))
        (t_4 (pow (floor h) 2.0))
        (t_5 (fma t_1 dY.u (* (* t_4 dY.v) dY.v))))
   (log2
    (if (>
         (/
          (fmax
           (+ (pow (* (floor w) dX.u) 2.0) (pow (* (floor h) dX.v) 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((fmaf(-dY_46_v, dX_46_u, (dY_46_u * dX_46_v)) * (floorf(w) * floorf(h))));
	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((floorf(w) * dX_46_u), 2.0f) + powf((floorf(h) * dX_46_v), 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(fma(Float32(-dY_46_v), dX_46_u, Float32(dY_46_u * dX_46_v)) * Float32(floor(w) * floor(h))))
	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(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? Float32(t_1 * dY_46_u) : ((Float32(t_1 * dY_46_u) != Float32(t_1 * dY_46_u)) ? Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ 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 80.8%

   \[\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 17.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)}} \cdot \left|\mathsf{fma}\left(-dY.v, dX.u, dY.u \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right|\\ \end{array}} \]
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 13.4%

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

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 15.1%

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

Reproduce

herbie shell --seed 2024319 
(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)))))))))