Anisotropic x16 LOD (LOD)

Percentage Accurate: 76.0% → 76.1%

Time: 23.7s

Alternatives: 2

Speedup: 1.2×

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: 76.0% 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: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot dY.v - dX.v \cdot dY.u\\ t_1 := {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\\ t_2 := {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\\ t_3 := \mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + t\_1, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + t\_2\right)\\ t_4 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\frac{t\_3}{\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor }}{\left|t\_0\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot t\_4 + t\_1, t\_2 + dY.u \cdot \left(dY.u \cdot t\_4\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot t\_0\right)\right| \cdot {t\_3}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (- (* dX.u dY.v) (* dX.v dY.u)))
        (t_1 (pow (* (floor h) dX.v) 2.0))
        (t_2 (pow (* (floor h) dY.v) 2.0))
        (t_3
         (fmax
          (+ (pow (* dX.u (floor w)) 2.0) t_1)
          (+ (pow (* (floor w) dY.u) 2.0) t_2)))
        (t_4 (pow (floor w) 2.0)))
   (log2
    (if (> (/ (/ t_3 (* (floor w) (floor h))) (fabs t_0)) (floor maxAniso))
      (/
       (sqrt
        (fmax (+ (* (* dX.u dX.u) t_4) t_1) (+ t_2 (* dY.u (* dY.u t_4)))))
       (floor maxAniso))
      (* (fabs (* (floor h) (* (floor w) t_0))) (pow t_3 -0.5))))))

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 = (dX_46_u * dY_46_v) - (dX_46_v * dY_46_u);
	float t_1 = powf((floorf(h) * dX_46_v), 2.0f);
	float t_2 = powf((floorf(h) * dY_46_v), 2.0f);
	float t_3 = fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + t_1), (powf((floorf(w) * dY_46_u), 2.0f) + t_2));
	float t_4 = powf(floorf(w), 2.0f);
	float tmp;
	if (((t_3 / (floorf(w) * floorf(h))) / fabsf(t_0)) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf((((dX_46_u * dX_46_u) * t_4) + t_1), (t_2 + (dY_46_u * (dY_46_u * t_4))))) / floorf(maxAniso);
	} else {
		tmp = fabsf((floorf(h) * (floorf(w) * t_0))) * powf(t_3, -0.5f);
	}
	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_u * dY_46_v) - Float32(dX_46_v * dY_46_u))
	t_1 = Float32(floor(h) * dX_46_v) ^ Float32(2.0)
	t_2 = Float32(floor(h) * dY_46_v) ^ Float32(2.0)
	t_3 = (Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_1) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_1)) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_2) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_2) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_2)) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_1) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_1), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + t_2)))
	t_4 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_3 / Float32(floor(w) * floor(h))) / abs(t_0)) > floor(maxAniso))
		tmp = Float32(sqrt(((Float32(Float32(Float32(dX_46_u * dX_46_u) * t_4) + t_1) != Float32(Float32(Float32(dX_46_u * dX_46_u) * t_4) + t_1)) ? Float32(t_2 + Float32(dY_46_u * Float32(dY_46_u * t_4))) : ((Float32(t_2 + Float32(dY_46_u * Float32(dY_46_u * t_4))) != Float32(t_2 + Float32(dY_46_u * Float32(dY_46_u * t_4)))) ? Float32(Float32(Float32(dX_46_u * dX_46_u) * t_4) + t_1) : max(Float32(Float32(Float32(dX_46_u * dX_46_u) * t_4) + t_1), Float32(t_2 + Float32(dY_46_u * Float32(dY_46_u * t_4))))))) / floor(maxAniso));
	else
		tmp = Float32(abs(Float32(floor(h) * Float32(floor(w) * t_0))) * (t_3 ^ Float32(-0.5)));
	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 = (dX_46_u * dY_46_v) - (dX_46_v * dY_46_u);
	t_1 = (floor(h) * dX_46_v) ^ single(2.0);
	t_2 = (floor(h) * dY_46_v) ^ single(2.0);
	t_3 = max((((dX_46_u * floor(w)) ^ single(2.0)) + t_1), (((floor(w) * dY_46_u) ^ single(2.0)) + t_2));
	t_4 = floor(w) ^ single(2.0);
	tmp = single(0.0);
	if (((t_3 / (floor(w) * floor(h))) / abs(t_0)) > floor(maxAniso))
		tmp = sqrt(max((((dX_46_u * dX_46_u) * t_4) + t_1), (t_2 + (dY_46_u * (dY_46_u * t_4))))) / floor(maxAniso);
	else
		tmp = abs((floor(h) * (floor(w) * t_0))) * (t_3 ^ single(-0.5));
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 79.8%

   \[\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. Simplified 79.8%

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

7. Applied egg-rr 79.8%

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

9. Applied egg-rr 79.9%

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

10. Taylor expanded in dX.u around 0

    \[\leadsto \mathsf{log2.f32}\left(\begin{array}{l} \mathbf{if}\;\mathsf{>.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), dY.u\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{floor.f32}\left(h\right)\right)\right), \mathsf{fabs.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right):\\ \;\;\;\;\mathsf{/.f32}\left(\color{blue}{\left(\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)}\right)}, \mathsf{floor.f32}\left(maxAniso\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \frac{-1}{2}\right)\right)\\ \end{array}\right) \]

12. Simplified 79.9%

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

13. Final simplification 79.9%

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

Alternative 2: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := dX.u \cdot dY.v - dX.v \cdot dY.u\\ t_2 := \mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\frac{t\_2}{\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor }}{\left|t\_1\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_3\right) + dX.v \cdot \left(dX.v \cdot t\_0\right), dY.u \cdot \left(dY.u \cdot t\_3\right) + dY.v \cdot \left(dY.v \cdot t\_0\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot t\_1\right)\right| \cdot {t\_2}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (- (* dX.u dY.v) (* dX.v dY.u)))
        (t_2
         (fmax
          (+ (pow (* dX.u (floor w)) 2.0) (pow (* (floor h) dX.v) 2.0))
          (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor h) dY.v) 2.0))))
        (t_3 (pow (floor w) 2.0)))
   (log2
    (if (> (/ (/ t_2 (* (floor w) (floor h))) (fabs t_1)) (floor maxAniso))
      (/
       (sqrt
        (fmax
         (+ (* dX.u (* dX.u t_3)) (* dX.v (* dX.v t_0)))
         (+ (* dY.u (* dY.u t_3)) (* dY.v (* dY.v t_0)))))
       (floor maxAniso))
      (* (fabs (* (floor h) (* (floor w) t_1))) (pow t_2 -0.5))))))

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(h), 2.0f);
	float t_1 = (dX_46_u * dY_46_v) - (dX_46_v * dY_46_u);
	float t_2 = fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f)), (powf((floorf(w) * dY_46_u), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f)));
	float t_3 = powf(floorf(w), 2.0f);
	float tmp;
	if (((t_2 / (floorf(w) * floorf(h))) / fabsf(t_1)) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(((dX_46_u * (dX_46_u * t_3)) + (dX_46_v * (dX_46_v * t_0))), ((dY_46_u * (dY_46_u * t_3)) + (dY_46_v * (dY_46_v * t_0))))) / floorf(maxAniso);
	} else {
		tmp = fabsf((floorf(h) * (floorf(w) * t_1))) * powf(t_2, -0.5f);
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u))
	t_2 = (Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))
	t_3 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_2 / Float32(floor(w) * floor(h))) / abs(t_1)) > floor(maxAniso))
		tmp = Float32(sqrt(((Float32(Float32(dX_46_u * Float32(dX_46_u * t_3)) + Float32(dX_46_v * Float32(dX_46_v * t_0))) != Float32(Float32(dX_46_u * Float32(dX_46_u * t_3)) + Float32(dX_46_v * Float32(dX_46_v * t_0)))) ? Float32(Float32(dY_46_u * Float32(dY_46_u * t_3)) + Float32(dY_46_v * Float32(dY_46_v * t_0))) : ((Float32(Float32(dY_46_u * Float32(dY_46_u * t_3)) + Float32(dY_46_v * Float32(dY_46_v * t_0))) != Float32(Float32(dY_46_u * Float32(dY_46_u * t_3)) + Float32(dY_46_v * Float32(dY_46_v * t_0)))) ? Float32(Float32(dX_46_u * Float32(dX_46_u * t_3)) + Float32(dX_46_v * Float32(dX_46_v * t_0))) : max(Float32(Float32(dX_46_u * Float32(dX_46_u * t_3)) + Float32(dX_46_v * Float32(dX_46_v * t_0))), Float32(Float32(dY_46_u * Float32(dY_46_u * t_3)) + Float32(dY_46_v * Float32(dY_46_v * t_0))))))) / floor(maxAniso));
	else
		tmp = Float32(abs(Float32(floor(h) * Float32(floor(w) * t_1))) * (t_2 ^ Float32(-0.5)));
	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) ^ single(2.0);
	t_1 = (dX_46_u * dY_46_v) - (dX_46_v * dY_46_u);
	t_2 = max((((dX_46_u * floor(w)) ^ single(2.0)) + ((floor(h) * dX_46_v) ^ single(2.0))), (((floor(w) * dY_46_u) ^ single(2.0)) + ((floor(h) * dY_46_v) ^ single(2.0))));
	t_3 = floor(w) ^ single(2.0);
	tmp = single(0.0);
	if (((t_2 / (floor(w) * floor(h))) / abs(t_1)) > floor(maxAniso))
		tmp = sqrt(max(((dX_46_u * (dX_46_u * t_3)) + (dX_46_v * (dX_46_v * t_0))), ((dY_46_u * (dY_46_u * t_3)) + (dY_46_v * (dY_46_v * t_0))))) / floor(maxAniso);
	else
		tmp = abs((floor(h) * (floor(w) * t_1))) * (t_2 ^ single(-0.5));
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 79.8%

   \[\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. Simplified 79.8%

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

7. Applied egg-rr 79.8%

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

9. Applied egg-rr 79.9%

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

    1. associate-*r* N/A

       \[\leadsto \mathsf{log2.f32}\left(\begin{array}{l} \mathbf{if}\;\mathsf{>.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), dY.u\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{floor.f32}\left(h\right)\right)\right), \mathsf{fabs.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right):\\ \;\;\;\;\mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{*.f32}\left(dX.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(dX.v, \mathsf{*.f32}\left(dX.v, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(h\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{*.f32}\left(dY.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \frac{-1}{2}\right)\right)\\ \end{array}\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{log2.f32}\left(\begin{array}{l} \mathbf{if}\;\mathsf{>.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), dY.u\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{floor.f32}\left(h\right)\right)\right), \mathsf{fabs.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right):\\ \;\;\;\;\mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{*.f32}\left(dX.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(dX.v, \mathsf{*.f32}\left(dX.v, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(h\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{*.f32}\left(dY.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right), dY.v\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \frac{-1}{2}\right)\right)\\ \end{array}\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{log2.f32}\left(\begin{array}{l} \mathbf{if}\;\mathsf{>.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), dY.u\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{floor.f32}\left(h\right)\right)\right), \mathsf{fabs.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right):\\ \;\;\;\;\mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{*.f32}\left(dX.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(dX.v, \mathsf{*.f32}\left(dX.v, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(h\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{*.f32}\left(dY.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v\right), dY.v\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \frac{-1}{2}\right)\right)\\ \end{array}\right) \]

    4. pow-lowering-pow.f32 N/A

       \[\leadsto \mathsf{log2.f32}\left(\begin{array}{l} \mathbf{if}\;\mathsf{>.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), dY.u\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{floor.f32}\left(h\right)\right)\right), \mathsf{fabs.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right):\\ \;\;\;\;\mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{*.f32}\left(dX.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(dX.v, \mathsf{*.f32}\left(dX.v, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(h\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{*.f32}\left(dY.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\left\lfloor h\right\rfloor \right), 2\right), dY.v\right), dY.v\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \frac{-1}{2}\right)\right)\\ \end{array}\right) \]

    5. floor-lowering-floor.f32 79.9%

       \[\leadsto \mathsf{log2.f32}\left(\begin{array}{l} \mathbf{if}\;\mathsf{>.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), dY.u\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{floor.f32}\left(h\right)\right)\right), \mathsf{fabs.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right):\\ \;\;\;\;\mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{*.f32}\left(dX.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(dX.v, \mathsf{*.f32}\left(dX.v, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(h\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{*.f32}\left(dY.u, \mathsf{pow.f32}\left(\mathsf{floor.f32}\left(w\right), 2\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{floor.f32}\left(h\right), 2\right), dY.v\right), dY.v\right)\right)\right)\right), \mathsf{floor.f32}\left(maxAniso\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), \mathsf{*.f32}\left(\mathsf{floor.f32}\left(w\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(dX.u, dY.v\right), \mathsf{*.f32}\left(dX.v, dY.u\right)\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{fmax.f32}\left(\mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dX.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dX.v\right), 2\right)\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(dY.u, \mathsf{floor.f32}\left(w\right)\right), 2\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{floor.f32}\left(h\right), dY.v\right), 2\right)\right)\right), \frac{-1}{2}\right)\right)\\ \end{array}\right) \]

11. Applied egg-rr 79.9%

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

12. Final simplification 79.9%

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

Reproduce

herbie shell --seed 2024192 
(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)))))))))