Anisotropic x16 LOD (LOD)

Percentage Accurate: 76.3% → 76.3%

Time: 30.8s

Alternatives: 5

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t_3 \cdot t_3 + t_0 \cdot t_0, t_1 \cdot t_1 + t_2 \cdot t_2\right)\\ t_5 := \sqrt{t_4}\\ t_6 := \left|t_3 \cdot t_2 - t_0 \cdot t_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_4}{t_6} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_5}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6}{t_5}\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t_3 \cdot t_3 + t_0 \cdot t_0, t_1 \cdot t_1 + t_2 \cdot t_2\right)\\ t_5 := \sqrt{t_4}\\ t_6 := \left|t_3 \cdot t_2 - t_0 \cdot t_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_4}{t_6} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_5}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6}{t_5}\\ \end{array} \end{array} \end{array} \]

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: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|\\ t_1 := \mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)\\ t_2 := \sqrt{t_1}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_1}{t_0} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_2}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

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

Derivation

1. Initial program 73.1%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]

Alternative 2: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_4 := \sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_1\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot t_3, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_2\right)\right)\right)}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{\mathsf{max}\left({t_0}^{2} + {t_1}^{2}, {\left(\mathsf{hypot}\left(t_2, t_3\right)\right)}^{2}\right)}{t_2 \cdot t_0}\right)}^{2}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_4}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|}{t_4}\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Derivation

1. Initial program 72.4%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}\right)}^{2}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := {t_1}^{2}\\ t_3 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_4 := \left\lfloorw\right\rfloor \cdot dY.u\\ \log_{2} \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{\mathsf{max}\left({t_0}^{2} + t_2, {\left(\mathsf{hypot}\left(t_3, t_4\right)\right)}^{2}\right)}{t_3 \cdot t_0}\right)}^{2}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_1\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot t_4, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_3\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left(t_2 + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {t_3}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \end{array} \end{array} \]

FPCore

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

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

Derivation

1. Initial program 70.8%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}\right)}^{2}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]

Alternative 4: 44.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := {t_0}^{2}\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + t_1, {\left(\mathsf{hypot}\left(t_2, t_3\right)\right)}^{2}\right)}{\left(dX.u \cdot dY.v\right) \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot t_3, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_2\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(t_1 + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {t_2}^{2}\right)\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Derivation

1. Initial program 44.0%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left(dX.u \cdot dY.v\right) \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}\\ \end{array} \]

Alternative 5: 39.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := {t_0}^{2}\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + t_1, {\left(\mathsf{hypot}\left(t_2, t_3\right)\right)}^{2}\right)}{\left(dX.u \cdot dY.v\right) \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot t_3, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_2\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left(t_1 + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {t_2}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \end{array} \end{array} \]

FPCore

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

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 = powf(t_0, 2.0f);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dY_46_u;
	float tmp;
	if ((fmaxf((powf((floorf(w) * dX_46_u), 2.0f) + t_1), powf(hypotf(t_2, t_3), 2.0f)) / ((dX_46_u * dY_46_v) * (floorf(w) * floorf(h)))) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(floorf(w), (floorf(w) * (dX_46_u * dX_46_u)), (floorf(h) * (dX_46_v * t_0))), fmaf(floorf(w), (dY_46_u * t_3), (floorf(h) * (dY_46_v * t_2))))) / floorf(maxAniso);
	} else {
		tmp = (floorf(h) * ((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u))) / (sqrtf(fmaxf((t_1 + ((dX_46_u * dX_46_u) * powf(floorf(w), 2.0f))), fmaf(floorf(w), (floorf(w) * (dY_46_u * dY_46_u)), powf(t_2, 2.0f)))) / floorf(w));
	}
	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 = t_0 ^ Float32(2.0)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (Float32(((Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + t_1) != Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + t_1)) ? (hypot(t_2, t_3) ^ Float32(2.0)) : (((hypot(t_2, t_3) ^ Float32(2.0)) != (hypot(t_2, t_3) ^ Float32(2.0))) ? Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + t_1) : max(Float32((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) + t_1), (hypot(t_2, t_3) ^ Float32(2.0))))) / Float32(Float32(dX_46_u * dY_46_v) * Float32(floor(w) * floor(h)))) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))) != fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0)))) ? fma(floor(w), Float32(dY_46_u * t_3), Float32(floor(h) * Float32(dY_46_v * t_2))) : ((fma(floor(w), Float32(dY_46_u * t_3), Float32(floor(h) * Float32(dY_46_v * t_2))) != fma(floor(w), Float32(dY_46_u * t_3), Float32(floor(h) * Float32(dY_46_v * t_2)))) ? fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))) : max(fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))), fma(floor(w), Float32(dY_46_u * t_3), Float32(floor(h) * Float32(dY_46_v * t_2))))))) / floor(maxAniso));
	else
		tmp = Float32(Float32(floor(h) * Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u))) / Float32(sqrt(((Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * (floor(w) ^ Float32(2.0)))) != Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * (floor(w) ^ Float32(2.0))))) ? fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), (t_2 ^ Float32(2.0))) : ((fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), (t_2 ^ Float32(2.0))) != fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), (t_2 ^ Float32(2.0)))) ? Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * (floor(w) ^ Float32(2.0)))) : max(Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * (floor(w) ^ Float32(2.0)))), fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), (t_2 ^ Float32(2.0))))))) / floor(w)));
	end
	return log2(tmp)
end

Derivation

1. Initial program 38.2%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left(dX.u \cdot dY.v\right) \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]

Reproduce

herbie shell --seed 2023297 
(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)))))))))