Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 67.7%

Time: 37.2s

Alternatives: 5

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(1 \leq d \land d \leq 4096\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.w\right| \land \left|dX.w\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.w\right| \land \left|dY.w\right| \leq 10^{+20}\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Initial Program: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Alternative 1: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t_1, t_1, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left\lfloord\right\rfloor\right)\right), \mathsf{fma}\left(t_0, t_0, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u)) (t_1 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+
       (fma t_1 t_1 (* (floor h) (* (floor h) (* dX.v dX.v))))
       (* dX.w (* (floor d) (* dX.w (floor d)))))
      (+
       (fma t_0 t_0 (* dY.v (* dY.v (* (floor h) (floor h)))))
       (* (floor d) (* (floor d) (* dY.w dY.w)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((fmaf(t_1, t_1, (floorf(h) * (floorf(h) * (dX_46_v * dX_46_v)))) + (dX_46_w * (floorf(d) * (dX_46_w * floorf(d))))), (fmaf(t_0, t_0, (dY_46_v * (dY_46_v * (floorf(h) * floorf(h))))) + (floorf(d) * (floorf(d) * (dY_46_w * dY_46_w)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(fma(t_1, t_1, Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))) + Float32(dX_46_w * Float32(floor(d) * Float32(dX_46_w * floor(d))))) != Float32(fma(t_1, t_1, Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))) + Float32(dX_46_w * Float32(floor(d) * Float32(dX_46_w * floor(d)))))) ? Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w)))) : ((Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w)))) != Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w))))) ? Float32(fma(t_1, t_1, Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))) + Float32(dX_46_w * Float32(floor(d) * Float32(dX_46_w * floor(d))))) : max(Float32(fma(t_1, t_1, Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))) + Float32(dX_46_w * Float32(floor(d) * Float32(dX_46_w * floor(d))))), Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w)))))))))
end

Derivation

1. Initial program 70.3%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

3. Simplified 70.3%

   \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right)} \]

4. Final simplification 70.3%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left\lfloord\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

Alternative 2: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}\right) + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (+
     (+ (pow (* (floor h) dX.v) 2.0) (pow (* dX.w (floor d)) 2.0))
     (pow (* (floor w) dX.u) 2.0))
    (+
     (pow (* (floor h) dY.v) 2.0)
     (+ (pow (* (floor d) dY.w) 2.0) (pow (* (floor w) dY.u) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(((powf((floorf(h) * dX_46_v), 2.0f) + powf((dX_46_w * floorf(d)), 2.0f)) + powf((floorf(w) * dX_46_u), 2.0f)), (powf((floorf(h) * dY_46_v), 2.0f) + (powf((floorf(d) * dY_46_w), 2.0f) + powf((floorf(w) * dY_46_u), 2.0f))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(((Float32(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) != Float32(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) + (Float32(floor(w) * dX_46_u) ^ Float32(2.0)))) ? Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))) : ((Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))) != Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))))) ? Float32(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) : max(Float32(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((((floor(h) * dX_46_v) ^ single(2.0)) + ((dX_46_w * floor(d)) ^ single(2.0))) + ((floor(w) * dX_46_u) ^ single(2.0))), (((floor(h) * dY_46_v) ^ single(2.0)) + (((floor(d) * dY_46_w) ^ single(2.0)) + ((floor(w) * dY_46_u) ^ single(2.0)))))));
end

Derivation

1. Initial program 70.3%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

2. Taylor expanded in w around 0 70.3%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + \left({dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right), {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + \left({dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right)\right)}\right)} \]

4. Simplified 70.3%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}\right) + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}\right)} \]

5. Final simplification 70.3%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}\right) + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}\right) \]

Alternative 3: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2} + \left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, \mathsf{fma}\left(t_0, t_0, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u)))
   (log2
    (sqrt
     (fmax
      (+ (pow (* dX.w (floor d)) 2.0) (* (* dX.v dX.v) (pow (floor h) 2.0)))
      (+
       (fma t_0 t_0 (* dY.v (* dY.v (* (floor h) (floor h)))))
       (* (floor d) (* (floor d) (* dY.w dY.w)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	return log2f(sqrtf(fmaxf((powf((dX_46_w * floorf(d)), 2.0f) + ((dX_46_v * dX_46_v) * powf(floorf(h), 2.0f))), (fmaf(t_0, t_0, (dY_46_v * (dY_46_v * (floorf(h) * floorf(h))))) + (floorf(d) * (floorf(d) * (dY_46_w * dY_46_w)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	return log2(sqrt(((Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32(Float32(dX_46_v * dX_46_v) * (floor(h) ^ Float32(2.0)))) != Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32(Float32(dX_46_v * dX_46_v) * (floor(h) ^ Float32(2.0))))) ? Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w)))) : ((Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w)))) != Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w))))) ? Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32(Float32(dX_46_v * dX_46_v) * (floor(h) ^ Float32(2.0)))) : max(Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32(Float32(dX_46_v * dX_46_v) * (floor(h) ^ Float32(2.0)))), Float32(fma(t_0, t_0, Float32(dY_46_v * Float32(dY_46_v * Float32(floor(h) * floor(h))))) + Float32(floor(d) * Float32(floor(d) * Float32(dY_46_w * dY_46_w)))))))))
end

Derivation

1. Initial program 70.3%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

3. Simplified 70.3%

   \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right)} \]

4. Taylor expanded in dX.u around 0 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

6. Simplified 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

7. Taylor expanded in dX.w around 0 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + \color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   2. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + \left(dX.w \cdot dX.w\right) \cdot \color{blue}{\left(\left\lfloord\right\rfloor \cdot \left\lfloord\right\rfloor\right)}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   3. swap-sqr 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + \color{blue}{\left(dX.w \cdot \left\lfloord\right\rfloor\right) \cdot \left(dX.w \cdot \left\lfloord\right\rfloor\right)}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   4. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + \color{blue}{{\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

9. Simplified 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + \color{blue}{{\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

10. Final simplification 61.2%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2} + \left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

Alternative 4: 60.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \left(dY.w \cdot dY.w\right) \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (+ (pow (* (floor h) dX.v) 2.0) (pow (* dX.w (floor d)) 2.0))
    (+
     (pow (* (floor w) dY.u) 2.0)
     (+
      (pow (* (floor h) dY.v) 2.0)
      (* (* dY.w dY.w) (pow (floor d) 2.0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf((powf((floorf(h) * dX_46_v), 2.0f) + powf((dX_46_w * floorf(d)), 2.0f)), (powf((floorf(w) * dY_46_u), 2.0f) + (powf((floorf(h) * dY_46_v), 2.0f) + ((dY_46_w * dY_46_w) * powf(floorf(d), 2.0f)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(((Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) != Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0)))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32(Float32(dY_46_w * dY_46_w) * (floor(d) ^ Float32(2.0))))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32(Float32(dY_46_w * dY_46_w) * (floor(d) ^ Float32(2.0))))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32(Float32(dY_46_w * dY_46_w) * (floor(d) ^ Float32(2.0)))))) ? Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) : max(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + Float32(Float32(dY_46_w * dY_46_w) * (floor(d) ^ Float32(2.0))))))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max((((floor(h) * dX_46_v) ^ single(2.0)) + ((dX_46_w * floor(d)) ^ single(2.0))), (((floor(w) * dY_46_u) ^ single(2.0)) + (((floor(h) * dY_46_v) ^ single(2.0)) + ((dY_46_w * dY_46_w) * (floor(d) ^ single(2.0))))))));
end

Derivation

1. Initial program 70.3%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

3. Simplified 70.3%

   \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right)} \]

4. Taylor expanded in dX.u around 0 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   2. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   3. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   4. swap-sqr 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   5. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   6. *-commutative 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

6. Simplified 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

7. Taylor expanded in dX.v around 0 61.2%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {dY.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2} + \mathsf{fma}\left(dY.u \cdot \left\lfloorw\right\rfloor, dY.u \cdot \left\lfloorw\right\rfloor, {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)\right)}\right)} \]

9. Simplified 61.2%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right) + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)} \]
10. Step-by-step derivation

    1. unpow-prod-down 61.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \color{blue}{{\left(\left\lfloord\right\rfloor\right)}^{2} \cdot {dY.w}^{2}}\right) + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right) \]

    2. pow2 61.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor\right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)}\right) + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right) \]

11. Applied egg-rr 61.2%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \color{blue}{{\left(\left\lfloord\right\rfloor\right)}^{2} \cdot \left(dY.w \cdot dY.w\right)}\right) + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right) \]

12. Final simplification 61.2%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + \left(dY.w \cdot dY.w\right) \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right)\right)}\right) \]

Alternative 5: 60.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (+ (pow (* (floor h) dX.v) 2.0) (pow (* dX.w (floor d)) 2.0))
    (+
     (pow (* (floor w) dY.u) 2.0)
     (+ (pow (* (floor h) dY.v) 2.0) (pow (* (floor d) dY.w) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf((powf((floorf(h) * dX_46_v), 2.0f) + powf((dX_46_w * floorf(d)), 2.0f)), (powf((floorf(w) * dY_46_u), 2.0f) + (powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(d) * dY_46_w), 2.0f))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(((Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) != Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0)))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0))))) ? Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) : max(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(dX_46_w * floor(d)) ^ Float32(2.0))), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max((((floor(h) * dX_46_v) ^ single(2.0)) + ((dX_46_w * floor(d)) ^ single(2.0))), (((floor(w) * dY_46_u) ^ single(2.0)) + (((floor(h) * dY_46_v) ^ single(2.0)) + ((floor(d) * dY_46_w) ^ single(2.0)))))));
end

Derivation

1. Initial program 70.3%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

3. Simplified 70.3%

   \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right)} \]

4. Taylor expanded in dX.u around 0 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   2. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   3. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   4. swap-sqr 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   5. unpow2 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

   6. *-commutative 61.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

6. Simplified 61.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}} + dX.w \cdot \left(\left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, dY.v \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(\left\lfloord\right\rfloor \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right) \]

7. Taylor expanded in dX.v around 0 61.2%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {dY.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2} + \mathsf{fma}\left(dY.u \cdot \left\lfloorw\right\rfloor, dY.u \cdot \left\lfloorw\right\rfloor, {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)\right)}\right)} \]

9. Simplified 61.2%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right) + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)} \]

10. Final simplification 61.2%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right) \]

Reproduce

herbie shell --seed 2023282 
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :name "Isotropic LOD (LOD)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1.0 d) (<= d 4096.0))) (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 dX.w)) (<= (fabs dX.w) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (and (<= 1e-20 (fabs dY.w)) (<= (fabs dY.w) 1e+20)))
  (log2 (sqrt (fmax (+ (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (* (* (floor d) dX.w) (* (floor d) dX.w))) (+ (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v))) (* (* (floor d) dY.w) (* (floor d) dY.w)))))))