Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 100.0%

Time: 17.7s

Alternatives: 3

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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \langle \left( \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right) + {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (log2
    (sqrt
     (fmax
      (+
       (+ (pow (* (floor h) dX.v) 2.0) (pow (* (floor w) dX.u) 2.0))
       (pow (* (floor d) dX.w) 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) {
	double tmp = log2(sqrt(fmax(((pow((floor(h) * ((double) dX_46_v)), 2.0) + pow((floor(w) * ((double) dX_46_u)), 2.0)) + pow((floor(d) * ((double) dX_46_w)), 2.0)), (pow((floor(w) * ((double) dY_46_u)), 2.0) + (pow((floor(h) * ((double) dY_46_v)), 2.0) + pow((floor(d) * ((double) dY_46_w)), 2.0))))));
	return (float) tmp;
}

Julia

function 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(((Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0)) != Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0))) ? Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0))) : ((Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0))) != Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0)))) ? Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0)) : max(Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0)), Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0))))))))
	return Float32(tmp)
end

MATLAB

function tmp_2 = 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) * double(dX_46_v)) ^ 2.0) + ((floor(w) * double(dX_46_u)) ^ 2.0)) + ((floor(d) * double(dX_46_w)) ^ 2.0)), (((floor(w) * double(dY_46_u)) ^ 2.0) + (((floor(h) * double(dY_46_v)) ^ 2.0) + ((floor(d) * double(dY_46_w)) ^ 2.0))))));
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 66.7%

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

   1. rewrite-binary32/binary64 100.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 100.0%

   \[\leadsto \color{blue}{\langle \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} \rangle_{\text{binary64}}} \]
4. Step-by-step derivation

5. Simplified 100.0%

   \[\leadsto \color{blue}{\langle \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right) + {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)} \rangle_{\text{binary64}}} \]

6. Final simplification 100.0%

   \[\leadsto \langle \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right) + {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right) \rangle_{\text{binary64}} \]

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \log_{2} \langle \left( \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right) + {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (cast
   (!
    :precision
    binary64
    (sqrt
     (fmax
      (+
       (+ (pow (* (floor h) dX.v) 2.0) (pow (* (floor w) dX.u) 2.0))
       (pow (* (floor d) dX.w) 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) {
	double tmp = sqrt(fmax(((pow((floor(h) * ((double) dX_46_v)), 2.0) + pow((floor(w) * ((double) dX_46_u)), 2.0)) + pow((floor(d) * ((double) dX_46_w)), 2.0)), (pow((floor(w) * ((double) dY_46_u)), 2.0) + (pow((floor(h) * ((double) dY_46_v)), 2.0) + pow((floor(d) * ((double) dY_46_w)), 2.0)))));
	return log2f(((float) tmp));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = sqrt(((Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0)) != Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0))) ? Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0))) : ((Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0))) != Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0)))) ? Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0)) : max(Float64(Float64((Float64(floor(h) * Float64(dX_46_v)) ^ 2.0) + (Float64(floor(w) * Float64(dX_46_u)) ^ 2.0)) + (Float64(floor(d) * Float64(dX_46_w)) ^ 2.0)), Float64((Float64(floor(w) * Float64(dY_46_u)) ^ 2.0) + Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (Float64(floor(d) * Float64(dY_46_w)) ^ 2.0)))))))
	return log2(Float32(tmp))
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = sqrt(max(((((floor(h) * double(dX_46_v)) ^ 2.0) + ((floor(w) * double(dX_46_u)) ^ 2.0)) + ((floor(d) * double(dX_46_w)) ^ 2.0)), (((floor(w) * double(dY_46_u)) ^ 2.0) + (((floor(h) * double(dY_46_v)) ^ 2.0) + ((floor(d) * double(dY_46_w)) ^ 2.0)))));
	tmp_2 = log2(single(tmp));
end

Derivation

1. Initial program 66.7%

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

   1. rewrite-binary32/binary64 100.0%

      \[\leadsto \color{blue}{\log_{2} \langle \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 100.0%

   \[\leadsto \log_{2} \color{blue}{\langle \color{blue}{\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} \rangle_{\text{binary64}}} \]
4. Step-by-step derivation

5. Simplified 100.0%

   \[\leadsto \log_{2} \color{blue}{\langle \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right) + {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)} \rangle_{\text{binary64}}} \]

6. Final simplification 100.0%

   \[\leadsto \log_{2} \langle \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right) + {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right) \rangle_{\text{binary64}} \]

Alternative 3: 67.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (+
     (pow (* dX.w (floor d)) 2.0)
     (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
    (+
     (pow (* dY.v (floor h)) 2.0)
     (+ (pow (* dY.w (floor d)) 2.0) (pow (* dY.u (floor 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((dX_46_w * floorf(d)), 2.0f) + (powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), (powf((dY_46_v * floorf(h)), 2.0f) + (powf((dY_46_w * floorf(d)), 2.0f) + powf((dY_46_u * floorf(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(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) != Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))))) ? Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) : ((Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) != Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))) ? Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) : max(Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(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((((dX_46_w * floor(d)) ^ single(2.0)) + (((dX_46_u * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), (((dY_46_v * floor(h)) ^ single(2.0)) + (((dY_46_w * floor(d)) ^ single(2.0)) + ((dY_46_u * floor(w)) ^ single(2.0)))))));
end

Derivation

1. Initial program 66.7%

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

2. Taylor expanded in w around 0 66.7%

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

4. Simplified 66.7%

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

5. Final simplification 66.7%

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

Reproduce

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