Isotropic LOD (LOD)

Percentage Accurate: 67.5% → 70.9%

Time: 13.7s

Alternatives: 14

Speedup: 0.5×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

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(fmax(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: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

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(fmax(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: 70.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, \left(t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4\right)}\right)\\ \mathbf{if}\;t\_6 \leq 63.97999954223633:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor d) dY.w))
        (t_5 (* (floor d) dX.w))
        (t_6
         (log2
          (sqrt
           (fmax
            (+ (+ (* t_0 t_0) (* t_3 t_3)) (* t_5 t_5))
            (+ (+ (* t_1 t_1) (* t_2 t_2)) (* t_4 t_4)))))))
   (if (<= t_6 63.97999954223633)
     t_6
     (log2 (sqrt (fmax (pow t_0 2.0) (pow (* dY.w (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) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_3 * t_3)) + (t_5 * t_5)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	float tmp;
	if (t_6 <= 63.97999954223633f) {
		tmp = t_6;
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf((dY_46_w * floorf(d)), 2.0f))));
	}
	return 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)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)))))
	tmp = Float32(0.0)
	if (t_6 <= Float32(63.97999954223633))
		tmp = t_6;
	else
		tmp = log2(sqrt(fmax((t_0 ^ Float32(2.0)), (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))));
	end
	return 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)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(h) * dX_46_v;
	t_4 = floor(d) * dY_46_w;
	t_5 = floor(d) * dX_46_w;
	t_6 = log2(sqrt(max((((t_0 * t_0) + (t_3 * t_3)) + (t_5 * t_5)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	tmp = single(0.0);
	if (t_6 <= single(63.97999954223633))
		tmp = t_6;
	else
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), ((dY_46_w * floor(d)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 63.9799995

   1. Initial program 99.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   if 63.9799995 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 7.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 12.3%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 16.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 16.5%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 17.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_4 := \left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + t\_3 \cdot t\_3\\ \mathbf{if}\;dY.w \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_4, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, t\_1 \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_4, \mathsf{fma}\left(t\_1, dY.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (pow (floor w) 2.0) dY.u))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dX.w))
        (t_4 (+ (+ (* t_0 t_0) (* t_2 t_2)) (* t_3 t_3))))
   (if (<= dY.w 200.0)
     (log2
      (sqrt (fmax t_4 (fma (* (pow (floor h) 2.0) dY.v) dY.v (* t_1 dY.u)))))
     (log2
      (sqrt
       (fmax t_4 (fma t_1 dY.u (* (* (pow (floor d) 2.0) 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) * dX_46_u;
	float t_1 = powf(floorf(w), 2.0f) * dY_46_u;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dX_46_w;
	float t_4 = ((t_0 * t_0) + (t_2 * t_2)) + (t_3 * t_3);
	float tmp;
	if (dY_46_w <= 200.0f) {
		tmp = log2f(sqrtf(fmaxf(t_4, fmaf((powf(floorf(h), 2.0f) * dY_46_v), dY_46_v, (t_1 * dY_46_u)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_4, fmaf(t_1, dY_46_u, ((powf(floorf(d), 2.0f) * dY_46_w) * dY_46_w)))));
	}
	return 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)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32((floor(w) ^ Float32(2.0)) * dY_46_u)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dX_46_w)
	t_4 = Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(200.0))
		tmp = log2(sqrt(fmax(t_4, fma(Float32((floor(h) ^ Float32(2.0)) * dY_46_v), dY_46_v, Float32(t_1 * dY_46_u)))));
	else
		tmp = log2(sqrt(fmax(t_4, fma(t_1, dY_46_u, Float32(Float32((floor(d) ^ Float32(2.0)) * dY_46_w) * dY_46_w)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 200

   1. Initial program 66.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.w around 0

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

   5. Applied rewrites 61.2%

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

   if 200 < dY.w

   1. Initial program 50.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

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

   5. Applied rewrites 52.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \mathsf{fma}\left(t\_6 \cdot dY.v, dY.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w, dX.w, \left(t\_6 \cdot dX.v\right) \cdot dX.v\right), \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right)\\ \end{array} \end{array} \]

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))
        (t_6 (pow (floor h) 2.0)))
   (if (<= dY.w 100.0)
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
        (fma (* t_6 dY.v) dY.v (* (* (pow (floor w) 2.0) dY.u) dY.u)))))
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor d) 2.0) dX.w) dX.w (* (* t_6 dX.v) dX.v))
        (+ (+ (* 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;
	float t_6 = powf(floorf(h), 2.0f);
	float tmp;
	if (dY_46_w <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), fmaf((t_6 * dY_46_v), dY_46_v, ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(d), 2.0f) * dX_46_w), dX_46_w, ((t_6 * dX_46_v) * dX_46_v)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
	}
	return 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)
	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)
	t_6 = floor(h) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(100.0))
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), fma(Float32(t_6 * dY_46_v), dY_46_v, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)))));
	else
		tmp = log2(sqrt(fmax(fma(Float32((floor(d) ^ Float32(2.0)) * dX_46_w), dX_46_w, Float32(Float32(t_6 * dX_46_v) * dX_46_v)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 100

   1. Initial program 66.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.w around 0

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

   5. Applied rewrites 61.0%

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

   if 100 < dY.w

   1. Initial program 51.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

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

   5. Applied rewrites 51.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w, dX.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_5 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_6 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dX.w \leq 50000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_5 \cdot dX.u, dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0\right) + t\_2 \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_4 \cdot t\_4 + t\_6 \cdot t\_6\right) + t\_1 \cdot t\_1, \left(t\_5 \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor d) dX.w))
        (t_2 (* (floor d) dY.w))
        (t_3 (* (floor w) dY.u))
        (t_4 (* (floor w) dX.u))
        (t_5 (pow (floor w) 2.0))
        (t_6 (* (floor h) dX.v)))
   (if (<= dX.w 50000.0)
     (log2
      (sqrt
       (fmax
        (fma (* t_5 dX.u) dX.u (* (* (pow (floor h) 2.0) dX.v) dX.v))
        (+ (+ (* t_3 t_3) (* t_0 t_0)) (* t_2 t_2)))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_4 t_4) (* t_6 t_6)) (* t_1 t_1))
        (* (* t_5 dY.u) dY.u)))))))

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(h) * dY_46_v;
	float t_1 = floorf(d) * dX_46_w;
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = powf(floorf(w), 2.0f);
	float t_6 = floorf(h) * dX_46_v;
	float tmp;
	if (dX_46_w <= 50000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_5 * dX_46_u), dX_46_u, ((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v)), (((t_3 * t_3) + (t_0 * t_0)) + (t_2 * t_2)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_4 * t_4) + (t_6 * t_6)) + (t_1 * t_1)), ((t_5 * dY_46_u) * dY_46_u))));
	}
	return 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)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dX_46_w)
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = floor(w) ^ Float32(2.0)
	t_6 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(50000.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_5 * dX_46_u), dX_46_u, Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)), Float32(Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) + Float32(t_2 * t_2)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_4 * t_4) + Float32(t_6 * t_6)) + Float32(t_1 * t_1)), Float32(Float32(t_5 * dY_46_u) * dY_46_u))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 5e4

   1. Initial program 67.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around 0

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

   5. Applied rewrites 62.0%

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

   if 5e4 < dX.w

   1. Initial program 44.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around inf

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

   5. Applied rewrites 44.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_5 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_6 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dX.u \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_5 \cdot dX.w, dX.w, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v\right), \left(t\_3 \cdot t\_3 + t\_1 \cdot t\_1\right) + t\_2 \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_4 \cdot t\_4 + t\_6 \cdot t\_6\right) + t\_0 \cdot t\_0, \left(t\_5 \cdot dY.w\right) \cdot dY.w\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor d) dY.w))
        (t_3 (* (floor w) dY.u))
        (t_4 (* (floor w) dX.u))
        (t_5 (pow (floor d) 2.0))
        (t_6 (* (floor h) dX.v)))
   (if (<= dX.u 500000.0)
     (log2
      (sqrt
       (fmax
        (fma (* t_5 dX.w) dX.w (* (* (pow (floor h) 2.0) dX.v) dX.v))
        (+ (+ (* t_3 t_3) (* t_1 t_1)) (* t_2 t_2)))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_4 t_4) (* t_6 t_6)) (* t_0 t_0))
        (* (* t_5 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(d) * dX_46_w;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = powf(floorf(d), 2.0f);
	float t_6 = floorf(h) * dX_46_v;
	float tmp;
	if (dX_46_u <= 500000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_5 * dX_46_w), dX_46_w, ((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v)), (((t_3 * t_3) + (t_1 * t_1)) + (t_2 * t_2)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_4 * t_4) + (t_6 * t_6)) + (t_0 * t_0)), ((t_5 * dY_46_w) * dY_46_w))));
	}
	return 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)
	t_0 = Float32(floor(d) * dX_46_w)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = floor(d) ^ Float32(2.0)
	t_6 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(500000.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_5 * dX_46_w), dX_46_w, Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)), Float32(Float32(Float32(t_3 * t_3) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_4 * t_4) + Float32(t_6 * t_6)) + Float32(t_0 * t_0)), Float32(Float32(t_5 * dY_46_w) * dY_46_w))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 5e5

   1. Initial program 62.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around 0

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

   5. Applied rewrites 58.8%

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

   if 5e5 < dX.u

   1. Initial program 63.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.w around inf

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

   5. Applied rewrites 57.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_6 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.w \leq 50000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot dX.u\right) \cdot dX.u, \left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right) + t\_6 \cdot t\_6\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_2 \cdot t\_2 + t\_4 \cdot t\_4\right) + t\_0 \cdot t\_0, \left(t\_5 \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (* (floor h) dY.v))
        (t_4 (* (floor h) dX.v))
        (t_5 (pow (floor w) 2.0))
        (t_6 (* (floor d) dY.w)))
   (if (<= dX.w 50000.0)
     (log2
      (sqrt
       (fmax
        (* (* t_5 dX.u) dX.u)
        (+ (+ (* t_1 t_1) (* t_3 t_3)) (* t_6 t_6)))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_2 t_2) (* t_4 t_4)) (* t_0 t_0))
        (* (* t_5 dY.u) dY.u)))))))

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(d) * dX_46_w;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = floorf(h) * dY_46_v;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = powf(floorf(w), 2.0f);
	float t_6 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_w <= 50000.0f) {
		tmp = log2f(sqrtf(fmaxf(((t_5 * dX_46_u) * dX_46_u), (((t_1 * t_1) + (t_3 * t_3)) + (t_6 * t_6)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_2 * t_2) + (t_4 * t_4)) + (t_0 * t_0)), ((t_5 * dY_46_u) * dY_46_u))));
	}
	return 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)
	t_0 = Float32(floor(d) * dX_46_w)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(h) * dY_46_v)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = floor(w) ^ Float32(2.0)
	t_6 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(50000.0))
		tmp = log2(sqrt(fmax(Float32(Float32(t_5 * dX_46_u) * dX_46_u), Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_6 * t_6)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)) + Float32(t_0 * t_0)), Float32(Float32(t_5 * dY_46_u) * dY_46_u))));
	end
	return 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)
	t_0 = floor(d) * dX_46_w;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = floor(h) * dY_46_v;
	t_4 = floor(h) * dX_46_v;
	t_5 = floor(w) ^ single(2.0);
	t_6 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_w <= single(50000.0))
		tmp = log2(sqrt(max(((t_5 * dX_46_u) * dX_46_u), (((t_1 * t_1) + (t_3 * t_3)) + (t_6 * t_6)))));
	else
		tmp = log2(sqrt(max((((t_2 * t_2) + (t_4 * t_4)) + (t_0 * t_0)), ((t_5 * dY_46_u) * dY_46_u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 5e4

   1. Initial program 67.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

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

   5. Applied rewrites 56.8%

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

   if 5e4 < dX.w

   1. Initial program 44.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around inf

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

   5. Applied rewrites 44.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u \leq 0.019999999552965164:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left(t\_1 + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0)) (t_1 (pow (* dY.v (floor h)) 2.0)))
   (if (<= dX.u 0.019999999552965164)
     (log2
      (sqrt
       (fmax
        t_0
        (+
         (pow (* dY.w (floor d)) 2.0)
         (+ t_1 (pow (* dY.u (floor w)) 2.0))))))
     (log2
      (sqrt
       (fmax
        (+ t_0 (+ (pow (* dX.v (floor h)) 2.0) (pow (* dX.u (floor w)) 2.0)))
        t_1))))))

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 = powf((dX_46_w * floorf(d)), 2.0f);
	float t_1 = powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dX_46_u <= 0.019999999552965164f) {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf((dY_46_w * floorf(d)), 2.0f) + (t_1 + powf((dY_46_u * floorf(w)), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 + (powf((dX_46_v * floorf(h)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f))), t_1)));
	}
	return 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)
	t_0 = Float32(dX_46_w * floor(d)) ^ Float32(2.0)
	t_1 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(0.019999999552965164))
		tmp = log2(sqrt(fmax(t_0, Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32(t_1 + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(Float32(t_0 + Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))), t_1)));
	end
	return 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)
	t_0 = (dX_46_w * floor(d)) ^ single(2.0);
	t_1 = (dY_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(0.019999999552965164))
		tmp = log2(sqrt(max(t_0, (((dY_46_w * floor(d)) ^ single(2.0)) + (t_1 + ((dY_46_u * floor(w)) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max((t_0 + (((dX_46_v * floor(h)) ^ single(2.0)) + ((dX_46_u * floor(w)) ^ single(2.0)))), t_1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.0199999996

   1. Initial program 64.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 55.1%

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

   if 0.0199999996 < dX.u

   1. Initial program 60.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around inf

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 55.6%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 180000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, t\_0 + \left({\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\right), t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0)))
   (if (<= dX.v 180000.0)
     (log2
      (sqrt
       (fmax
        (pow (* dX.w (floor d)) 2.0)
        (+
         t_0
         (+ (pow (* dY.v (floor h)) 2.0) (pow (* dY.u (floor w)) 2.0))))))
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor h) 2.0) dX.v) dX.v (pow (* (floor w) dX.u) 2.0))
        t_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) {
	float t_0 = powf((dY_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_v <= 180000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_w * floorf(d)), 2.0f), (t_0 + (powf((dY_46_v * floorf(h)), 2.0f) + powf((dY_46_u * floorf(w)), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(h), 2.0f) * dX_46_v), dX_46_v, powf((floorf(w) * dX_46_u), 2.0f)), t_0)));
	}
	return 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)
	t_0 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(180000.0))
		tmp = log2(sqrt(fmax((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), Float32(t_0 + Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32((floor(h) ^ Float32(2.0)) * dX_46_v), dX_46_v, (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.8e5

   1. Initial program 65.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 54.5%

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

   if 1.8e5 < dX.v

   1. Initial program 53.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 27.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 27.3%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.w around 0

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

   13. Applied rewrites 50.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\right)}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(t\_0 \cdot dY.u, dY.u, \left(t\_1 \cdot dY.v\right) \cdot dY.v\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1 \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (pow (floor h) 2.0)))
   (if (<= dY.w 200.0)
     (log2
      (sqrt
       (fmax
        (* (* t_0 dX.u) dX.u)
        (fma (* t_0 dY.u) dY.u (* (* t_1 dY.v) dY.v)))))
     (log2
      (sqrt
       (fmax
        (fma (* t_1 dX.v) dX.v (pow (* (floor d) dX.w) 2.0))
        (* (* (pow (floor d) 2.0) 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 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(h), 2.0f);
	float tmp;
	if (dY_46_w <= 200.0f) {
		tmp = log2f(sqrtf(fmaxf(((t_0 * dX_46_u) * dX_46_u), fmaf((t_0 * dY_46_u), dY_46_u, ((t_1 * dY_46_v) * dY_46_v)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_1 * dX_46_v), dX_46_v, powf((floorf(d) * dX_46_w), 2.0f)), ((powf(floorf(d), 2.0f) * dY_46_w) * dY_46_w))));
	}
	return 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)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(h) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(200.0))
		tmp = log2(sqrt(fmax(Float32(Float32(t_0 * dX_46_u) * dX_46_u), fma(Float32(t_0 * dY_46_u), dY_46_u, Float32(Float32(t_1 * dY_46_v) * dY_46_v)))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_1 * dX_46_v), dX_46_v, (Float32(floor(d) * dX_46_w) ^ Float32(2.0))), Float32(Float32((floor(d) ^ Float32(2.0)) * dY_46_w) * dY_46_w))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 200

   1. Initial program 66.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in dY.w around 0

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

   8. Applied rewrites 49.0%

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

   if 200 < dY.w

   1. Initial program 50.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

   9. Taylor expanded in dX.u around 0

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

   11. Applied rewrites 46.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right)}, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u \leq 1:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.w, dX.w, {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_1 \cdot dX.u\right) \cdot dX.u, \mathsf{fma}\left(t\_1 \cdot dY.u, dY.u, \left(t\_0 \cdot dY.w\right) \cdot dY.w\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)) (t_1 (pow (floor w) 2.0)))
   (if (<= dY.u 1.0)
     (log2
      (sqrt
       (fmax
        (fma (* t_0 dX.w) dX.w (pow (* (floor w) dX.u) 2.0))
        (pow (* dY.w (floor d)) 2.0))))
     (log2
      (sqrt
       (fmax
        (* (* t_1 dX.u) dX.u)
        (fma (* t_1 dY.u) dY.u (* (* t_0 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 = powf(floorf(d), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_u <= 1.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_w), dX_46_w, powf((floorf(w) * dX_46_u), 2.0f)), powf((dY_46_w * floorf(d)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((t_1 * dX_46_u) * dX_46_u), fmaf((t_1 * dY_46_u), dY_46_u, ((t_0 * dY_46_w) * dY_46_w)))));
	}
	return 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)
	t_0 = floor(d) ^ Float32(2.0)
	t_1 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(1.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_w), dX_46_w, (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(t_1 * dX_46_u) * dX_46_u), fma(Float32(t_1 * dY_46_u), dY_46_u, Float32(Float32(t_0 * dY_46_w) * dY_46_w)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1

   1. Initial program 64.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 34.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 34.9%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around 0

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

   13. Applied rewrites 44.7%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w, dX.w, {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\right)}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   if 1 < dY.u

   1. Initial program 57.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 49.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.u, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, dX.u, {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left(t\_0 \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right), \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0)))
   (if (<= dY.w 200.0)
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor w) 2.0) dX.u) dX.u (pow (* dX.v (floor h)) 2.0))
        (* (* t_0 dY.v) dY.v))))
     (log2
      (sqrt
       (fmax
        (fma (* t_0 dX.v) dX.v (pow (* (floor d) dX.w) 2.0))
        (* (* (pow (floor d) 2.0) 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 = powf(floorf(h), 2.0f);
	float tmp;
	if (dY_46_w <= 200.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(w), 2.0f) * dX_46_u), dX_46_u, powf((dX_46_v * floorf(h)), 2.0f)), ((t_0 * dY_46_v) * dY_46_v))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_v), dX_46_v, powf((floorf(d) * dX_46_w), 2.0f)), ((powf(floorf(d), 2.0f) * dY_46_w) * dY_46_w))));
	}
	return 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)
	t_0 = floor(h) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(200.0))
		tmp = log2(sqrt(fmax(fma(Float32((floor(w) ^ Float32(2.0)) * dX_46_u), dX_46_u, (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), Float32(Float32(t_0 * dY_46_v) * dY_46_v))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_v), dX_46_v, (Float32(floor(d) * dX_46_w) ^ Float32(2.0))), Float32(Float32((floor(d) ^ Float32(2.0)) * dY_46_w) * dY_46_w))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 200

   1. Initial program 66.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around inf

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in dX.w around 0

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

   8. Applied rewrites 45.5%

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

   10. Applied rewrites 45.5%

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

   if 200 < dY.w

   1. Initial program 50.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]

   9. Taylor expanded in dX.u around 0

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

   11. Applied rewrites 46.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}\right)}, \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ \mathbf{if}\;dY.u \leq 60:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w, dX.w, t\_0\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dX.u) 2.0)))
   (if (<= dY.u 60.0)
     (log2
      (sqrt
       (fmax
        (fma (* (pow (floor d) 2.0) dX.w) dX.w t_0)
        (pow (* dY.w (floor d)) 2.0))))
     (log2 (sqrt (fmax t_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) {
	float t_0 = powf((floorf(w) * dX_46_u), 2.0f);
	float tmp;
	if (dY_46_u <= 60.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((powf(floorf(d), 2.0f) * dX_46_w), dX_46_w, t_0), powf((dY_46_w * floorf(d)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((dY_46_u * floorf(w)), 2.0f))));
	}
	return 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)
	t_0 = Float32(floor(w) * dX_46_u) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(60.0))
		tmp = log2(sqrt(fmax(fma(Float32((floor(d) ^ Float32(2.0)) * dX_46_w), dX_46_w, t_0), (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))));
	else
		tmp = log2(sqrt(fmax(t_0, (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 60

   1. Initial program 64.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 35.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 35.1%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around 0

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

   13. Applied rewrites 45.0%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w, dX.w, {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\right)}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   if 60 < dY.u

   1. Initial program 54.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 49.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.w \leq 120000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.w 120000.0)
   (log2
    (sqrt (fmax (pow (* (floor w) dX.u) 2.0) (pow (* dY.u (floor w)) 2.0))))
   (log2
    (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* dY.w (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) {
	float tmp;
	if (dY_46_w <= 120000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), powf((dY_46_u * floorf(w)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf((dY_46_w * floorf(d)), 2.0f))));
	}
	return 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 = Float32(0.0)
	if (dY_46_w <= Float32(120000.0))
		tmp = log2(sqrt(fmax((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))));
	else
		tmp = log2(sqrt(fmax((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (Float32(dY_46_w * floor(d)) ^ Float32(2.0)))));
	end
	return 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 = single(0.0);
	if (dY_46_w <= single(120000.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), ((dY_46_u * floor(w)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((dY_46_w * floor(d)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 1.2e5

   1. Initial program 66.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 39.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 39.1%

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

   if 1.2e5 < dY.w

   1. Initial program 46.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 44.2%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 35.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* dY.w (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(d) * dX_46_w), 2.0f), powf((dY_46_w * 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(fmax((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (Float32(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(d) * dX_46_w) ^ single(2.0)), ((dY_46_w * floor(d)) ^ single(2.0)))));
end

Derivation

1. Initial program 63.1%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
2. Add Preprocessing

3. Taylor expanded in dX.w around inf

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in dY.w around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 34.2%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 34.2%

    \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(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)))))))