UniformSampleCone 2

Percentage Accurate: 99.0% → 99.1%

Time: 9.3s

Alternatives: 14

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\\ \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot t\_0, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right), xi \cdot t\_0, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* (* (- ux 1.0) (* maxCos ux)) (- 1.0 ux))
           (* maxCos ux)
           1.0))))
   (fma
    (* (sin (* PI (+ uy uy))) t_0)
    yi
    (fma
     (sin (fma (- PI) (+ uy uy) (* PI 0.5)))
     (* xi t_0)
     (* zi (* (* maxCos (- 1.0 ux)) ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((((ux - 1.0f) * (maxCos * ux)) * (1.0f - ux)), (maxCos * ux), 1.0f));
	return fmaf((sinf((((float) M_PI) * (uy + uy))) * t_0), yi, fmaf(sinf(fmaf(-((float) M_PI), (uy + uy), (((float) M_PI) * 0.5f))), (xi * t_0), (zi * ((maxCos * (1.0f - ux)) * ux))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(Float32(Float32(ux - Float32(1.0)) * Float32(maxCos * ux)) * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(1.0)))
	return fma(Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * t_0), yi, fma(sin(fma(Float32(-Float32(pi)), Float32(uy + uy), Float32(Float32(pi) * Float32(0.5)))), Float32(xi * t_0), Float32(zi * Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\cos \left(\pi \cdot \left(uy + uy\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(uy + uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   6. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy + uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   7. count-2 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(2 \cdot uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(uy \cdot 2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\pi\right), uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   12. lower-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\color{blue}{-\pi}, uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy \cdot 2}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{2 \cdot uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   15. count-2 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy + uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   16. lift-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy + uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   17. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \frac{\color{blue}{\pi}}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   18. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot \color{blue}{\frac{1}{2}}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   20. lower-*.f32 99.1

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \color{blue}{\pi \cdot 0.5}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ t_1 := \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\\ \mathsf{fma}\left(\sin t\_0 \cdot t\_1, yi, \mathsf{fma}\left(\cos t\_0, xi \cdot t\_1, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy)))
        (t_1
         (sqrt
          (fma
           (* (* (- ux 1.0) (* maxCos ux)) (- 1.0 ux))
           (* maxCos ux)
           1.0))))
   (fma
    (* (sin t_0) t_1)
    yi
    (fma (cos t_0) (* xi t_1) (* zi (* (* maxCos (- 1.0 ux)) ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float t_1 = sqrtf(fmaf((((ux - 1.0f) * (maxCos * ux)) * (1.0f - ux)), (maxCos * ux), 1.0f));
	return fmaf((sinf(t_0) * t_1), yi, fmaf(cosf(t_0), (xi * t_1), (zi * ((maxCos * (1.0f - ux)) * ux))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	t_1 = sqrt(fma(Float32(Float32(Float32(ux - Float32(1.0)) * Float32(maxCos * ux)) * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(1.0)))
	return fma(Float32(sin(t_0) * t_1), yi, fma(cos(t_0), Float32(xi * t_1), Float32(zi * Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]
4. Add Preprocessing

Alternative 3: 98.8% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right), xi \cdot 1, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (* (sin (* PI (+ uy uy))) 1.0)
  yi
  (fma
   (sin (fma (- PI) (+ uy uy) (* PI 0.5)))
   (* xi 1.0)
   (* zi (* (* maxCos (- 1.0 ux)) ux)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((sinf((((float) M_PI) * (uy + uy))) * 1.0f), yi, fmaf(sinf(fmaf(-((float) M_PI), (uy + uy), (((float) M_PI) * 0.5f))), (xi * 1.0f), (zi * ((maxCos * (1.0f - ux)) * ux))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * Float32(1.0)), yi, fma(sin(fma(Float32(-Float32(pi)), Float32(uy + uy), Float32(Float32(pi) * Float32(0.5)))), Float32(xi * Float32(1.0)), Float32(zi * Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\cos \left(\pi \cdot \left(uy + uy\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(uy + uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   6. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy + uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   7. count-2 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(2 \cdot uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(uy \cdot 2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\pi\right), uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   12. lower-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\color{blue}{-\pi}, uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy \cdot 2}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{2 \cdot uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   15. count-2 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy + uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   16. lift-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy + uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   17. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \frac{\color{blue}{\pi}}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   18. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot \color{blue}{\frac{1}{2}}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   20. lower-*.f32 99.1

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \color{blue}{\pi \cdot 0.5}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

6. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot \frac{1}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

9. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot \frac{1}{2}\right)\right), xi \cdot \color{blue}{1}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right), xi \cdot \color{blue}{1}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot uy\\ \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \mathsf{fma}\left(1 \cdot yi, \sin t\_0, \left(1 \cdot xi\right) \cdot \cos t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ PI PI) uy)))
   (fma
    (- 1.0 ux)
    (* (* zi ux) maxCos)
    (fma (* 1.0 yi) (sin t_0) (* (* 1.0 xi) (cos t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) + ((float) M_PI)) * uy;
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), fmaf((1.0f * yi), sinf(t_0), ((1.0f * xi) * cosf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) + Float32(pi)) * uy)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), fma(Float32(Float32(1.0) * yi), sin(t_0), Float32(Float32(Float32(1.0) * xi) * cos(t_0))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{1}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Step-by-step derivation

4. Applied rewrites 98.8%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{1}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{1}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
6. Step-by-step derivation

7. Applied rewrites 98.7%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{1}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi + \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot yi\right)} \]

9. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \mathsf{fma}\left(1 \cdot yi, \sin \left(\left(\pi + \pi\right) \cdot uy\right), \left(1 \cdot xi\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot uy\right)\right)\right)} \]
10. Add Preprocessing

Alternative 5: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma maxCos (* ux (* zi (- 1.0 ux))) (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma maxCos (* ux zi) (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * zi), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * zi), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 1.7× speedup

Math

\[\mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma xi (sin (fma -2.0 (* uy PI) (* 0.5 PI))) (* yi (sin (* 2.0 (* uy PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(xi, sinf(fmaf(-2.0f, (uy * ((float) M_PI)), (0.5f * ((float) M_PI)))), (yi * sinf((2.0f * (uy * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(xi, sin(fma(Float32(-2.0), Float32(uy * Float32(pi)), Float32(Float32(0.5) * Float32(pi)))), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\cos \left(\pi \cdot \left(uy + uy\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(uy + uy\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(uy + uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   6. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy + uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   7. count-2 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(2 \cdot uy\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\left(\mathsf{neg}\left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(uy \cdot 2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\pi\right), uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   12. lower-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\color{blue}{-\pi}, uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy \cdot 2}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{2 \cdot uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   15. count-2 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy + uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   16. lift-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy + uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   17. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \frac{\color{blue}{\pi}}{2}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   18. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot \color{blue}{\frac{1}{2}}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

   20. lower-*.f32 99.1

       \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \color{blue}{\pi \cdot 0.5}\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right)}, xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right) \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{xi \cdot \sin \left(-2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\sin \left(-2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \sin \left(-2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \mathsf{PI}\left(\right), \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \mathsf{PI}\left(\right), \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, \frac{1}{2} \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

8. Applied rewrites 90.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
9. Add Preprocessing

Alternative 8: 90.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))) (fma xi (cos t_0) (* yi (sin t_0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(xi, cos(t_0), Float32(yi * sin(t_0)))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lower-PI.f32 90.2

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

4. Applied rewrites 90.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ t_1 := \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\\ t_2 := 1 \cdot \left(\left(xi \cdot t\_1\right) \cdot \cos t\_0\right)\\ \mathbf{if}\;xi \leq -1.400000027358074 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_0 \cdot t\_1, yi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy)))
        (t_1
         (sqrt
          (fma (* (* (- ux 1.0) (* maxCos ux)) (- 1.0 ux)) (* maxCos ux) 1.0)))
        (t_2 (* 1.0 (* (* xi t_1) (cos t_0)))))
   (if (<= xi -1.400000027358074e-24)
     t_2
     (if (<= xi 1.9999999920083944e-12)
       (fma (* (sin t_0) t_1) yi (* maxCos (* ux (* zi (- 1.0 ux)))))
       t_2))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float t_1 = sqrtf(fmaf((((ux - 1.0f) * (maxCos * ux)) * (1.0f - ux)), (maxCos * ux), 1.0f));
	float t_2 = 1.0f * ((xi * t_1) * cosf(t_0));
	float tmp;
	if (xi <= -1.400000027358074e-24f) {
		tmp = t_2;
	} else if (xi <= 1.9999999920083944e-12f) {
		tmp = fmaf((sinf(t_0) * t_1), yi, (maxCos * (ux * (zi * (1.0f - ux)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	t_1 = sqrt(fma(Float32(Float32(Float32(ux - Float32(1.0)) * Float32(maxCos * ux)) * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(1.0)))
	t_2 = Float32(Float32(1.0) * Float32(Float32(xi * t_1) * cos(t_0)))
	tmp = Float32(0.0)
	if (xi <= Float32(-1.400000027358074e-24))
		tmp = t_2;
	elseif (xi <= Float32(1.9999999920083944e-12))
		tmp = fma(Float32(sin(t_0) * t_1), yi, Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = t_2;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if xi < -1.40000003e-24 or 1.99999999e-12 < xi

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(1 + \frac{\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)}{\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)}\right) \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)} \]

   4. Taylor expanded in xi around inf

      \[\leadsto \color{blue}{1} \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \color{blue}{1} \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right) \]

   if -1.40000003e-24 < xi < 1.99999999e-12

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]

   4. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right)\right) \]

      4. lower--.f32 43.9

         \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right)\right) \]

   6. Applied rewrites 43.9%

      \[\leadsto \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, yi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0001294999965466559:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0001294999965466559)
   (+ xi (* maxCos (* ux (* zi (- 1.0 ux)))))
   (*
    1.0
    (*
     (*
      xi
      (sqrt
       (fma (* (* (- ux 1.0) (* maxCos ux)) (- 1.0 ux)) (* maxCos ux) 1.0)))
     (cos (* PI (+ uy uy)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0001294999965466559f) {
		tmp = xi + (maxCos * (ux * (zi * (1.0f - ux))));
	} else {
		tmp = 1.0f * ((xi * sqrtf(fmaf((((ux - 1.0f) * (maxCos * ux)) * (1.0f - ux)), (maxCos * ux), 1.0f))) * cosf((((float) M_PI) * (uy + uy))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0001294999965466559))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(Float32(1.0) * Float32(Float32(xi * sqrt(fma(Float32(Float32(Float32(ux - Float32(1.0)) * Float32(maxCos * ux)) * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(1.0)))) * cos(Float32(Float32(pi) * Float32(uy + uy)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.29499997e-4

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right) \]

      5. lower--.f32 52.4

         \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

   7. Applied rewrites 52.4%

      \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   if 1.29499997e-4 < uy

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(1 + \frac{\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)}{\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)}\right) \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)} \]

   4. Taylor expanded in xi around inf

      \[\leadsto \color{blue}{1} \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.5%

      \[\leadsto \color{blue}{1} \cdot \left(\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right), maxCos \cdot ux, 1\right)}\right) \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.4% accurate, 10.4× speedup

Math

\[xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (* maxCos (* ux (* zi (- 1.0 ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * (zi * (1.0f - ux))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (maxcos * (ux * (zi * (1.0e0 - ux))))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right) \]

   5. lower--.f32 52.4

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

7. Applied rewrites 52.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
8. Add Preprocessing

Alternative 12: 50.4% accurate, 17.3× speedup

Math

\[\mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (fma (* maxCos ux) zi xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * ux), zi, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), zi, xi)
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 50.4

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 50.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. +-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   3. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   4. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   5. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   6. lift-*.f32 N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   7. lower-fma.f32 50.4

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

9. Applied rewrites 50.4%

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]
10. Add Preprocessing

Alternative 13: 12.1% accurate, 22.3× speedup

Math

\[\left(ux \cdot maxCos\right) \cdot zi \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* (* ux maxCos) zi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (ux * maxCos) * zi;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (ux * maxcos) * zi
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(ux * maxCos) * zi)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (ux * maxCos) * zi;
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 50.4

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 50.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 12.1

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.1%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

    2. lift-*.f32 N/A

       \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

    3. associate-*r* N/A

       \[\leadsto \left(maxCos \cdot ux\right) \cdot zi \]

    4. lift-*.f32 N/A

       \[\leadsto \left(maxCos \cdot ux\right) \cdot zi \]

    5. lower-*.f32 12.1

       \[\leadsto \left(maxCos \cdot ux\right) \cdot zi \]

    6. lift-*.f32 N/A

       \[\leadsto \left(maxCos \cdot ux\right) \cdot zi \]

    7. *-commutative N/A

       \[\leadsto \left(ux \cdot maxCos\right) \cdot zi \]

    8. lower-*.f32 12.1

       \[\leadsto \left(ux \cdot maxCos\right) \cdot zi \]

12. Applied rewrites 12.1%

    \[\leadsto \left(ux \cdot maxCos\right) \cdot zi \]
13. Add Preprocessing

Alternative 14: 12.1% accurate, 22.3× speedup

Math

\[maxCos \cdot \left(ux \cdot zi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 50.4

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 50.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 12.1

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.1%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))