UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 8.6s

Alternatives: 19

Speedup: 1.4×

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

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

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: 98.9% accurate, 1.0× speedup

Math

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

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: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)) (t_1 (sin (acos t_0))))
   (+
    (fma
     (* (sin (* PI (* 2.0 uy))) t_1)
     yi
     (* (sin (fma PI (* 2.0 uy) (/ PI 2.0))) (* t_1 xi)))
    (* 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 = sinf(acosf(t_0));
	return fmaf((sinf((((float) M_PI) * (2.0f * uy))) * t_1), yi, (sinf(fmaf(((float) M_PI), (2.0f * uy), (((float) M_PI) / 2.0f))) * (t_1 * xi))) + (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 = sin(acos(t_0))
	return Float32(fma(Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * t_1), yi, Float32(sin(fma(Float32(pi), Float32(Float32(2.0) * uy), Float32(Float32(pi) / Float32(2.0)))) * Float32(t_1 * xi))) + Float32(t_0 * zi))
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

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

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

   3. lower-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.9

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

5. Applied rewrites 98.9%

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

Alternative 2: 98.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* PI (* 2.0 uy)))
        (t_2 (sin (acos t_0))))
   (+ (fma (* (sin t_1) t_2) yi (* (cos t_1) (* t_2 xi))) (* 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 = ((float) M_PI) * (2.0f * uy);
	float t_2 = sinf(acosf(t_0));
	return fmaf((sinf(t_1) * t_2), yi, (cosf(t_1) * (t_2 * xi))) + (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 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	t_2 = sin(acos(t_0))
	return Float32(fma(Float32(sin(t_1) * t_2), yi, Float32(cos(t_1) * Float32(t_2 * xi))) + Float32(t_0 * zi))
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

Alternative 3: 98.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* PI (* 2.0 uy)))
        (t_2 (sin (acos t_0))))
   (fma (* (cos t_1) t_2) xi (fma (sin t_1) (* t_2 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 = ((float) M_PI) * (2.0f * uy);
	float t_2 = sinf(acosf(t_0));
	return fmaf((cosf(t_1) * t_2), xi, fmaf(sinf(t_1), (t_2 * 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 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	t_2 = sin(acos(t_0))
	return fma(Float32(cos(t_1) * t_2), xi, fma(sin(t_1), Float32(t_2 * yi), Float32(t_0 * zi)))
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 99.0%

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

Alternative 4: 98.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around inf

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

4. Applied rewrites 98.7%

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

Alternative 5: 98.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \mathsf{fma}\left(\sin \left(\left(-\left(\pi \cdot uy\right) \cdot 2\right) + \frac{\pi}{2}\right), xi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

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

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

   3. lower-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-sin.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-/.f32 N/A

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

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

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

   8. cos-neg-rev N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

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

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

   12. lower-sin.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-+.f32 N/A

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

8. Applied rewrites 98.8%

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

Alternative 6: 98.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

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

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

   3. lower-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-sin.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-/.f32 N/A

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

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

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

8. Applied rewrites 98.8%

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

Alternative 7: 98.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

Alternative 8: 97.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;uy \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \mathsf{fma}\left(t\_1, xi, \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(t\_1, xi, \sin t\_0 \cdot yi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))) (t_1 (cos t_0)))
   (if (<= uy 9.999999747378752e-5)
     (fma
      (* maxCos ux)
      (* (- 1.0 ux) zi)
      (fma t_1 xi (* (* 2.0 (* uy PI)) yi)))
     (fma (* maxCos ux) zi (fma t_1 xi (* (sin t_0) yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float t_1 = cosf(t_0);
	float tmp;
	if (uy <= 9.999999747378752e-5f) {
		tmp = fmaf((maxCos * ux), ((1.0f - ux) * zi), fmaf(t_1, xi, ((2.0f * (uy * ((float) M_PI))) * yi)));
	} else {
		tmp = fmaf((maxCos * ux), zi, fmaf(t_1, xi, (sinf(t_0) * yi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	t_1 = cos(t_0)
	tmp = Float32(0.0)
	if (uy <= Float32(9.999999747378752e-5))
		tmp = fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), fma(t_1, xi, Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi)));
	else
		tmp = fma(Float32(maxCos * ux), zi, fma(t_1, xi, Float32(sin(t_0) * yi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.99999975e-5

   1. Initial program 99.3%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right), xi, \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 9.99999975e-5 < uy

   1. Initial program 98.3%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

   4. Applied rewrites 95.4%

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

Alternative 9: 97.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;uy \leq 9.600000339560211 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= uy 9.600000339560211e-5)
     (fma (* maxCos ux) (* (- 1.0 ux) zi) (+ xi (* 2.0 (* uy (* yi PI)))))
     (fma (* maxCos ux) zi (fma (cos t_0) xi (* (sin t_0) yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if (uy <= 9.600000339560211e-5f) {
		tmp = fmaf((maxCos * ux), ((1.0f - ux) * zi), (xi + (2.0f * (uy * (yi * ((float) M_PI))))));
	} else {
		tmp = fmaf((maxCos * ux), zi, fmaf(cosf(t_0), xi, (sinf(t_0) * yi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (uy <= Float32(9.600000339560211e-5))
		tmp = fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))));
	else
		tmp = fma(Float32(maxCos * ux), zi, fma(cos(t_0), xi, Float32(sin(t_0) * yi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.60000034e-5

   1. Initial program 99.3%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 9.60000034e-5 < uy

   1. Initial program 98.3%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

   4. Applied rewrites 95.5%

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

Alternative 10: 95.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot uy\right) \cdot 2\\ \mathbf{if}\;uy \leq 9.600000339560211 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \frac{\sin t\_0}{xi}, \cos t\_0\right) \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* PI uy) 2.0)))
   (if (<= uy 9.600000339560211e-5)
     (fma (* maxCos ux) (* (- 1.0 ux) zi) (+ xi (* 2.0 (* uy (* yi PI)))))
     (* (fma yi (/ (sin t_0) xi) (cos t_0)) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) * uy) * 2.0f;
	float tmp;
	if (uy <= 9.600000339560211e-5f) {
		tmp = fmaf((maxCos * ux), ((1.0f - ux) * zi), (xi + (2.0f * (uy * (yi * ((float) M_PI))))));
	} else {
		tmp = fmaf(yi, (sinf(t_0) / xi), cosf(t_0)) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) * uy) * Float32(2.0))
	tmp = Float32(0.0)
	if (uy <= Float32(9.600000339560211e-5))
		tmp = fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))));
	else
		tmp = Float32(fma(yi, Float32(sin(t_0) / xi), cos(t_0)) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.60000034e-5

   1. Initial program 99.3%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 9.60000034e-5 < uy

   1. Initial program 98.3%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in xi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left(yi, \frac{\sin \left(\left(\pi \cdot uy\right) \cdot 2\right)}{xi}, \cos \left(\left(\pi \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{xi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 95.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot uy\right) \cdot 2\\ \mathbf{if}\;uy \leq 9.600000339560211 \cdot 10^{-5}:\\ \;\;\;\;xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \frac{\sin t\_0}{xi}, \cos t\_0\right) \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* PI uy) 2.0)))
   (if (<= uy 9.600000339560211e-5)
     (+ xi (fma 2.0 (* uy (* yi PI)) (* maxCos (* ux (* zi (- 1.0 ux))))))
     (* (fma yi (/ (sin t_0) xi) (cos t_0)) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) * uy) * 2.0f;
	float tmp;
	if (uy <= 9.600000339560211e-5f) {
		tmp = xi + fmaf(2.0f, (uy * (yi * ((float) M_PI))), (maxCos * (ux * (zi * (1.0f - ux)))));
	} else {
		tmp = fmaf(yi, (sinf(t_0) / xi), cosf(t_0)) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) * uy) * Float32(2.0))
	tmp = Float32(0.0)
	if (uy <= Float32(9.600000339560211e-5))
		tmp = Float32(xi + fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))));
	else
		tmp = Float32(fma(yi, Float32(sin(t_0) / xi), cos(t_0)) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.60000034e-5

   1. Initial program 99.3%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift--.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 9.60000034e-5 < uy

   1. Initial program 98.3%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in xi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left(yi, \frac{\sin \left(\left(\pi \cdot uy\right) \cdot 2\right)}{xi}, \cos \left(\left(\pi \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{xi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 93.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot uy\right) \cdot 2\\ \mathbf{if}\;uy \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\left(yi \cdot \pi\right) \cdot uy, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \frac{\sin t\_0}{xi}, \cos t\_0\right) \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* PI uy) 2.0)))
   (if (<= uy 3.9999998989515007e-5)
     (fma (* maxCos ux) zi (fma (* (* yi PI) uy) 2.0 xi))
     (* (fma yi (/ (sin t_0) xi) (cos t_0)) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) * uy) * 2.0f;
	float tmp;
	if (uy <= 3.9999998989515007e-5f) {
		tmp = fmaf((maxCos * ux), zi, fmaf(((yi * ((float) M_PI)) * uy), 2.0f, xi));
	} else {
		tmp = fmaf(yi, (sinf(t_0) / xi), cosf(t_0)) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) * uy) * Float32(2.0))
	tmp = Float32(0.0)
	if (uy <= Float32(3.9999998989515007e-5))
		tmp = fma(Float32(maxCos * ux), zi, fma(Float32(Float32(yi * Float32(pi)) * uy), Float32(2.0), xi));
	else
		tmp = Float32(fma(yi, Float32(sin(t_0) / xi), cos(t_0)) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-5

   1. Initial program 99.4%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 95.9

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

   10. Applied rewrites 95.9%

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

   if 3.9999999e-5 < uy

   1. Initial program 98.4%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in xi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 90.2%

      \[\leadsto \mathsf{fma}\left(yi, \frac{\sin \left(\left(\pi \cdot uy\right) \cdot 2\right)}{xi}, \cos \left(\left(\pi \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{xi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 93.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;uy \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\left(yi \cdot \pi\right) \cdot uy, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= uy 3.9999998989515007e-5)
     (fma (* maxCos ux) zi (fma (* (* yi PI) uy) 2.0 xi))
     (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if (uy <= 3.9999998989515007e-5f) {
		tmp = fmaf((maxCos * ux), zi, fmaf(((yi * ((float) M_PI)) * uy), 2.0f, xi));
	} else {
		tmp = fmaf(cosf(t_0), xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (uy <= Float32(3.9999998989515007e-5))
		tmp = fma(Float32(maxCos * ux), zi, fma(Float32(Float32(yi * Float32(pi)) * uy), Float32(2.0), xi));
	else
		tmp = fma(cos(t_0), xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-5

   1. Initial program 99.4%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 95.9

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

   10. Applied rewrites 95.9%

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

   if 3.9999999e-5 < uy

   1. Initial program 98.4%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right), xi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 85.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\left(yi \cdot \pi\right) \cdot uy, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right), xi, \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 3.9999998989515007e-5)
   (fma (* maxCos ux) zi (fma (* (* yi PI) uy) 2.0 xi))
   (fma (cos (* PI (* 2.0 uy))) xi (* (* (* PI uy) 2.0) yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 3.9999998989515007e-5f) {
		tmp = fmaf((maxCos * ux), zi, fmaf(((yi * ((float) M_PI)) * uy), 2.0f, xi));
	} else {
		tmp = fmaf(cosf((((float) M_PI) * (2.0f * uy))), xi, (((((float) M_PI) * uy) * 2.0f) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(3.9999998989515007e-5))
		tmp = fma(Float32(maxCos * ux), zi, fma(Float32(Float32(yi * Float32(pi)) * uy), Float32(2.0), xi));
	else
		tmp = fma(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))), xi, Float32(Float32(Float32(Float32(pi) * uy) * Float32(2.0)) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-5

   1. Initial program 99.4%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 95.9

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

   10. Applied rewrites 95.9%

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

   if 3.9999999e-5 < uy

   1. Initial program 98.4%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right), xi, \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 72.9

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

   7. Applied rewrites 72.9%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right), xi, \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot yi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\left(yi \cdot \pi\right) \cdot uy, 2, xi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 79.3

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

10. Applied rewrites 79.3%

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

Alternative 16: 77.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.017999999225139618:\\ \;\;\;\;\mathsf{fma}\left(\left(yi \cdot \pi\right) \cdot uy, 2, xi\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.017999999225139618)
   (fma (* (* yi PI) uy) 2.0 xi)
   (* xi (cos (* 2.0 (* uy PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.017999999225139618f) {
		tmp = fmaf(((yi * ((float) M_PI)) * uy), 2.0f, xi);
	} else {
		tmp = xi * cosf((2.0f * (uy * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.017999999225139618))
		tmp = fma(Float32(Float32(yi * Float32(pi)) * uy), Float32(2.0), xi);
	else
		tmp = Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0179999992

   1. Initial program 99.2%

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

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

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

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi \]

      2. *-commutative N/A

         \[\leadsto \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 + xi \]

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 83.9

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

   7. Applied rewrites 83.9%

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

   if 0.0179999992 < uy

   1. Initial program 97.5%

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

   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. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 45.2

         \[\leadsto xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 45.2%

      \[\leadsto xi \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 74.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(yi \cdot \pi\right) \cdot uy, 2, xi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in uy around 0

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

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi \]

   2. *-commutative N/A

      \[\leadsto \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 + xi \]

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 74.3

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

7. Applied rewrites 74.3%

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

Alternative 18: 50.2% accurate, 10.8× speedup

Math

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

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 98.9%

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

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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]
9. Step-by-step derivation

10. Applied rewrites 50.2%

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

Alternative 19: 45.9% accurate, 65.0× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

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
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in xi around inf

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

   1. lower-*.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 53.1

      \[\leadsto xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

7. Applied rewrites 53.1%

   \[\leadsto xi \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto xi \]
9. Step-by-step derivation

   1. cos-2 45.9

      \[\leadsto xi \]

10. Applied rewrites 45.9%

    \[\leadsto xi \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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)))