UniformSampleCone 2

Percentage Accurate: 98.9% → 98.8%

Time: 10.3s

Alternatives: 22

Speedup: N/A×

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: 98.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (fma (* maxCos ux) (- zi (* 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), (zi - (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(zi - Float32(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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.4× 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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

Alternative 3: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in maxCos around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-fma.f32 N/A

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

8. Applied rewrites 98.9%

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

Alternative 4: 95.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (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);
	return fmaf((maxCos * 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), 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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

Alternative 5: 95.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;uy \leq 0.0015999999595806003:\\ \;\;\;\;\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\right) \cdot xi + \left(t\_1 \cdot 1\right) \cdot yi\right) + t\_0 \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_1, xi, \sin t\_1 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)) (t_1 (* PI (* 2.0 uy))))
   (if (<= uy 0.0015999999595806003)
     (+
      (+
       (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0)))) xi)
       (* (* t_1 1.0) yi))
      (* t_0 zi))
     (fma (cos t_1) xi (* (sin t_1) yi)))))

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 tmp;
	if (uy <= 0.0015999999595806003f) {
		tmp = (((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) * xi) + ((t_1 * 1.0f) * yi)) + (t_0 * zi);
	} else {
		tmp = fmaf(cosf(t_1), xi, (sinf(t_1) * yi));
	}
	return tmp;
}

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))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0015999999595806003))
		tmp = Float32(Float32(Float32(Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) * xi) + Float32(Float32(t_1 * Float32(1.0)) * yi)) + Float32(t_0 * zi));
	else
		tmp = fma(cos(t_1), xi, Float32(sin(t_1) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00159999996

   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. Add Preprocessing

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 98.9

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.9%

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

   if 0.00159999996 < uy

   1. Initial program 97.8%

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

   3. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   4. 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

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

   5. Applied rewrites 92.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0015999999595806003:\\ \;\;\;\;\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot 1\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right), xi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)))
   (+
    (+
     (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0)))) xi)
     (* (* (* PI (* 2.0 uy)) 1.0) 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;
	return (((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) * xi) + (((((float) M_PI) * (2.0f * uy)) * 1.0f) * yi)) + (t_0 * zi);
}

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 89.9

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 89.9

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

5. Applied rewrites 89.9%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 89.9%

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

9. Final simplification 89.9%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot 1\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Add Preprocessing

Alternative 7: 90.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)))
   (+
    (+
     (* (cos (* 2.0 (* uy PI))) xi)
     (* (* (* PI (* 2.0 uy)) (sqrt (- 1.0 (* t_0 t_0)))) 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;
	return ((cosf((2.0f * (uy * ((float) M_PI)))) * xi) + (((((float) M_PI) * (2.0f * uy)) * sqrtf((1.0f - (t_0 * t_0)))) * yi)) + (t_0 * zi);
}

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 89.9

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 89.9

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

5. Applied rewrites 89.9%

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

6. Taylor expanded in ux around 0

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

   1. lower-cos.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 89.8

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

8. Applied rewrites 89.8%

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

9. Final simplification 89.8%

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + \left(\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Add Preprocessing

Alternative 8: 87.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), zi, fma(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))), xi, Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * 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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 86.7

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

8. Applied rewrites 86.7%

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

Alternative 9: 88.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi - ux \cdot zi, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot xi\right) \cdot uy, -2, \left(yi \cdot \pi\right) \cdot 2\right), uy, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(1, xi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.019999999552965164)
   (fma
    (* maxCos ux)
    (- zi (* ux zi))
    (fma (fma (* (* (* PI PI) xi) uy) -2.0 (* (* yi PI) 2.0)) uy xi))
   (fma (* maxCos ux) zi (fma 1.0 xi (* (sin (* PI (* 2.0 uy))) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.019999999552965164f) {
		tmp = fmaf((maxCos * ux), (zi - (ux * zi)), fmaf(fmaf((((((float) M_PI) * ((float) M_PI)) * xi) * uy), -2.0f, ((yi * ((float) M_PI)) * 2.0f)), uy, xi));
	} else {
		tmp = fmaf((maxCos * ux), zi, fmaf(1.0f, xi, (sinf((((float) M_PI) * (2.0f * uy))) * yi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.019999999552965164))
		tmp = fma(Float32(maxCos * ux), Float32(zi - Float32(ux * zi)), fma(fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * xi) * uy), Float32(-2.0), Float32(Float32(yi * Float32(pi)) * Float32(2.0))), uy, xi));
	else
		tmp = fma(Float32(maxCos * ux), zi, fma(Float32(1.0), xi, Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * yi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   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. Add Preprocessing

   3. Taylor expanded in maxCos around 0

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

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in uy around 0

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

       1. *-commutative N/A

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

       2. lift-*.f32 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f32 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lift-*.f32 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f32 N/A

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

       10. associate-*r* N/A

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

       11. +-commutative N/A

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

   11. Applied rewrites 96.3%

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

   if 0.0199999996 < uy

   1. Initial program 97.1%

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

   3. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   4. 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

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 56.0%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(1, xi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi - ux \cdot zi, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot xi\right) \cdot uy, -2, \left(yi \cdot \pi\right) \cdot 2\right), uy, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(1, xi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.9% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))))
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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 84.7%

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

Alternative 11: 85.9% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), Float32(zi - Float32(ux * zi)), fma(fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * xi) * uy), Float32(-2.0), Float32(Float32(yi * Float32(pi)) * Float32(2.0))), uy, 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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in uy around 0

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

    1. *-commutative N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lift-*.f32 N/A

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

    10. associate-*r* N/A

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

    11. +-commutative N/A

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

11. Applied rewrites 84.8%

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

12. Final simplification 84.8%

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

Alternative 12: 83.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(fma(fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * xi) * uy), Float32(-2.0), Float32(Float32(yi * Float32(pi)) * Float32(2.0))), uy, Float32(Float32(zi * ux) * maxCos)) + 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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 48.2%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

11. Applied rewrites 81.6%

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

Alternative 13: 83.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(maxCos, Float32(ux * zi), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))))
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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-fma.f32 N/A

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

8. Applied rewrites 81.6%

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

Alternative 14: 81.7% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))))
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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
7. 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 80.9

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

8. Applied rewrites 80.9%

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

Alternative 15: 81.7% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))))
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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
7. 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 80.8

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

8. Applied rewrites 80.8%

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

Alternative 16: 81.7% accurate, 9.8× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* maxCos ux) (- zi (* 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 - (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), Float32(zi - Float32(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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in uy around 0

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

    1. *-commutative N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lift-*.f32 N/A

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

    10. associate-*r* N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. lower-fma.f32 N/A

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

11. Applied rewrites 80.9%

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

12. Final simplification 80.9%

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

Alternative 17: 79.0% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(maxCos * Float32(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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 77.8

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

8. Applied rewrites 77.8%

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

Alternative 18: 51.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi\right) \end{array} \]

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - 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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 50.2%

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

Alternative 19: 51.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, xi\right) \end{array} \]

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(Float32(1.0) - ux) * ux) * maxCos), 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. Add Preprocessing

3. Taylor expanded in maxCos around 0

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

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-*r* N/A

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

    6. lower-fma.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f32 N/A

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

    11. lift--.f32 50.1

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

11. Applied rewrites 50.1%

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

Alternative 20: 49.6% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(zi * ux), maxCos, 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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 48.2%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 48.2

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

10. Applied rewrites 48.2%

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

11. Final simplification 48.2%

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

Alternative 21: 49.6% accurate, 29.4× 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. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. 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

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 48.2%

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

Alternative 22: 12.2% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

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 = (zi * ux) * maxcos
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (zi * ux) * maxCos;
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. Add Preprocessing

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift--.f32 14.3

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

5. Applied rewrites 14.3%

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

6. Taylor expanded in ux around 0

   \[\leadsto \left(zi \cdot ux\right) \cdot maxCos \]
7. Step-by-step derivation

8. Applied rewrites 12.7%

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

Reproduce

herbie shell --seed 2025072 
(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)))