UniformSampleCone, y

Percentage Accurate: 57.9% → 98.4%

Time: 6.3s

Alternatives: 9

Speedup: 3.2×

Specification

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

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

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

MATLAB

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

Initial Program: 57.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.7× speedup

Math

\[\sin \left(\mathsf{fma}\left(uy \cdot 2.145029306411743, \sqrt[3]{\pi}, \pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (fma (* uy 2.145029306411743) (cbrt PI) (* PI uy)))
  (sqrt
   (*
    ux
    (- (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0)))) (* 2.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(fmaf((uy * 2.145029306411743f), cbrtf(((float) M_PI)), (((float) M_PI) * uy))) * sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(uy * Float32(2.145029306411743)), cbrt(Float32(pi)), Float32(Float32(pi) * uy))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos)))))
end

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   6. lower-pow.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   7. lower--.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   8. lower-*.f32 98.3%

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

4. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. count-2 N/A

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

   5. distribute-lft-out N/A

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

   6. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

      \[\leadsto \sin \left(uy \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} + uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   8. associate-*r* N/A

      \[\leadsto \sin \left(\color{blue}{\left(uy \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} + uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)} + \color{blue}{uy \cdot \pi}\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   10. lower-fma.f32 N/A

       \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(uy \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right), \sqrt[3]{\mathsf{PI}\left(\right)}, uy \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

6. Applied rewrites 98.4%

   \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(uy \cdot \sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi}, \pi \cdot uy\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

7. Evaluated real constant 98.4%

   \[\leadsto \sin \left(\mathsf{fma}\left(uy \cdot \color{blue}{2.145029306411743}, \sqrt[3]{\pi}, \pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\sqrt{\left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right) \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- ux (fma maxCos ux 2.0)) (* ux (- maxCos 1.0))))
  (sin (* (+ PI PI) uy))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(ux - fma(maxCos, ux, Float32(2.0))) * Float32(ux * Float32(maxCos - Float32(1.0))))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)))
end

Derivation

1. Initial program 57.9%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

   3. pow-to-exp N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   4. lower-unsound-exp.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

3. Applied rewrites 96.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   3. lower-*.f32 96.4%

      \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

5. Applied rewrites 98.3%

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

6. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3%

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

8. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

\[\sqrt{\left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right) \cdot \left(maxCos \cdot ux - ux\right)} \cdot \sin \left(6.2831854820251465 \cdot uy\right) \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- ux (fma maxCos ux 2.0)) (- (* maxCos ux) ux)))
  (sin (* 6.2831854820251465 uy))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux - fmaf(maxCos, ux, 2.0f)) * ((maxCos * ux) - ux))) * sinf((6.2831854820251465f * uy));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(ux - fma(maxCos, ux, Float32(2.0))) * Float32(Float32(maxCos * ux) - ux))) * sin(Float32(Float32(6.2831854820251465) * uy)))
end

Derivation

1. Initial program 57.9%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

   3. pow-to-exp N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   4. lower-unsound-exp.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

3. Applied rewrites 96.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   3. lower-*.f32 96.4%

      \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

5. Applied rewrites 98.3%

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

6. Evaluated real constant 98.3%

   \[\leadsto \sqrt{\left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right) \cdot \left(maxCos \cdot ux - ux\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot uy\right) \]
7. Add Preprocessing

Alternative 4: 95.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00043200000072829425:\\ \;\;\;\;\sqrt{\left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right) \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00043200000072829425)
   (*
    (sqrt (* (- ux (fma maxCos ux 2.0)) (* ux (- maxCos 1.0))))
    (* (+ uy uy) PI))
   (* (sqrt (* -1.0 (* ux (- ux 2.0)))) (sin (* (+ PI PI) uy)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00043200000072829425f) {
		tmp = sqrtf(((ux - fmaf(maxCos, ux, 2.0f)) * (ux * (maxCos - 1.0f)))) * ((uy + uy) * ((float) M_PI));
	} else {
		tmp = sqrtf((-1.0f * (ux * (ux - 2.0f)))) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00043200000072829425))
		tmp = Float32(sqrt(Float32(Float32(ux - fma(maxCos, ux, Float32(2.0))) * Float32(ux * Float32(maxCos - Float32(1.0))))) * Float32(Float32(uy + uy) * Float32(pi)));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 4.32000001e-4

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-unsound-exp.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 96.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      3. lower-PI.f32 80.5%

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   8. Applied rewrites 81.6%

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

   if 4.32000001e-4 < uy

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-unsound-exp.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 96.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

      3. lower-*.f32 96.4%

         \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 92.3%

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

   8. Applied rewrites 92.3%

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

Alternative 5: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := ux \cdot \left(maxCos - 1\right)\\ \mathbf{if}\;uy \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{\left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right) \cdot t\_0} \cdot \left(\left(uy + uy\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot t\_0} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- maxCos 1.0))))
   (if (<= uy 0.0020000000949949026)
     (* (sqrt (* (- ux (fma maxCos ux 2.0)) t_0)) (* (+ uy uy) PI))
     (* (sqrt (* -2.0 t_0)) (sin (* (+ PI PI) uy))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (maxCos - 1.0f);
	float tmp;
	if (uy <= 0.0020000000949949026f) {
		tmp = sqrtf(((ux - fmaf(maxCos, ux, 2.0f)) * t_0)) * ((uy + uy) * ((float) M_PI));
	} else {
		tmp = sqrtf((-2.0f * t_0)) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(maxCos - Float32(1.0)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(Float32(ux - fma(maxCos, ux, Float32(2.0))) * t_0)) * Float32(Float32(uy + uy) * Float32(pi)));
	else
		tmp = Float32(sqrt(Float32(Float32(-2.0) * t_0)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00200000009

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-unsound-exp.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 96.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      3. lower-PI.f32 80.5%

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   8. Applied rewrites 81.6%

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

   if 0.00200000009 < uy

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-unsound-exp.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 96.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

      3. lower-*.f32 96.4%

         \[\leadsto \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 76.2%

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

   8. Applied rewrites 76.2%

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

Alternative 6: 81.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(ux - fma(maxCos, ux, Float32(2.0))) * Float32(ux * Float32(maxCos - Float32(1.0))))) * Float32(Float32(uy + uy) * Float32(pi)))
end

Derivation

1. Initial program 57.9%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

   3. pow-to-exp N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   4. lower-unsound-exp.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

3. Applied rewrites 96.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]

4. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

   3. lower-PI.f32 80.5%

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

6. Applied rewrites 80.5%

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

8. Applied rewrites 81.6%

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

Alternative 7: 79.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= maxCos 4.999999987376214e-7)
     (* t_0 (sqrt (* -1.0 (* ux (- ux 2.0)))))
     (* t_0 (sqrt (* -2.0 (* ux (- maxCos 1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (maxCos <= 4.999999987376214e-7f) {
		tmp = t_0 * sqrtf((-1.0f * (ux * (ux - 2.0f))));
	} else {
		tmp = t_0 * sqrtf((-2.0f * (ux * (maxCos - 1.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999987376214e-7))
		tmp = Float32(t_0 * sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))));
	else
		tmp = Float32(t_0 * sqrt(Float32(Float32(-2.0) * Float32(ux * Float32(maxCos - Float32(1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = single(0.0);
	if (maxCos <= single(4.999999987376214e-7))
		tmp = t_0 * sqrt((single(-1.0) * (ux * (ux - single(2.0)))));
	else
		tmp = t_0 * sqrt((single(-2.0) * (ux * (maxCos - single(1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999999e-7

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-unsound-exp.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 96.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      3. lower-PI.f32 80.5%

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \]

      4. lower--.f32 77.3%

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \]

   9. Applied rewrites 77.3%

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}} \]

   if 4.99999999e-7 < maxCos

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-unsound-exp.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 96.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

      3. lower-PI.f32 80.5%

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

   7. Taylor expanded in ux around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

      4. lower--.f32 65.8%

         \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

   9. Applied rewrites 65.8%

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

Alternative 8: 65.8% accurate, 3.2× speedup

Math

\[\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* 2.0 (* uy PI)) (sqrt (* -2.0 (* ux (- maxCos 1.0))))))

C

float code(float ux, float uy, float maxCos) {
	return (2.0f * (uy * ((float) M_PI))) * sqrtf((-2.0f * (ux * (maxCos - 1.0f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(-2.0) * Float32(ux * Float32(maxCos - Float32(1.0))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(-2.0) * (ux * (maxCos - single(1.0)))));
end

Derivation

1. Initial program 57.9%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{\frac{1}{2}}} \]

   3. pow-to-exp N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   4. lower-unsound-exp.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot e^{\color{blue}{\log \left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \frac{1}{2}}} \]

3. Applied rewrites 96.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5}} \]

4. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot \frac{1}{2}} \]

   3. lower-PI.f32 80.5%

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

6. Applied rewrites 80.5%

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot e^{\log \left(\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 2\right)\right) \cdot 0.5} \]

7. Taylor expanded in ux around 0

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)}} \]
8. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

   4. lower--.f32 65.8%

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]

9. Applied rewrites 65.8%

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{-2 \cdot \left(ux \cdot \left(maxCos - 1\right)\right)}} \]
10. Add Preprocessing

Alternative 9: 7.1% accurate, 4.7× speedup

Math

\[\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (+ uy uy) PI) (sqrt (- 1.0 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return ((uy + uy) * ((float) M_PI)) * sqrtf((1.0f - 1.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(Float32(1.0) - Float32(1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ((uy + uy) * single(pi)) * sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.9%

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{1}} \]
6. Step-by-step derivation

7. Applied rewrites 7.1%

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{1}} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(uy \cdot \pi\right)}\right) \cdot \sqrt{1 - 1} \]

   2. lift-*.f32 N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{1 - 1} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \color{blue}{\pi}\right) \cdot \sqrt{1 - 1} \]

   4. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   6. lift-*.f32 7.1%

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \color{blue}{\pi}\right) \cdot \sqrt{1 - 1} \]

   7. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   8. *-commutative N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   9. count-2-rev N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   10. lower-+.f32 7.1%

       \[\leadsto \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

9. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))