UniformSampleCone, y

Percentage Accurate: 58.1% → 98.3%

Time: 6.2s

Alternatives: 13

Speedup: 3.3×

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

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

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: 58.1% accurate, 1.0× speedup

Math

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

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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(\left(uy + uy\right) \cdot \pi\right)}{{\left(\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)\right)}^{-0.5}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (/
  (sin (* (+ uy uy) PI))
  (pow (* (- (fma maxCos ux 2.0) ux) (- ux (* maxCos ux))) -0.5)))

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy + uy) * ((float) M_PI))) / powf(((fmaf(maxCos, ux, 2.0f) - ux) * (ux - (maxCos * ux))), -0.5f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) / (Float32(Float32(fma(maxCos, ux, Float32(2.0)) - ux) * Float32(ux - Float32(maxCos * ux))) ^ Float32(-0.5)))
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. 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 50.7

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

4. Applied rewrites 50.7%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lower-pow.f32 50.7

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \frac{1}{\color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{-0.5}}} \]

6. Applied rewrites 80.8%

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\frac{1}{{\left(\left(0 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(2 - \left(ux - maxCos \cdot ux\right)\right)\right)}^{-0.5}}} \]

7. Taylor expanded in uy around inf

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 98.2

       \[\leadsto \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{{\left(\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)\right)}^{-0.5}} \]

9. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{{\left(\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)\right)}^{-0.5}}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 98.2

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. count-2 N/A

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

    10. lift-+.f32 98.2

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

    11. lift-*.f32 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f32 98.2

        \[\leadsto \frac{\sin \left(\left(uy + uy\right) \cdot \pi\right)}{{\left(\left(\left(2 + maxCos \cdot ux\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)\right)}^{-0.5}} \]

    14. lift-+.f32 N/A

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

    15. +-commutative N/A

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

    16. lift-*.f32 N/A

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

    17. lower-fma.f32 98.2

        \[\leadsto \frac{\sin \left(\left(uy + uy\right) \cdot \pi\right)}{{\left(\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)\right)}^{-0.5}} \]

11. Applied rewrites 98.2%

    \[\leadsto \frac{\sin \left(\left(uy + uy\right) \cdot \pi\right)}{\color{blue}{{\left(\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)\right)}^{-0.5}}} \]
12. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 58.1

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 58.1

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

3. Applied rewrites 58.1%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 98.3

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

9. Applied rewrites 98.3%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 58.1

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 58.1

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

3. Applied rewrites 58.1%

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

4. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 98.3

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

6. Applied rewrites 98.3%

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

Alternative 4: 97.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 58.1

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 58.1

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

3. Applied rewrites 58.1%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. lower--.f32 97.2

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

9. Applied rewrites 97.2%

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

Alternative 5: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 58.1

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 58.1

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

3. Applied rewrites 58.1%

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

4. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 92.5

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

6. Applied rewrites 92.5%

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

Alternative 6: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* 2.0 (* uy PI))
  (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 (2.0f * (uy * ((float) M_PI))) * sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * 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

MATLAB

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

Derivation

1. Initial program 58.1%

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

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. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 80.9

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

7. Applied rewrites 80.9%

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

Alternative 7: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux - ux \cdot maxCos\\ \frac{\pi \cdot \left(uy + uy\right)}{{\left(\left(2 - t\_0\right) \cdot t\_0\right)}^{-0.5}} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- ux (* ux maxCos))))
   (/ (* PI (+ uy uy)) (pow (* (- 2.0 t_0) t_0) -0.5))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (ux * maxCos);
	return (((float) M_PI) * (uy + uy)) / powf(((2.0f - t_0) * t_0), -0.5f);
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(ux * maxCos))
	return Float32(Float32(Float32(pi) * Float32(uy + uy)) / (Float32(Float32(Float32(2.0) - t_0) * t_0) ^ Float32(-0.5)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = ux - (ux * maxCos);
	tmp = (single(pi) * (uy + uy)) / (((single(2.0) - t_0) * t_0) ^ single(-0.5));
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. 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 50.7

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

4. Applied rewrites 50.7%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lower-pow.f32 50.7

      \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \frac{1}{\color{blue}{{\left(1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}^{-0.5}}} \]

6. Applied rewrites 80.8%

   \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \color{blue}{\frac{1}{{\left(\left(0 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(2 - \left(ux - maxCos \cdot ux\right)\right)\right)}^{-0.5}}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lower-/.f32 80.9

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

   5. lift-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. distribute-rgt-in N/A

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

   10. lift-+.f32 N/A

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

   11. lift-*.f32 80.9

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

8. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(uy + uy\right)}{{\left(\left(2 - \left(ux - ux \cdot maxCos\right)\right) \cdot \left(ux - ux \cdot maxCos\right)\right)}^{-0.5}}} \]
9. Add Preprocessing

Alternative 8: 80.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \sqrt{\left(0 + t\_0\right) \cdot \left(2 - t\_0\right)} \cdot \left(\pi \cdot \left(uy + uy\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- ux (* maxCos ux))))
   (* (sqrt (* (+ 0.0 t_0) (- 2.0 t_0))) (* PI (+ uy uy)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	return sqrtf(((0.0f + t_0) * (2.0f - t_0))) * (((float) M_PI) * (uy + uy));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	return Float32(sqrt(Float32(Float32(Float32(0.0) + t_0) * Float32(Float32(2.0) - t_0))) * Float32(Float32(pi) * Float32(uy + uy)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = sqrt(((single(0.0) + t_0) * (single(2.0) - t_0))) * (single(pi) * (uy + uy));
end

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. 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 50.7

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

4. Applied rewrites 50.7%

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

6. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\sqrt{\left(0 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(2 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\pi \cdot \left(uy + uy\right)\right)} \]
7. Add Preprocessing

Alternative 9: 80.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 58.1

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 58.1

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

3. Applied rewrites 58.1%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 80.9

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

9. Applied rewrites 80.9%

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

Alternative 10: 80.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. difference-of-squares N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower--.f32 58.1

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 58.1

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

3. Applied rewrites 58.1%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 80.9

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

6. Applied rewrites 80.9%

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

Alternative 11: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996500015258789:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996500015258789)
     (* (* 2.0 (* uy PI)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))
     (* (* PI (+ uy uy)) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9996500015258789f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	} else {
		tmp = (((float) M_PI) * (uy + uy)) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9996500015258789))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if ((t_0 * t_0) <= single(0.9996500015258789))
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	else
		tmp = (single(pi) * (uy + uy)) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99965

   1. Initial program 58.1%

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

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   3. 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 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.4

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.2

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

   10. Applied rewrites 49.2%

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

   if 0.99965 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 58.1%

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

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   3. 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 50.7

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

   4. Applied rewrites 50.7%

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

   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. count-2-rev N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 7.1

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

   9. Applied rewrites 7.1%

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

   10. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 65.4

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

   12. Applied rewrites 65.4%

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

Alternative 12: 65.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. 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 50.7

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

4. Applied rewrites 50.7%

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

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. count-2-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-rgt-in N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 7.1

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

9. Applied rewrites 7.1%

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

10. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 65.4

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

12. Applied rewrites 65.4%

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

Alternative 13: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. 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 50.7

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

4. Applied rewrites 50.7%

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

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. count-2-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-rgt-in N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 7.1

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025140 
(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)))))))