UniformSampleCone, y

Percentage Accurate: 57.5% → 98.4%

Time: 15.5s

Alternatives: 16

Speedup: 2.0×

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: 57.5% 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: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(2 + \left(maxCos \cdot \mathsf{fma}\left(ux, 2 - maxCos, -2\right) - ux\right)\right)\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* 2.0 (* uy PI))) 3.0)
   (pow (* ux (+ 2.0 (- (* maxCos (fma ux (- 2.0 maxCos) -2.0)) ux))) 1.5))))

C

float code(float ux, float uy, float maxCos) {
	return cbrtf((powf(sinf((2.0f * (uy * ((float) M_PI)))), 3.0f) * powf((ux * (2.0f + ((maxCos * fmaf(ux, (2.0f - maxCos), -2.0f)) - ux))), 1.5f)));
}

Julia

function code(ux, uy, maxCos)
	return cbrt(Float32((sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) ^ Float32(3.0)) * (Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * fma(ux, Float32(Float32(2.0) - maxCos), Float32(-2.0))) - ux))) ^ Float32(1.5))))
end

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

8. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right)}^{3} \cdot {\left(ux \cdot \left(2 + \mathsf{fma}\left(maxCos, \mathsf{fma}\left(-ux, maxCos, \mathsf{fma}\left(2, ux, -2\right)\right), -ux\right)\right)\right)}^{1.5}}} \]

10. Simplified 98.7%

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(2 + \left(maxCos \cdot \mathsf{fma}\left(ux, 2 - maxCos, -2\right) - ux\right)\right)\right)}^{1.5}}} \]
11. Add Preprocessing

Alternative 2: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

   1. pow1/2 98.4%

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

   2. *-commutative 98.4%

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

   3. unpow-prod-down 98.5%

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

   4. pow1/2 98.5%

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

   5. distribute-lft-out 98.5%

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

   6. fma-define 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

   7. sub-neg 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

   8. metadata-eval 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

   9. +-commutative 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

   10. fma-neg 98.5%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

   11. metadata-eval 98.5%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

   12. pow1/2 98.5%

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

7. Applied egg-rr 98.5%

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

   1. mul-1-neg 98.5%

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

9. Simplified 98.5%

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

10. Taylor expanded in maxCos around 0 98.5%

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

    1. associate--l+ 98.5%

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

    2. mul-1-neg 98.5%

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

    3. *-commutative 98.5%

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

    4. metadata-eval 98.5%

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

    5. cancel-sign-sub-inv 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 3: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot ux\right)\right)} + maxCos \cdot \left(\left(-1 \cdot \left(maxCos \cdot ux\right) + 2 \cdot ux\right) - 2\right)\right)\right)} \]

   2. log1p-define 98.4%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + -1 \cdot ux\right)}\right) + maxCos \cdot \left(\left(-1 \cdot \left(maxCos \cdot ux\right) + 2 \cdot ux\right) - 2\right)\right)\right)} \]

   3. mul-1-neg 98.4%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(\mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(-ux\right)}\right)\right) + maxCos \cdot \left(\left(-1 \cdot \left(maxCos \cdot ux\right) + 2 \cdot ux\right) - 2\right)\right)\right)} \]

   4. sub-neg 98.4%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 - ux\right)}\right) + maxCos \cdot \left(\left(-1 \cdot \left(maxCos \cdot ux\right) + 2 \cdot ux\right) - 2\right)\right)\right)} \]

   5. expm1-undefine 98.4%

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

   6. add-exp-log 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Taylor expanded in uy around inf 98.4%

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

   1. *-commutative 98.4%

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

   2. neg-mul-1 98.4%

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

   3. associate-+r+ 98.4%

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

   4. sub-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. neg-mul-1 98.4%

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

   7. +-commutative 98.4%

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

   8. distribute-lft-neg-in 98.4%

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

   9. distribute-rgt-in 98.4%

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

   10. sub-neg 98.4%

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

   11. metadata-eval 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 7: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot ux\right) \cdot \sqrt{-1 + 2 \cdot \frac{1}{ux}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (* (sin (* 2.0 (* uy PI))) ux) (sqrt (+ -1.0 (* 2.0 (/ 1.0 ux)))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (*
       ux
       (- (+ 2.0 (* maxCos (- (* 2.0 ux) (+ 2.0 (* ux maxCos))))) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = (sinf((2.0f * (uy * ((float) M_PI)))) * ux) * sqrtf((-1.0f + (2.0f * (1.0f / ux))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((2.0f + (maxCos * ((2.0f * ux) - (2.0f + (ux * maxCos))))) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * ux) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) * Float32(Float32(1.0) / ux)))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(Float32(2.0) + Float32(ux * maxCos))))) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(9.999999747378752e-6))
		tmp = (sin((single(2.0) * (uy * single(pi)))) * ux) * sqrt((single(-1.0) + (single(2.0) * (single(1.0) / ux))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((single(2.0) + (maxCos * ((single(2.0) * ux) - (single(2.0) + (ux * maxCos))))) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. sub-neg 58.7%

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

      3. +-commutative 58.7%

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

      4. distribute-rgt-neg-in 58.7%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in maxCos around 0 58.7%

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

   6. Taylor expanded in ux around inf 97.9%

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

      1. sub-neg 97.9%

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

      2. associate-*r/ 97.9%

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

      3. metadata-eval 97.9%

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

      4. metadata-eval 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in uy around inf 98.0%

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

   if 9.99999975e-6 < maxCos

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. sub-neg 58.0%

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

      3. +-commutative 58.0%

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

      4. distribute-rgt-neg-in 58.0%

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

      5. fma-define 58.1%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in ux around 0 98.6%

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

      1. pow1/2 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow-prod-down 98.2%

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

      4. pow1/2 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. fma-define 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

      7. sub-neg 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      8. metadata-eval 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      9. +-commutative 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      10. fma-neg 98.2%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

      11. metadata-eval 98.2%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

      12. pow1/2 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. mul-1-neg 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in maxCos around 0 98.3%

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

       1. associate--l+ 98.4%

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

       2. mul-1-neg 98.4%

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

       3. *-commutative 98.4%

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

       4. metadata-eval 98.4%

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

       5. cancel-sign-sub-inv 98.4%

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

   12. Simplified 98.4%

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

   13. Taylor expanded in uy around 0 88.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot ux\right) \cdot \sqrt{-1 + 2 \cdot \frac{1}{ux}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (sin (* 2.0 (* uy PI))) (sqrt (- (* 2.0 ux) (* ux ux))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (*
       ux
       (- (+ 2.0 (* maxCos (- (* 2.0 ux) (+ 2.0 (* ux maxCos))))) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((2.0f + (maxCos * ((2.0f * ux) - (2.0f + (ux * maxCos))))) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(Float32(2.0) + Float32(ux * maxCos))))) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(9.999999747378752e-6))
		tmp = sin((single(2.0) * (uy * single(pi)))) * sqrt(((single(2.0) * ux) - (ux * ux)));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((single(2.0) + (maxCos * ((single(2.0) * ux) - (single(2.0) + (ux * maxCos))))) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. sub-neg 58.7%

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

      3. +-commutative 58.7%

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

      4. distribute-rgt-neg-in 58.7%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in ux around 0 98.4%

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

   6. Taylor expanded in maxCos around 0 97.8%

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

      1. neg-mul-1 97.8%

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

   8. Simplified 97.8%

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

      1. distribute-lft-in 97.9%

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

   10. Applied egg-rr 97.9%

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

   if 9.99999975e-6 < maxCos

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. sub-neg 58.0%

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

      3. +-commutative 58.0%

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

      4. distribute-rgt-neg-in 58.0%

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

      5. fma-define 58.1%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in ux around 0 98.6%

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

      1. pow1/2 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow-prod-down 98.2%

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

      4. pow1/2 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. fma-define 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

      7. sub-neg 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      8. metadata-eval 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      9. +-commutative 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      10. fma-neg 98.2%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

      11. metadata-eval 98.2%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

      12. pow1/2 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. mul-1-neg 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in maxCos around 0 98.3%

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

       1. associate--l+ 98.4%

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

       2. mul-1-neg 98.4%

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

       3. *-commutative 98.4%

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

       4. metadata-eval 98.4%

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

       5. cancel-sign-sub-inv 98.4%

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

   12. Simplified 98.4%

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

   13. Taylor expanded in uy around 0 88.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (sin (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (*
       ux
       (- (+ 2.0 (* maxCos (- (* 2.0 ux) (+ 2.0 (* ux maxCos))))) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((2.0f + (maxCos * ((2.0f * ux) - (2.0f + (ux * maxCos))))) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(Float32(2.0) + Float32(ux * maxCos))))) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(9.999999747378752e-6))
		tmp = sin((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((single(2.0) + (maxCos * ((single(2.0) * ux) - (single(2.0) + (ux * maxCos))))) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. sub-neg 58.7%

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

      3. +-commutative 58.7%

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

      4. distribute-rgt-neg-in 58.7%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in ux around 0 98.4%

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

      1. pow1/2 98.4%

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

      2. *-commutative 98.4%

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

      3. unpow-prod-down 98.5%

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

      4. pow1/2 98.5%

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

      5. distribute-lft-out 98.5%

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

      6. fma-define 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

      7. sub-neg 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      8. metadata-eval 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      9. +-commutative 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      10. fma-neg 98.5%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

      11. metadata-eval 98.5%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

      12. pow1/2 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. mul-1-neg 98.5%

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

   9. Simplified 98.5%

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

   10. Taylor expanded in maxCos around 0 97.8%

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

   if 9.99999975e-6 < maxCos

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. sub-neg 58.0%

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

      3. +-commutative 58.0%

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

      4. distribute-rgt-neg-in 58.0%

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

      5. fma-define 58.1%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in ux around 0 98.6%

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

      1. pow1/2 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow-prod-down 98.2%

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

      4. pow1/2 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. fma-define 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

      7. sub-neg 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      8. metadata-eval 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      9. +-commutative 98.2%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      10. fma-neg 98.2%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

      11. metadata-eval 98.2%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

      12. pow1/2 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. mul-1-neg 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in maxCos around 0 98.3%

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

       1. associate--l+ 98.4%

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

       2. mul-1-neg 98.4%

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

       3. *-commutative 98.4%

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

       4. metadata-eval 98.4%

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

       5. cancel-sign-sub-inv 98.4%

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

   12. Simplified 98.4%

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

   13. Taylor expanded in uy around 0 88.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.004800000227987766:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.004800000227987766)
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (* ux (- (+ 2.0 (* maxCos (- (* 2.0 ux) (+ 2.0 (* ux maxCos))))) ux)))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.004800000227987766f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((2.0f + (maxCos * ((2.0f * ux) - (2.0f + (ux * maxCos))))) - ux))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.004800000227987766))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(Float32(2.0) + Float32(ux * maxCos))))) - ux)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.004800000227987766))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((single(2.0) + (maxCos * ((single(2.0) * ux) - (single(2.0) + (ux * maxCos))))) - ux))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00480000023

   1. Initial program 58.8%

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

      1. associate-*l* 58.8%

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

      2. sub-neg 58.8%

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

      3. +-commutative 58.8%

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

      4. distribute-rgt-neg-in 58.8%

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

      5. fma-define 58.9%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in ux around 0 98.5%

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

      1. pow1/2 98.5%

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

      2. *-commutative 98.5%

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

      3. unpow-prod-down 98.5%

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

      4. pow1/2 98.5%

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

      5. distribute-lft-out 98.5%

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

      6. fma-define 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

      7. sub-neg 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      8. metadata-eval 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      9. +-commutative 98.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

      10. fma-neg 98.5%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

      11. metadata-eval 98.5%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

      12. pow1/2 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. mul-1-neg 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in maxCos around 0 98.5%

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

       1. associate--l+ 98.6%

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

       2. mul-1-neg 98.6%

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

       3. *-commutative 98.6%

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

       4. metadata-eval 98.6%

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

       5. cancel-sign-sub-inv 98.6%

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

   12. Simplified 98.6%

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

   13. Taylor expanded in uy around 0 95.9%

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

   if 0.00480000023 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. sub-neg 58.2%

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

      3. +-commutative 58.2%

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

      4. distribute-rgt-neg-in 58.2%

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

      5. fma-define 58.1%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in maxCos around 0 56.3%

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

   6. Taylor expanded in ux around 0 74.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.004800000227987766:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot \left(2 \cdot ux - \left(2 + ux \cdot maxCos\right)\right)\right) - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

   1. pow1/2 98.4%

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

   2. *-commutative 98.4%

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

   3. unpow-prod-down 98.5%

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

   4. pow1/2 98.5%

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

   5. distribute-lft-out 98.5%

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

   6. fma-define 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}} \cdot {ux}^{0.5}\right) \]

   7. sub-neg 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

   8. metadata-eval 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(maxCos + \color{blue}{-1}\right)}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

   9. +-commutative 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\color{blue}{\left(-1 + maxCos\right)}}^{2}, 2 \cdot maxCos - 2\right)} \cdot {ux}^{0.5}\right) \]

   10. fma-neg 98.5%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right)} \cdot {ux}^{0.5}\right) \]

   11. metadata-eval 98.5%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{-1 \cdot \mathsf{fma}\left(ux, {\left(-1 + maxCos\right)}^{2}, \mathsf{fma}\left(2, maxCos, \color{blue}{-2}\right)\right)} \cdot {ux}^{0.5}\right) \]

   12. pow1/2 98.5%

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

7. Applied egg-rr 98.5%

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

   1. mul-1-neg 98.5%

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

9. Simplified 98.5%

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

10. Taylor expanded in maxCos around 0 98.5%

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

    1. associate--l+ 98.5%

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

    2. mul-1-neg 98.5%

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

    3. *-commutative 98.5%

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

    4. metadata-eval 98.5%

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

    5. cancel-sign-sub-inv 98.5%

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

12. Simplified 98.5%

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

13. Taylor expanded in uy around 0 80.5%

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

14. Final simplification 80.5%

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

Alternative 12: 81.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Taylor expanded in uy around 0 80.5%

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

   1. *-commutative 80.5%

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

   2. neg-mul-1 80.5%

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

   3. associate-+r+ 80.4%

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

   4. sub-neg 80.4%

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

   5. sub-neg 80.4%

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

   6. neg-mul-1 80.4%

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

   7. +-commutative 80.4%

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

   8. distribute-lft-neg-in 80.4%

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

   9. distribute-rgt-in 80.4%

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

   10. sub-neg 80.4%

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

   11. metadata-eval 80.4%

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

9. Simplified 80.4%

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

10. Final simplification 80.4%

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

Alternative 13: 77.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in maxCos around 0 55.8%

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

6. Taylor expanded in ux around inf 91.9%

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

   1. sub-neg 91.9%

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

   2. associate-*r/ 91.9%

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

   3. metadata-eval 91.9%

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

   4. metadata-eval 91.9%

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

8. Simplified 91.9%

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

9. Taylor expanded in uy around 0 75.5%

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

10. Final simplification 75.5%

    \[\leadsto 2 \cdot \left(\sqrt{-1 + 2 \cdot \frac{1}{ux}} \cdot \left(\left(uy \cdot \pi\right) \cdot ux\right)\right) \]
11. Add Preprocessing

Alternative 14: 77.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 91.9%

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

   1. neg-mul-1 91.9%

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

8. Simplified 91.9%

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

9. Taylor expanded in uy around 0 75.4%

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

    1. *-commutative 75.4%

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

11. Simplified 75.4%

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

Alternative 15: 63.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around 0 98.4%

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

6. Taylor expanded in maxCos around 0 91.9%

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

   1. neg-mul-1 91.9%

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

8. Simplified 91.9%

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

9. Taylor expanded in ux around 0 71.7%

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

10. Taylor expanded in uy around 0 61.5%

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

11. Final simplification 61.5%

    \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux} \]
12. Add Preprocessing

Alternative 16: 7.1% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in uy around 0 50.8%

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

7. Simplified 50.8%

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

8. Taylor expanded in ux around 0 7.1%

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

9. Taylor expanded in uy around 0 7.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))))))