Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.3%

Time: 5.4s

Alternatives: 15

Speedup: 4.7×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)))
end

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. flip3-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lower-log.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. unpow3 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. lower-log.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. lower-+.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   18. lower-*.f32 54.8

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Applied rewrites 54.8%

   \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{u1 \cdot \left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. +-commutative N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + u1 \cdot \color{blue}{\left(1 + u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lower-+.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. +-commutative N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + 1 \cdot u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. *-lft-identity N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-fma.f32 96.6

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Applied rewrites 96.6%

   \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. lift--.f32 N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lift-fma.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. lift-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \left(u1 \cdot u1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. diff-log N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. *-lft-identity N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{-\log \left(\frac{\color{blue}{{1}^{3}} - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. pow3 N/A

       \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - \color{blue}{{u1}^{3}}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{\color{blue}{1 \cdot 1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. flip3-- N/A

       \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. *-lft-identity N/A

       \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. mul-1-neg N/A

       \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   18. lift-neg.f32 N/A

       \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
8. Step-by-step derivation

   1. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot u2\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

   3. count-2-rev N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

   4. lift-+.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

   5. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\color{blue}{\pi} + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lift-PI.f32 98.3

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \color{blue}{\pi}\right) \cdot u2\right) \]

9. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]
10. Add Preprocessing

Alternative 2: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.003000000026077032:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.003000000026077032)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.003000000026077032f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003000000026077032))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00300000003

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot u2\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

      3. count-2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

      4. lower-+.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\color{blue}{\pi} + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-PI.f32 57.7

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\pi + \color{blue}{\pi}\right) \cdot u2\right) \]

   3. Applied rewrites 57.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   if -0.00300000003 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-fma.f32 87.9

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.019999999552965164)
   (*
    (sqrt (- (log1p (- u1))))
    (* (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) u2))
   (* (sqrt (fma (* u1 u1) 0.5 u1)) (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.019999999552965164f) {
		tmp = sqrtf(-log1pf(-u1)) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf(fmaf((u1 * u1), 0.5f, u1)) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.019999999552965164))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(fma(Float32(u1 * u1), Float32(0.5), u1)) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0199999996

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. flip3-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. unpow3 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-log.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. lower-+.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lower-*.f32 54.8

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 54.8%

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{u1 \cdot \left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + u1 \cdot \color{blue}{\left(1 + u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-+.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + 1 \cdot u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-fma.f32 96.6

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 96.6%

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lift--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lift-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lift-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \left(u1 \cdot u1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(\frac{\color{blue}{{1}^{3}} - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. pow3 N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - \color{blue}{{u1}^{3}}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{\color{blue}{1 \cdot 1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. flip3-- N/A

          \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. *-lft-identity N/A

          \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. mul-1-neg N/A

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lift-neg.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   10. Applied rewrites 89.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]

   if 0.0199999996 < u2

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-fma.f32 87.9

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around inf

      \[\leadsto \sqrt{{u1}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{u1}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + {u1}^{2} \cdot \color{blue}{\frac{1}{u1}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. inv-pow N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + {u1}^{2} \cdot {u1}^{-1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. pow-prod-up N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + {u1}^{\left(2 + \color{blue}{-1}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + {u1}^{1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. unpow1 N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left({u1}^{2}, \frac{1}{2}, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. pow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \frac{1}{2}, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lift-*.f32 87.9

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Applied rewrites 87.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{0.5}, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.019999999552965164)
   (*
    (sqrt (- (log1p (- u1))))
    (* (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) u2))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.019999999552965164f) {
		tmp = sqrtf(-log1pf(-u1)) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.019999999552965164))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0199999996

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. flip3-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. unpow3 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-log.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. lower-+.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lower-*.f32 54.8

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 54.8%

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{u1 \cdot \left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + u1 \cdot \color{blue}{\left(1 + u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-+.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + 1 \cdot u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-fma.f32 96.6

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 96.6%

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lift--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lift-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lift-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \left(u1 \cdot u1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(\frac{\color{blue}{{1}^{3}} - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. pow3 N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - \color{blue}{{u1}^{3}}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{\color{blue}{1 \cdot 1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. flip3-- N/A

          \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. *-lft-identity N/A

          \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. mul-1-neg N/A

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lift-neg.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   10. Applied rewrites 89.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]

   if 0.0199999996 < u2

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-fma.f32 87.9

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.026000000536441803:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.026000000536441803)
   (*
    (sqrt (- (log1p (- u1))))
    (* (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) u2))
   (* (sqrt u1) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.026000000536441803f) {
		tmp = sqrtf(-log1pf(-u1)) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.026000000536441803))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0260000005

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. flip3-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. unpow3 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-log.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. lower-+.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lower-*.f32 54.8

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 54.8%

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{u1 \cdot \left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + u1 \cdot \color{blue}{\left(1 + u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-+.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + 1 \cdot u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. *-lft-identity N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-fma.f32 96.6

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 96.6%

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lift--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lift-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lift-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \left(u1 \cdot u1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(\frac{\color{blue}{{1}^{3}} - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. pow3 N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - \color{blue}{{u1}^{3}}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{\color{blue}{1 \cdot 1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. flip3-- N/A

          \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. *-lft-identity N/A

          \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. mul-1-neg N/A

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lift-neg.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   10. Applied rewrites 89.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]

   if 0.0260000005 < u2

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{u1}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. lift-sin.f32 76.5

         \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{u1} \cdot \sin \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \sin \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. lift-PI.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \sin \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lift-PI.f32 76.5

          \[\leadsto \sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (* (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2))
end

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. flip3-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lower-log.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. unpow3 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. lower-log.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. lower-+.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   18. lower-*.f32 54.8

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Applied rewrites 54.8%

   \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{u1 \cdot \left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. +-commutative N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + u1 \cdot \color{blue}{\left(1 + u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lower-+.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot \left(1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. +-commutative N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + 1 \cdot u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. *-lft-identity N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-fma.f32 96.6

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Applied rewrites 96.6%

   \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. lift--.f32 N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - \left(u1 \cdot u1\right) \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lift-fma.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot u1 + u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. lift-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 + \left(u1 \cdot u1 + u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. diff-log N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. *-lft-identity N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + \color{blue}{1 \cdot u1}\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{-\log \left(\frac{\color{blue}{{1}^{3}} - \left(u1 \cdot u1\right) \cdot u1}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. pow3 N/A

       \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - \color{blue}{{u1}^{3}}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{\color{blue}{1 \cdot 1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. flip3-- N/A

       \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. *-lft-identity N/A

       \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. mul-1-neg N/A

       \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   18. lift-neg.f32 N/A

       \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

10. Applied rewrites 89.3%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
11. Add Preprocessing

Alternative 7: 88.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq -0.003000000026077032:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1)))
        (t_1
         (*
          (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
          u2)))
   (if (<= t_0 -0.003000000026077032)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2;
	float tmp;
	if (t_0 <= -0.003000000026077032f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003000000026077032))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00300000003

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]

   if -0.00300000003 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-fma.f32 87.9

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   7. Applied rewrites 80.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.002199999988079071)
   (*
    (sqrt (* (fma 0.5 u1 1.0) u1))
    (* (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) u2))
   (* (sqrt (- (log1p (- u1)))) (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.002199999988079071f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf(-log1pf(-u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0022

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-fma.f32 87.9

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   7. Applied rewrites 80.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]

   if 0.0022 < u1

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-PI.f32 50.9

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]
   5. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. *-lft-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower-neg.f32 81.5

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   6. Applied rewrites 81.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 6.4000000747910235 \cdot 10^{-6}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \left(\pi + \pi\right) \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 6.4000000747910235e-6)
   (*
    u2
    (fma
     (* (* u2 u2) -1.3333333333333333)
     (* (sqrt u1) (* (* PI PI) PI))
     (* (+ PI PI) (sqrt u1))))
   (* (sqrt (- (log1p (- u1)))) (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 6.4000000747910235e-6f) {
		tmp = u2 * fmaf(((u2 * u2) * -1.3333333333333333f), (sqrtf(u1) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), ((((float) M_PI) + ((float) M_PI)) * sqrtf(u1)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(6.4000000747910235e-6))
		tmp = Float32(u2 * fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(sqrt(u1) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(Float32(pi) + Float32(pi)) * sqrt(u1))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 6.40000007e-6

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. flip3-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. unpow3 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-log.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. lower-+.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lower-*.f32 54.8

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 54.8%

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + \left(u1 + {u1}^{2}\right)\right) - \log \left(1 - {u1}^{3}\right)}\right)\right)} \]

   6. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\left(2 \cdot u2\right) \cdot \left(\sqrt{\log \left(\frac{\mathsf{fma}\left(u1 + 1, u1, 1\right)}{1 - \left(u1 \cdot u1\right) \cdot u1}\right)} \cdot \pi\right)} \]

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{2}}{1 + u1}\right)\right)}\right)\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{2}}{1 + u1}\right)\right)}\right)\right)} \]

   9. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}, \left(\pi + \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto u2 \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{u1}\right)\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right)} \]
   11. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{u1}\right)\right) + \color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto u2 \cdot \left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{u1}\right) + 2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{u1}\right)\right) \]

       3. lower-fma.f32 N/A

          \[\leadsto u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\sqrt{u1}}, 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

   12. Applied rewrites 70.7%

       \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \left(\pi + \pi\right) \cdot \sqrt{u1}\right)} \]

   if 6.40000007e-6 < u1

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-PI.f32 50.9

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]
   5. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. *-lft-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower-neg.f32 81.5

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   6. Applied rewrites 81.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (* (+ PI PI) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2))
end

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. lift-PI.f32 50.9

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   3. *-lft-identity N/A

      \[\leadsto \sqrt{-\log \left(1 - \color{blue}{1 \cdot u1}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + -1 \cdot u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   8. lower-neg.f32 81.5

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

6. Applied rewrites 81.5%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
7. Add Preprocessing

Alternative 11: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot u2\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_1 \leq 0.0560000017285347:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (+ PI PI) u2)) (t_1 (sqrt (- (log (- 1.0 u1))))))
   (if (<= t_1 0.0560000017285347)
     (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) t_0)
     (* t_1 t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) + ((float) M_PI)) * u2;
	float t_1 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if (t_1 <= 0.0560000017285347f) {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) + Float32(pi)) * u2)
	t_1 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0560000017285347))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * t_0);
	else
		tmp = Float32(t_1 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (single(pi) + single(pi)) * u2;
	t_1 = sqrt(-log((single(1.0) - u1)));
	tmp = single(0.0);
	if (t_1 <= single(0.0560000017285347))
		tmp = sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1)) * t_0;
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) < 0.0560000017

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-PI.f32 50.9

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. flip3-- N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. pow3 N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{-\color{blue}{u1} \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      9. lower-*.f32 74.2

         \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   if 0.0560000017 < (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))))

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-PI.f32 50.9

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.003100000089034438:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \pi\right)\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.003100000089034438)
   (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (* (+ PI PI) u2))
   (* (* (sqrt (- (log (- 1.0 u1)))) (+ PI PI)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.003100000089034438f) {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	} else {
		tmp = (sqrtf(-logf((1.0f - u1))) * (((float) M_PI) + ((float) M_PI))) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.003100000089034438))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	else
		tmp = Float32(Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(pi) + Float32(pi))) * u2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.003100000089034438))
		tmp = sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
	else
		tmp = (sqrt(-log((single(1.0) - u1))) * (single(pi) + single(pi))) * u2;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00310000009

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-PI.f32 50.9

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. flip3-- N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. pow3 N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{-\color{blue}{u1} \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      9. lower-*.f32 74.2

         \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   if 0.00310000009 < u1

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. flip3-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-log.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. unpow3 N/A

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-log.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. lower-+.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      18. lower-*.f32 54.8

          \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 54.8%

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + \left(u1 + {u1}^{2}\right)\right) - \log \left(1 - {u1}^{3}\right)}\right)\right)} \]

   6. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\left(2 \cdot u2\right) \cdot \left(\sqrt{\log \left(\frac{\mathsf{fma}\left(u1 + 1, u1, 1\right)}{1 - \left(u1 \cdot u1\right) \cdot u1}\right)} \cdot \pi\right)} \]

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{2}}{1 + u1}\right)\right)}\right)\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{2}}{1 + u1}\right)\right)}\right)\right)} \]

   9. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}, \left(\pi + \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]

   10. Taylor expanded in u2 around 0

       \[\leadsto \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right) \cdot u2 \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \cdot u2 \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       3. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       4. lift-log.f32 N/A

          \[\leadsto \left(\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       5. lift--.f32 N/A

          \[\leadsto \left(\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       6. lift-neg.f32 N/A

          \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       7. lift-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       8. count-2-rev N/A

          \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       9. lift-+.f32 N/A

          \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       10. lift-PI.f32 N/A

           \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \mathsf{PI}\left(\right)\right)\right) \cdot u2 \]

       11. lift-PI.f32 50.9

           \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \pi\right)\right) \cdot u2 \]

   12. Applied rewrites 50.9%

       \[\leadsto \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \pi\right)\right) \cdot u2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (* (+ PI PI) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
end

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{u2}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. lift-PI.f32 50.9

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   3. pow3 N/A

      \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{-u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   5. diff-log N/A

      \[\leadsto \sqrt{-\color{blue}{u1} \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{-\left(\frac{-1}{2} \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   9. lower-*.f32 74.2

      \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

7. Applied rewrites 74.2%

   \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
8. Add Preprocessing

Alternative 14: 66.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (+ u2 u2) (* (sqrt u1) PI)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 + u2) * (sqrtf(u1) * ((float) M_PI));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 + u2) * Float32(sqrt(u1) * Float32(pi)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 + u2) * (sqrt(u1) * single(pi));
end

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. flip3-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lower-log.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. unpow3 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. lower-log.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. lower-+.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   18. lower-*.f32 54.8

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Applied rewrites 54.8%

   \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + \left(u1 + {u1}^{2}\right)\right) - \log \left(1 - {u1}^{3}\right)}\right)\right)} \]

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\left(2 \cdot u2\right) \cdot \left(\sqrt{\log \left(\frac{\mathsf{fma}\left(u1 + 1, u1, 1\right)}{1 - \left(u1 \cdot u1\right) \cdot u1}\right)} \cdot \pi\right)} \]

7. Taylor expanded in u1 around 0

   \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{u1}}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{u1}}\right) \]

   3. count-2-rev N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\color{blue}{u1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\color{blue}{u1}}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-sqrt.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

   8. lift-PI.f32 66.3

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

9. Applied rewrites 66.3%

   \[\leadsto \left(u2 + u2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 15: 66.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \pi \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (* (+ u2 u2) (sqrt u1)) PI))

C

float code(float cosTheta_i, float u1, float u2) {
	return ((u2 + u2) * sqrtf(u1)) * ((float) M_PI);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(u2 + u2) * sqrt(u1)) * Float32(pi))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((u2 + u2) * sqrt(u1)) * single(pi);
end

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. flip3-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. lower-log.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\log \left({1}^{3} - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 - {u1}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. unpow3 N/A

      \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2}} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{{u1}^{2} \cdot u1}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot u1\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   14. lower-log.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \color{blue}{\log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. lower-+.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \color{blue}{\left(1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   18. lower-*.f32 54.8

       \[\leadsto \sqrt{-\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, \color{blue}{1 \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Applied rewrites 54.8%

   \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - \left(u1 \cdot u1\right) \cdot u1\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + \left(u1 + {u1}^{2}\right)\right) - \log \left(1 - {u1}^{3}\right)}\right)\right)} \]

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\left(2 \cdot u2\right) \cdot \left(\sqrt{\log \left(\frac{\mathsf{fma}\left(u1 + 1, u1, 1\right)}{1 - \left(u1 \cdot u1\right) \cdot u1}\right)} \cdot \pi\right)} \]

7. Taylor expanded in u1 around 0

   \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{u1}}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \left(2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{u1}}\right) \]

   3. count-2-rev N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\color{blue}{u1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\color{blue}{u1}}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-sqrt.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

   8. lift-PI.f32 66.3

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

9. Applied rewrites 66.3%

   \[\leadsto \left(u2 + u2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot \pi\right)} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \color{blue}{\pi}\right) \]

    2. lift-PI.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

    3. lift-*.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

    4. lift-sqrt.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \mathsf{PI}\left(\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \mathsf{PI}\left(\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \mathsf{PI}\left(\right) \]

    7. lower-*.f32 N/A

       \[\leadsto \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \mathsf{PI}\left(\right) \]

    8. lift-sqrt.f32 N/A

       \[\leadsto \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \mathsf{PI}\left(\right) \]

    9. lift-PI.f32 66.3

       \[\leadsto \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \pi \]

11. Applied rewrites 66.3%

    \[\leadsto \left(\left(u2 + u2\right) \cdot \sqrt{u1}\right) \cdot \pi \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))