Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.2%

Time: 4.4s

Alternatives: 13

Speedup: 1.6×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(u2 \cdot -2, \pi, \frac{\pi}{2}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (fma (* u2 -2.0) PI (/ PI 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(fmaf((u2 * -2.0f), ((float) M_PI), (((float) M_PI) / 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(fma(Float32(u2 * Float32(-2.0)), Float32(pi), Float32(Float32(pi) / Float32(2.0)))))
end

Derivation

1. Initial program 57.6%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f32 N/A

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

   9. lift--.f32 N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. flip-- N/A

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

   15. *-rgt-identity N/A

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

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

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

   17. lower-log1p.f32 N/A

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

   18. lower-*.f32 N/A

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

   19. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

   1. lift-fma.f32 N/A

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

7. Applied rewrites 99.2%

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

   1. lift-*.f32 N/A

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

   2. *-rgt-identity 99.2

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

9. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \sin \left(\mathsf{fma}\left(u2 \cdot -2, \pi, \frac{\pi}{2}\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \pi, \frac{\pi}{2}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (fma -2.0 (* u2 PI) (/ PI 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(fmaf(-2.0f, (u2 * ((float) M_PI)), (((float) M_PI) / 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(fma(Float32(-2.0), Float32(u2 * Float32(pi)), Float32(Float32(pi) / Float32(2.0)))))
end

Derivation

1. Initial program 57.6%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f32 N/A

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

   9. lift--.f32 N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. flip-- N/A

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

   15. *-rgt-identity N/A

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

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

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

   17. lower-log1p.f32 N/A

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

   18. lower-*.f32 N/A

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

   19. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

   1. lift-*.f32 N/A

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

   2. *-rgt-identity 99.1

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

7. Applied rewrites 99.1%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \pi, \frac{\pi}{2}\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f32 N/A

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

   9. lift--.f32 N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. flip-- N/A

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

   15. *-rgt-identity N/A

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

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

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

   17. lower-log1p.f32 N/A

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

   18. lower-*.f32 N/A

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

   19. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)}} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-rgt-identity 99.1

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

7. Applied rewrites 99.1%

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

Alternative 4: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.006500000134110451:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.006500000134110451)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (+ PI PI) u2)))
   (*
    (sqrt (- (log1p (* (- u1) 1.0))))
    (fma
     (fma
      (* (* (* PI PI) (* PI PI)) (* u2 u2))
      0.6666666666666666
      (* (* PI PI) -2.0))
     (* u2 u2)
     1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.006500000134110451f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-log1pf((-u1 * 1.0f))) * fmaf(fmaf((((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * (u2 * u2)), 0.6666666666666666f, ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.006500000134110451))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(Float32(-u1) * Float32(1.0))))) * fma(fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(u2 * u2)), Float32(0.6666666666666666), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.00650000013

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.2

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

   8. Applied rewrites 88.2%

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

   if 0.00650000013 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left({u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) \cdot {u2}^{2} + 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right), \color{blue}{{u2}^{2}}, 1\right) \]

   8. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (cos (* (+ 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 = cosf(((((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 = cos(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 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

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

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 6: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.007000000216066837:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.007000000216066837)
   (* (sqrt (- (log1p (* (- u1) 1.0)))) (fma (* (* u2 u2) -2.0) (* PI PI) 1.0))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.007000000216066837f) {
		tmp = sqrtf(-log1pf((-u1 * 1.0f))) * fmaf(((u2 * u2) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.007000000216066837))
		tmp = Float32(sqrt(Float32(-log1p(Float32(Float32(-u1) * Float32(1.0))))) * fma(Float32(Float32(u2 * u2) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00700000022

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 88.5

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

   8. Applied rewrites 88.5%

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

   if 0.00700000022 < u2

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 7: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9900000095367432:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2 \cdot -2, \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.9900000095367432)
   (* (sqrt u1) (sin (fma (* u2 -2.0) PI (/ PI 2.0))))
   (*
    (sqrt (- (log1p (* (- u1) 1.0))))
    (fma (* (* u2 u2) -2.0) (* PI PI) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.9900000095367432f) {
		tmp = sqrtf(u1) * sinf(fmaf((u2 * -2.0f), ((float) M_PI), (((float) M_PI) / 2.0f)));
	} else {
		tmp = sqrtf(-log1pf((-u1 * 1.0f))) * fmaf(((u2 * u2) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.9900000095367432))
		tmp = Float32(sqrt(u1) * sin(fma(Float32(u2 * Float32(-2.0)), Float32(pi), Float32(Float32(pi) / Float32(2.0)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(Float32(-u1) * Float32(1.0))))) * fma(Float32(Float32(u2 * u2) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.99000001

   1. Initial program 57.6%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f32 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-fma.f32 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 76.7%

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

   if 0.99000001 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 88.5

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

   8. Applied rewrites 88.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9900000095367432:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, -2, 0.5 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.9900000095367432)
   (* (sqrt u1) (sin (fma (* PI u2) -2.0 (* 0.5 PI))))
   (*
    (sqrt (- (log1p (* (- u1) 1.0))))
    (fma (* (* u2 u2) -2.0) (* PI PI) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.9900000095367432f) {
		tmp = sqrtf(u1) * sinf(fmaf((((float) M_PI) * u2), -2.0f, (0.5f * ((float) M_PI))));
	} else {
		tmp = sqrtf(-log1pf((-u1 * 1.0f))) * fmaf(((u2 * u2) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.9900000095367432))
		tmp = Float32(sqrt(u1) * sin(fma(Float32(Float32(pi) * u2), Float32(-2.0), Float32(Float32(0.5) * Float32(pi)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(Float32(-u1) * Float32(1.0))))) * fma(Float32(Float32(u2 * u2) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.99000001

   1. Initial program 57.6%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f32 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-fma.f32 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 76.6

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

   10. Applied rewrites 76.6%

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

   if 0.99000001 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 88.5

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

   8. Applied rewrites 88.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9900000095367432:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.9900000095367432)
   (* (sqrt u1) (cos (* (+ PI PI) u2)))
   (*
    (sqrt (- (log1p (* (- u1) 1.0))))
    (fma (* (* u2 u2) -2.0) (* PI PI) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.9900000095367432f) {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-log1pf((-u1 * 1.0f))) * fmaf(((u2 * u2) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.9900000095367432))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(Float32(-u1) * Float32(1.0))))) * fma(Float32(Float32(u2 * u2) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.99000001

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.6%

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

   if 0.99000001 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.6

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

   3. Applied rewrites 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. *-rgt-identity N/A

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

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

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

      17. lower-log1p.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 88.5

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

   8. Applied rewrites 88.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \pi \cdot \pi, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (* (- u1) 1.0)))) (fma (* (* u2 u2) -2.0) (* PI PI) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf((-u1 * 1.0f))) * fmaf(((u2 * u2) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(Float32(-u1) * Float32(1.0))))) * fma(Float32(Float32(u2 * u2) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 57.6%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f32 N/A

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

   9. lift--.f32 N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. flip-- N/A

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

   15. *-rgt-identity N/A

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

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

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

   17. lower-log1p.f32 N/A

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

   18. lower-*.f32 N/A

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

   19. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r* N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\left(-u1\right) \cdot 1\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 88.5

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

8. Applied rewrites 88.5%

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

Alternative 11: 49.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right) - 2, u2 \cdot u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log (- 1.0 u1))))
  (fma (- (* 0.6666666666666666 (* u2 u2)) 2.0) (* u2 u2) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * fmaf(((0.6666666666666666f * (u2 * u2)) - 2.0f), (u2 * u2), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * fma(Float32(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)) - Float32(2.0)), Float32(u2 * u2), Float32(1.0)))
end

Derivation

1. Initial program 57.6%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. flip-+ N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. +-inverses N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \frac{\color{blue}{0}}{\pi - \pi}\right) \]

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \frac{1 \cdot 1 - 1 \cdot 1}{\color{blue}{0}}\right) \]

   14. metadata-eval N/A

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

   15. flip-+ N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 49.4

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

5. Applied rewrites 49.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 49.4

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

7. Applied rewrites 49.4%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\frac{2}{3} \cdot {u2}^{2} - 2, {u2}^{2}, 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right) - 2, {u2}^{2}, 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right) - 2, {u2}^{2}, 1\right) \]

   8. unpow2 N/A

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

   9. lower-*.f32 49.4

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right) - 2, u2 \cdot \color{blue}{u2}, 1\right) \]

10. Applied rewrites 49.4%

    \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right) - 2, u2 \cdot u2, 1\right)} \]
11. Add Preprocessing

Alternative 12: 49.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. flip-+ N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. +-inverses N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \frac{\color{blue}{0}}{\pi - \pi}\right) \]

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \frac{1 \cdot 1 - 1 \cdot 1}{\color{blue}{0}}\right) \]

   14. metadata-eval N/A

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

   15. flip-+ N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 49.4

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

5. Applied rewrites 49.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 49.4

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

7. Applied rewrites 49.4%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 49.4

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

10. Applied rewrites 49.4%

    \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, -2, 1\right)} \]
11. Add Preprocessing

Alternative 13: 49.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log (- 1.0 u1)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-log((1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
end

MATLAB

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

Derivation

1. Initial program 57.6%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 57.6

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

3. Applied rewrites 57.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. flip-+ N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. +-inverses N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \frac{\color{blue}{0}}{\pi - \pi}\right) \]

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

       \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \frac{1 \cdot 1 - 1 \cdot 1}{\color{blue}{0}}\right) \]

   14. metadata-eval N/A

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

   15. flip-+ N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 49.4

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

5. Applied rewrites 49.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 49.4

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

7. Applied rewrites 49.4%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}} \]
9. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]

   4. lift-sqrt.f32 49.6

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]

10. Applied rewrites 49.6%

    \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))