Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 99.2%

Time: 7.0s

Alternatives: 18

Speedup: 3.1×

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: 57.4% 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

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (fma -2.0 u2 0.5) PI))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((fmaf(-2.0f, u2, 0.5f) * ((float) M_PI)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(fma(Float32(-2.0), u2, Float32(0.5)) * Float32(pi))))
end

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-in N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lower-/.f32 N/A

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

   18. lift-PI.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-fma.f32 99.2

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, \color{blue}{-2}, 0.5\right)\right) \]

8. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)} \]
9. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-PI.f32 99.2

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right) \]

10. Applied rewrites 99.2%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right)} \]
11. Add Preprocessing

Alternative 2: 99.0% 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 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-in N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.030500000342726707:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \pi, \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.6666666666666666\right) \cdot u2\right) \cdot u2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.030500000342726707)
   (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (sin (* PI (fma u2 -2.0 0.5))))
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (fma
      (* -2.0 PI)
      PI
      (* (* (* (* (* PI PI) (* PI PI)) 0.6666666666666666) u2) u2))
     (* 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))) <= 0.030500000342726707f) {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(fmaf((-2.0f * ((float) M_PI)), ((float) M_PI), (((((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * 0.6666666666666666f) * u2) * u2)), (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(0.030500000342726707))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(fma(Float32(Float32(-2.0) * Float32(pi)), Float32(pi), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(0.6666666666666666)) * u2) * u2)), 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))) < 0.0305000003

   1. Initial program 42.5%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-/.f32 N/A

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

      18. lift-PI.f32 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-fma.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, \color{blue}{-2}, 0.5\right)\right) \]

   8. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)} \]

   9. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. distribute-rgt-neg-in N/A

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

       6. metadata-eval N/A

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

       7. *-rgt-identity N/A

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

       8. *-commutative N/A

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

       9. lower-*.f32 N/A

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

       10. lower--.f32 N/A

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

       11. lower-*.f32 98.1

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

   11. Applied rewrites 98.1%

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

   if 0.0305000003 < (*.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 92.9%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 92.8

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in u2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 N/A

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

   11. Applied rewrites 96.1%

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

Alternative 4: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.030500000342726707:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \pi, \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.6666666666666666\right) \cdot u2\right) \cdot u2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.030500000342726707)
     (* (sqrt (fma (* 0.5 u1) u1 u1)) t_0)
     (*
      (sqrt (- (log1p (- u1))))
      (fma
       (fma
        (* -2.0 PI)
        PI
        (* (* (* (* (* PI PI) (* PI PI)) 0.6666666666666666) u2) u2))
       (* u2 u2)
       1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * t_0) <= 0.030500000342726707f) {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1)) * t_0;
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(fmaf((-2.0f * ((float) M_PI)), ((float) M_PI), (((((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * 0.6666666666666666f) * u2) * u2)), (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0) <= Float32(0.030500000342726707))
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(fma(Float32(Float32(-2.0) * Float32(pi)), Float32(pi), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(0.6666666666666666)) * u2) * u2)), 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))) < 0.0305000003

   1. Initial program 42.5%

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

   2. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + \color{blue}{1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) + \color{blue}{u1 \cdot 1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1 + \color{blue}{u1} \cdot 1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1 + u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. lower-fma.f32 98.6

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 98.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1, u1, u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 98.1%

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

   if 0.0305000003 < (*.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 92.9%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 92.8

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in u2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 N/A

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

   11. Applied rewrites 96.1%

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

Alternative 5: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.030500000342726707:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \pi, \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.6666666666666666\right) \cdot u2\right) \cdot u2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.030500000342726707)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (+ PI PI) u2)))
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (fma
      (* -2.0 PI)
      PI
      (* (* (* (* (* PI PI) (* PI PI)) 0.6666666666666666) u2) u2))
     (* 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))) <= 0.030500000342726707f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(fmaf((-2.0f * ((float) M_PI)), ((float) M_PI), (((((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * 0.6666666666666666f) * u2) * u2)), (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(0.030500000342726707))
		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(-u1)))) * fma(fma(Float32(Float32(-2.0) * Float32(pi)), Float32(pi), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(0.6666666666666666)) * u2) * u2)), 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))) < 0.0305000003

   1. Initial program 42.5%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 97.9

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

   4. Applied rewrites 97.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 97.9

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

   6. Applied rewrites 97.9%

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

   if 0.0305000003 < (*.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 92.9%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 92.8

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in u2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 N/A

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

   11. Applied rewrites 96.1%

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

Alternative 6: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \pi, \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.6666666666666666\right) \cdot u2\right) \cdot u2\right), u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.05000000074505806)
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (fma
      (* -2.0 PI)
      PI
      (* (* (* (* (* PI PI) (* PI PI)) 0.6666666666666666) u2) u2))
     (* u2 u2)
     1.0))
   (* (sqrt u1) (sin (* PI (fma u2 -2.0 0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.05000000074505806f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(fmaf((-2.0f * ((float) M_PI)), ((float) M_PI), (((((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * 0.6666666666666666f) * u2) * u2)), (u2 * u2), 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(fma(Float32(Float32(-2.0) * Float32(pi)), Float32(pi), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(0.6666666666666666)) * u2) * u2)), Float32(u2 * u2), Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0500000007

   1. Initial program 57.3%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.4

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 97.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 97.4

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

   8. Applied rewrites 97.4%

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

   9. Taylor expanded in u2 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 N/A

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

   11. Applied rewrites 99.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \pi, \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.6666666666666666\right) \cdot u2\right) \cdot u2\right), u2 \cdot u2, 1\right)} \]

   if 0.0500000007 < u2

   1. Initial program 57.7%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 96.8

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

   3. Applied rewrites 96.8%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-/.f32 N/A

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

      18. lift-PI.f32 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-fma.f32 97.9

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, \color{blue}{-2}, 0.5\right)\right) \]

   8. Applied rewrites 97.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)} \]

   9. Taylor expanded in u1 around 0

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

       1. mul-1-neg 76.0

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

       2. *-commutative 76.0

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

       3. fp-cancel-sign-sub-inv 76.0

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

       4. distribute-lft-neg-in 76.0

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

       5. distribute-rgt-neg-in 76.0

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

       6. metadata-eval 76.0

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

       7. *-rgt-identity 76.0

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

   11. Applied rewrites 76.0%

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

Alternative 7: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.001500000013038516)
   (* (sqrt u1) (sin (* PI (fma u2 -2.0 0.5))))
   (* (sqrt (- (log1p (- u1)))) (fma (* (* (* PI u2) -2.0) PI) 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))) <= 0.001500000013038516f) {
		tmp = sqrtf(u1) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((((((float) M_PI) * u2) * -2.0f) * ((float) M_PI)), 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(0.001500000013038516))
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(Float32(Float32(pi) * u2) * Float32(-2.0)) * Float32(pi)), 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))) < 0.00150000001

   1. Initial program 28.1%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-/.f32 N/A

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

      18. lift-PI.f32 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-fma.f32 99.1

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, \color{blue}{-2}, 0.5\right)\right) \]

   8. Applied rewrites 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)} \]

   9. Taylor expanded in u1 around 0

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

       1. mul-1-neg 95.7

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

       2. *-commutative 95.7

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

       3. fp-cancel-sign-sub-inv 95.7

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

       4. distribute-lft-neg-in 95.7

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

       5. distribute-rgt-neg-in 95.7

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

       6. metadata-eval 95.7

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

       7. *-rgt-identity 95.7

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

   11. Applied rewrites 95.7%

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

   if 0.00150000001 < (*.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 81.1%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 92.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 8: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.001500000013038516:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.001500000013038516)
   (* (cos (* u2 (+ PI PI))) (sqrt u1))
   (* (sqrt (- (log1p (- u1)))) (fma (* (* (* PI u2) -2.0) PI) 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))) <= 0.001500000013038516f) {
		tmp = cosf((u2 * (((float) M_PI) + ((float) M_PI)))) * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((((((float) M_PI) * u2) * -2.0f) * ((float) M_PI)), 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(0.001500000013038516))
		tmp = Float32(cos(Float32(u2 * Float32(Float32(pi) + Float32(pi)))) * sqrt(u1));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(Float32(Float32(pi) * u2) * Float32(-2.0)) * Float32(pi)), 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))) < 0.00150000001

   1. Initial program 28.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 26.3

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

   3. Applied rewrites 26.3%

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

   4. Taylor expanded in u1 around 0

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

      1. neg-log N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. *-rgt-identity N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

   6. Applied rewrites 95.5%

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

   if 0.00150000001 < (*.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 81.1%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 92.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 9: 88.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-in N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f32 N/A

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

6. Applied rewrites 88.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

8. Applied rewrites 88.6%

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

Alternative 10: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot u2\right) \cdot u2, -2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.054999999701976776)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (fma (* (* (* PI PI) -2.0) u2) u2 1.0))
     (* t_0 (fma (* (* (* PI PI) u2) u2) -2.0 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.054999999701976776f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * fmaf((((((float) M_PI) * ((float) M_PI)) * -2.0f) * u2), u2, 1.0f);
	} else {
		tmp = t_0 * fmaf((((((float) M_PI) * ((float) M_PI)) * u2) * u2), -2.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(t_0 * fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * u2) * u2), Float32(-2.0), 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))) < 0.0549999997

   1. Initial program 45.3%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f32 86.5

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

   9. Applied rewrites 86.5%

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

   if 0.0549999997 < (*.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 94.8%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 92.2%

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in u2 around 0

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

   9. Applied rewrites 89.4%

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

Alternative 11: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1)))))
        (t_1 (fma (* (* (* PI PI) -2.0) u2) u2 1.0)))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.054999999701976776)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1)
     (* t_0 t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = fmaf((((((float) M_PI) * ((float) M_PI)) * -2.0f) * u2), u2, 1.0f);
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.054999999701976776f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_1;
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)) * u2), u2, Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	else
		tmp = Float32(t_0 * t_1);
	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))) < 0.0549999997

   1. Initial program 45.3%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f32 86.5

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

   9. Applied rewrites 86.5%

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

   if 0.0549999997 < (*.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 94.8%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-PI.f32 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 12: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \left(\pi \cdot \pi\right) \cdot -2\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(t\_1 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(t\_1, u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (* (* PI PI) -2.0)))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.054999999701976776)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (fma (* t_1 u2) u2 1.0))
     (* t_0 (fma t_1 (* u2 u2) 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = (((float) M_PI) * ((float) M_PI)) * -2.0f;
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.054999999701976776f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * fmaf((t_1 * u2), u2, 1.0f);
	} else {
		tmp = t_0 * fmaf(t_1, (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * fma(Float32(t_1 * u2), u2, Float32(1.0)));
	else
		tmp = Float32(t_0 * fma(t_1, 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))) < 0.0549999997

   1. Initial program 45.3%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      12. lower-fma.f32 86.5

          \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

   9. Applied rewrites 86.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

   if 0.0549999997 < (*.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 94.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left({u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) \cdot {u2}^{2} + 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right), \color{blue}{{u2}^{2}}, 1\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \pi, \pi, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.6666666666666666\right) \cdot u2\right) \cdot u2\right), u2 \cdot u2, 1\right)} \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}, \color{blue}{u2} \cdot u2, 1\right) \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), u2 \cdot u2, 1\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), u2 \cdot u2, 1\right) \]

      3. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right), u2 \cdot u2, 1\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(\pi \cdot \pi\right), u2 \cdot u2, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, u2 \cdot u2, 1\right) \]

      6. lower-*.f32 89.4

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, u2 \cdot u2, 1\right) \]

   7. Applied rewrites 89.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, \color{blue}{u2} \cdot u2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.04399999976158142:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.04399999976158142)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (fma (* (* (* PI PI) -2.0) u2) u2 1.0))
   (* (sqrt (- (log1p (- u1)))) 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))) <= 0.04399999976158142f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * fmaf((((((float) M_PI) * ((float) M_PI)) * -2.0f) * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(-log1pf(-u1)) * 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(0.04399999976158142))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * 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))) < 0.0439999998

   1. Initial program 44.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-log1p.f32 N/A

          \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-neg.f32 99.0

          \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 99.0%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. pow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \pi\right)\right) + 1\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot {u2}^{2}\right) + 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot {u2}^{2} + 1\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \pi\right)\right) \cdot {u2}^{2} + 1\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {u2}^{2} + 1\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {u2}^{2} + 1\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {u2}^{2} + 1\right) \]

      12. pow2 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot u2\right) \cdot u2 + 1\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

   6. Applied rewrites 87.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      8. mul-1-neg N/A

         \[\leadsto \sqrt{\color{blue}{u1} \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot \color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      12. lower-fma.f32 86.6

          \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

   9. Applied rewrites 86.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

   if 0.0439999998 < (*.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 94.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-log1p.f32 N/A

          \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 84.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999300241470337:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.9999300241470337)
   (* (sqrt u1) (fma (* (* (* PI PI) -2.0) u2) u2 1.0))
   (* (sqrt (- (log1p (- u1)))) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.9999300241470337f) {
		tmp = sqrtf(u1) * fmaf((((((float) M_PI) * ((float) M_PI)) * -2.0f) * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(-log1pf(-u1)) * 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(0.9999300241470337))
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * 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)) < 0.999930024

   1. Initial program 58.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-log1p.f32 N/A

          \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-neg.f32 97.9

          \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 97.9%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. pow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \pi\right)\right) + 1\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot {u2}^{2}\right) + 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot {u2}^{2} + 1\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \pi\right)\right) \cdot {u2}^{2} + 1\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {u2}^{2} + 1\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {u2}^{2} + 1\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {u2}^{2} + 1\right) \]

      12. pow2 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot u2\right) \cdot u2 + 1\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

   6. Applied rewrites 60.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      2. *-commutative 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      3. fp-cancel-sign-sub-inv 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      4. distribute-lft-neg-in 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      5. distribute-rgt-neg-in 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      6. metadata-eval 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      7. *-rgt-identity 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      8. mul-1-neg 51.4

         \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

   9. Applied rewrites 51.4%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

   if 0.999930024 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-log1p.f32 N/A

          \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-neg.f32 99.5

          \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 96.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999300241470337:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.9999300241470337)
   (* (sqrt u1) (fma (* (* (* PI u2) -2.0) PI) u2 1.0))
   (* (sqrt (- (log1p (- u1)))) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.9999300241470337f) {
		tmp = sqrtf(u1) * fmaf((((((float) M_PI) * u2) * -2.0f) * ((float) M_PI)), u2, 1.0f);
	} else {
		tmp = sqrtf(-log1pf(-u1)) * 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(0.9999300241470337))
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(Float32(Float32(pi) * u2) * Float32(-2.0)) * Float32(pi)), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * 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)) < 0.999930024

   1. Initial program 58.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-log1p.f32 N/A

          \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-neg.f32 97.9

          \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 97.9%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. pow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \pi\right)\right) + 1\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot {u2}^{2}\right) + 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot {u2}^{2} + 1\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \pi\right)\right) \cdot {u2}^{2} + 1\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {u2}^{2} + 1\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {u2}^{2} + 1\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {u2}^{2} + 1\right) \]

      12. pow2 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot u2\right) \cdot u2 + 1\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

   6. Applied rewrites 60.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot u2, u2, 1\right) \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(\left(\pi \cdot \pi\right) \cdot u2\right), u2, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), u2, 1\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot \left(\pi \cdot \pi\right), u2, 1\right) \]

      7. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \pi\right), u2, 1\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), u2, 1\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), u2, 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(-2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      15. *-commutative N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      16. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      17. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), u2, 1\right) \]

      18. lift-PI.f32 60.6

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

   8. Applied rewrites 60.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

   9. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

       2. *-commutative 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

       3. fp-cancel-sign-sub-inv 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

       4. distribute-lft-neg-in 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

       5. distribute-rgt-neg-in 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

       6. metadata-eval 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

       7. *-rgt-identity 51.4

          \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

   11. Applied rewrites 51.4%

       \[\leadsto \sqrt{\color{blue}{u1}} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot u2\right) \cdot -2\right) \cdot \pi, u2, 1\right) \]

   if 0.999930024 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-log1p.f32 N/A

          \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. lower-neg.f32 99.5

          \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 96.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 80.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * 1.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0))
end

Derivation

1. Initial program 57.4%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \sqrt{-\log \left(1 - \color{blue}{u1 \cdot 1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{-\log \left(1 - u1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot -1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{-\log \left(1 - \color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. fp-cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + u1 \cdot -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{-\log \left(1 + \color{blue}{-1 \cdot u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. mul-1-neg N/A

      \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. lower-log1p.f32 N/A

       \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. lower-neg.f32 99.0

       \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Applied rewrites 99.0%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
5. Step-by-step derivation

6. Applied rewrites 80.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
7. Add Preprocessing

Alternative 17: 49.3% 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 57.4%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
3. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   5. lift-log.f32 N/A

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   6. lift--.f32 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

4. Applied rewrites 49.3%

   \[\leadsto \color{blue}{\sqrt{-1 \cdot \log \left(1 - u1\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   2. mul-1-neg 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   3. *-rgt-identity 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   4. metadata-eval 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   5. distribute-rgt-neg-in 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   6. distribute-lft-neg-in 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   7. fp-cancel-sign-sub-inv 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   8. *-commutative 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   9. mul-1-neg 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   11. lift--.f32 N/A

       \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   12. lift-log.f32 N/A

       \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]

   14. lift-log.f32 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]

   15. lift--.f32 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]

6. Applied rewrites 49.3%

   \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
7. Add Preprocessing

Alternative 18: 6.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log 1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf(1.0f));
}

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))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log(Float32(1.0))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log(single(1.0)));
end

Derivation

1. Initial program 57.4%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
3. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   5. lift-log.f32 N/A

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

   6. lift--.f32 49.3

      \[\leadsto \sqrt{-1 \cdot \log \left(1 - u1\right)} \]

4. Applied rewrites 49.3%

   \[\leadsto \color{blue}{\sqrt{-1 \cdot \log \left(1 - u1\right)}} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-1 \cdot \log 1} \]
6. Step-by-step derivation

7. Applied rewrites 6.6%

   \[\leadsto \sqrt{-1 \cdot \log 1} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{-1 \cdot \log 1} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log 1\right)} \]

   3. lower-neg.f32 6.6

      \[\leadsto \sqrt{-\log 1} \]

9. Applied rewrites 6.6%

   \[\leadsto \sqrt{-\log 1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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))))