Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.0%

Time: 5.8s

Alternatives: 21

Speedup: 18.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% 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 54.6%

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

   1. lift-cos.f32 N/A

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

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

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

   3. lower-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lift-PI.f32 54.6

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

4. Applied rewrites 54.6%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

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

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

   7. mul-1-neg N/A

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

   8. lower-log1p.f32 N/A

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

   9. lower-neg.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-sin.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. count-2-rev N/A

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

   9. *-commutative N/A

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

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

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

   11. count-2-rev N/A

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

   12. lift-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

8. Applied rewrites 99.0%

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

Alternative 2: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.003000000026077032:\\ \;\;\;\;\sqrt{-t\_0} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(0.5 \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.003000000026077032)
     (* (sqrt (- t_0)) (cos (* (+ PI PI) u2)))
     (* (sqrt (fma u1 1.0 (* u1 (* 0.5 u1)))) (cos (* (* 2.0 PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.003000000026077032f) {
		tmp = sqrtf(-t_0) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (0.5f * u1)))) * cosf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003000000026077032))
		tmp = Float32(sqrt(Float32(-t_0)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(Float32(0.5) * u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 94.7

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

   4. Applied rewrites 94.7%

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

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

   1. Initial program 43.4%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 99.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 98.5%

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

Alternative 3: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.003000000026077032:\\ \;\;\;\;\sqrt{-t\_0} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.003000000026077032)
     (* (sqrt (- t_0)) (cos (* (+ PI PI) u2)))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (* 2.0 PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.003000000026077032f) {
		tmp = sqrtf(-t_0) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003000000026077032))
		tmp = Float32(sqrt(Float32(-t_0)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 94.7

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

   4. Applied rewrites 94.7%

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

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

   1. Initial program 43.4%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. 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 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 4: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 PI) u2))))
   (if (<= (* t_0 t_1) 0.09000000357627869)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1)
     (*
      t_0
      (fma
       (fma
        (* 0.6666666666666666 (* u2 u2))
        (* (* PI PI) (* PI PI))
        (* (* PI PI) -2.0))
       (* 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 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if ((t_0 * t_1) <= 0.09000000357627869f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_1;
	} else {
		tmp = t_0 * fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (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 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.09000000357627869))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	else
		tmp = Float32(t_0 * fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0900000036

   1. Initial program 45.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. 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 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.0900000036 < (*.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 95.6%

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

   3. Taylor expanded in u2 around 0

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

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

   5. Applied rewrites 91.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ t_2 := t\_1 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_2 \leq 0.00031999999191612005:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{elif}\;t\_2 \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (fma
          (fma
           (* 0.6666666666666666 (* u2 u2))
           (* (* PI PI) (* PI PI))
           (* (* PI PI) -2.0))
          (* u2 u2)
          1.0))
        (t_1 (sqrt (- (log (- 1.0 u1)))))
        (t_2 (* t_1 (cos (* (* 2.0 PI) u2)))))
   (if (<= t_2 0.00031999999191612005)
     (* (sqrt u1) (cos (* (+ PI PI) u2)))
     (if (<= t_2 0.054999999701976776)
       (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) t_0)
       (* t_1 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	float t_1 = sqrtf(-logf((1.0f - u1)));
	float t_2 = t_1 * cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_2 <= 0.00031999999191612005f) {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else if (t_2 <= 0.054999999701976776f) {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0))
	t_1 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_2 = Float32(t_1 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
	tmp = Float32(0.0)
	if (t_2 <= Float32(0.00031999999191612005))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	elseif (t_2 <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * t_0);
	else
		tmp = Float32(t_1 * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 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))) < 3.19999992e-4

   1. Initial program 21.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 21.7

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

   4. Applied rewrites 21.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 20.9

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

   6. Applied rewrites 20.9%

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

   7. Taylor expanded in u1 around 0

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

      1. lift-sqrt.f32 94.5

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

   9. Applied rewrites 94.5%

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

   if 3.19999992e-4 < (*.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 61.4%

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

   3. Taylor expanded in u2 around 0

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

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 97.0

         \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   8. Applied rewrites 97.0%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   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. Add Preprocessing

   3. Taylor expanded in u2 around 0

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

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

   5. Applied rewrites 90.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.00031999999191612005:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{elif}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ t_2 := \left(\pi \cdot \pi\right) \cdot -2\\ \mathbf{if}\;t\_1 \leq 0.00031999999191612005:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{elif}\;t\_1 \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), t\_2\right), u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, t\_2\right) \cdot u2, u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1)))))
        (t_1 (* t_0 (cos (* (* 2.0 PI) u2))))
        (t_2 (* (* PI PI) -2.0)))
   (if (<= t_1 0.00031999999191612005)
     (* (sqrt u1) (cos (* (+ PI PI) u2)))
     (if (<= t_1 0.054999999701976776)
       (*
        (sqrt (- (* (- (* -0.5 u1) 1.0) u1)))
        (fma
         (fma (* 0.6666666666666666 (* u2 u2)) (* (* PI PI) (* PI PI)) t_2)
         (* u2 u2)
         1.0))
       (*
        t_0
        (fma
         (*
          (fma (* (* (* (* PI PI) PI) PI) (* u2 u2)) 0.6666666666666666 t_2)
          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 = t_0 * cosf(((2.0f * ((float) M_PI)) * u2));
	float t_2 = (((float) M_PI) * ((float) M_PI)) * -2.0f;
	float tmp;
	if (t_1 <= 0.00031999999191612005f) {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else if (t_1 <= 0.054999999701976776f) {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), t_2), (u2 * u2), 1.0f);
	} else {
		tmp = t_0 * fmaf((fmaf(((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI)) * (u2 * u2)), 0.6666666666666666f, t_2) * 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(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
	t_2 = Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.00031999999191612005))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	elseif (t_1 <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), t_2), Float32(u2 * u2), Float32(1.0)));
	else
		tmp = Float32(t_0 * fma(Float32(fma(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)) * Float32(u2 * u2)), Float32(0.6666666666666666), t_2) * u2), u2, Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 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))) < 3.19999992e-4

   1. Initial program 21.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 21.7

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

   4. Applied rewrites 21.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 20.9

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

   6. Applied rewrites 20.9%

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

   7. Taylor expanded in u1 around 0

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

      1. lift-sqrt.f32 94.5

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

   9. Applied rewrites 94.5%

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

   if 3.19999992e-4 < (*.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 61.4%

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

   3. Taylor expanded in u2 around 0

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

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 97.0

         \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   8. Applied rewrites 97.0%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   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. Add Preprocessing

   3. Taylor expanded in u2 around 0

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

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.00031999999191612005:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{elif}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right) \cdot u2, u2, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0215000007301569:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0215000007301569)
   (*
    (sqrt
     (fma u1 1.0 (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
    (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0))
   (* (sqrt u1) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0215000007301569f) {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1)))) * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0215000007301569))
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1)))) * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0215000007

   1. Initial program 54.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 95.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 95.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 95.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 94.4

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 94.4%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)} \]

   if 0.0215000007 < u2

   1. Initial program 53.8%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 53.8

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

   4. Applied rewrites 53.8%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 52.7

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

   6. Applied rewrites 52.7%

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

   7. Taylor expanded in u1 around 0

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

      1. lift-sqrt.f32 77.1

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

   9. Applied rewrites 77.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0215000007301569:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.19499999284744263:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot \left(u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_0\right)\right), -2, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.19499999284744263)
     (*
      (sqrt
       (fma
        u1
        1.0
        (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
      (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0))
     (fma (* u2 (* u2 (* (* PI PI) t_0))) -2.0 t_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))) <= 0.19499999284744263f) {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1)))) * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = fmaf((u2 * (u2 * ((((float) M_PI) * ((float) M_PI)) * t_0))), -2.0f, t_0);
	}
	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(0.19499999284744263))
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1)))) * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = fma(Float32(u2 * Float32(u2 * Float32(Float32(Float32(pi) * Float32(pi)) * t_0))), Float32(-2.0), t_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))) < 0.194999993

   1. Initial program 49.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 98.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 87.3

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 87.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)} \]

   if 0.194999993 < (*.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 97.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 67.8

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 67.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

   10. Applied rewrites 90.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.19499999284744263:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot \left(u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right), -2, \sqrt{-\log \left(1 - u1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\left(-u1\right)}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot \left(u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_0\right)\right), -2, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.10999999940395355)
     (fma
      (* (* u2 u2) (* (* PI PI) (sqrt (- (- u1)))))
      -2.0
      (sqrt (+ u1 (* (* (fma 0.3333333333333333 u1 0.5) u1) u1))))
     (fma (* u2 (* u2 (* (* PI PI) t_0))) -2.0 t_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))) <= 0.10999999940395355f) {
		tmp = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(-(-u1)))), -2.0f, sqrtf((u1 + ((fmaf(0.3333333333333333f, u1, 0.5f) * u1) * u1))));
	} else {
		tmp = fmaf((u2 * (u2 * ((((float) M_PI) * ((float) M_PI)) * t_0))), -2.0f, t_0);
	}
	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(0.10999999940395355))
		tmp = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(-Float32(-u1))))), Float32(-2.0), sqrt(Float32(u1 + Float32(Float32(fma(Float32(0.3333333333333333), u1, Float32(0.5)) * u1) * u1))));
	else
		tmp = fma(Float32(u2 * Float32(u2 * Float32(Float32(Float32(pi) * Float32(pi)) * t_0))), Float32(-2.0), t_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))) < 0.109999999

   1. Initial program 46.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 84.6

         \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\right) \]

   8. Applied rewrites 84.6%

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lift-fma.f32 84.8

          \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   10. Applied rewrites 84.8%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   11. Taylor expanded in u1 around 0

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-1 \cdot u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(\frac{1}{3}, u1, \frac{1}{2}\right) \cdot u1\right) \cdot u1}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lift-neg.f32 87.5

          \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   13. Applied rewrites 87.5%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   if 0.109999999 < (*.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 96.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 75.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 75.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 75.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

   10. Applied rewrites 88.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\left(-u1\right)}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot \left(u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right), -2, \sqrt{-\log \left(1 - u1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\left(-u1\right)}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \left(u2 \cdot u2\right)\right) \cdot -2, t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.10999999940395355)
     (fma
      (* (* u2 u2) (* (* PI PI) (sqrt (- (- u1)))))
      -2.0
      (sqrt (+ u1 (* (* (fma 0.3333333333333333 u1 0.5) u1) u1))))
     (fma (* (* (* PI PI) (* u2 u2)) -2.0) t_0 t_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))) <= 0.10999999940395355f) {
		tmp = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(-(-u1)))), -2.0f, sqrtf((u1 + ((fmaf(0.3333333333333333f, u1, 0.5f) * u1) * u1))));
	} else {
		tmp = fmaf((((((float) M_PI) * ((float) M_PI)) * (u2 * u2)) * -2.0f), t_0, t_0);
	}
	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(0.10999999940395355))
		tmp = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(-Float32(-u1))))), Float32(-2.0), sqrt(Float32(u1 + Float32(Float32(fma(Float32(0.3333333333333333), u1, Float32(0.5)) * u1) * u1))));
	else
		tmp = fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(u2 * u2)) * Float32(-2.0)), t_0, t_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))) < 0.109999999

   1. Initial program 46.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 84.6

         \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\right) \]

   8. Applied rewrites 84.6%

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lift-fma.f32 84.8

          \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   10. Applied rewrites 84.8%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   11. Taylor expanded in u1 around 0

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-1 \cdot u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(\frac{1}{3}, u1, \frac{1}{2}\right) \cdot u1\right) \cdot u1}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lift-neg.f32 87.5

          \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   13. Applied rewrites 87.5%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   if 0.109999999 < (*.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 96.2%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
   4. 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. lower-*.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. lift--.f32 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

         \[\leadsto {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{-1}{6} \cdot \sqrt{u1}\right) + \sqrt{u1} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{-1}{6} \cdot \sqrt{u1}\right) \cdot {u1}^{2} + \sqrt{u1} \cdot {\color{blue}{\left(\sqrt{-1}\right)}}^{2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{-1}{6} \cdot \sqrt{u1}, {u1}^{\color{blue}{2}}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot \sqrt{u1} + \frac{-1}{4} \cdot \sqrt{\frac{1}{u1}}, {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{-1}{4} \cdot \sqrt{\frac{1}{u1}}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{-1}{4} \cdot \sqrt{\frac{1}{u1}}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \sqrt{\frac{1}{u1}} \cdot \frac{-1}{4}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \sqrt{\frac{1}{u1}} \cdot \frac{-1}{4}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{\sqrt{1}}{\sqrt{u1}} \cdot \frac{-1}{4}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), {u1}^{2}, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), u1 \cdot u1, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), u1 \cdot u1, \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      15. sqrt-pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), u1 \cdot u1, \sqrt{u1} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), u1 \cdot u1, \sqrt{u1} \cdot {-1}^{1}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot \frac{-1}{4}\right), u1 \cdot u1, \sqrt{u1} \cdot -1\right) \]

   8. Applied rewrites 3.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \sqrt{u1}, \frac{1}{\sqrt{u1}} \cdot -0.25\right), \color{blue}{u1 \cdot u1}, \sqrt{u1} \cdot -1\right) \]

   9. Taylor expanded in u1 around 0

      \[\leadsto -1 \cdot \sqrt{u1} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lower-neg.f32 N/A

          \[\leadsto -\sqrt{u1} \]

       3. lift-sqrt.f32 3.2

          \[\leadsto -\sqrt{u1} \]

   11. Applied rewrites 3.2%

       \[\leadsto -\sqrt{u1} \]

   12. Taylor expanded in u2 around 0

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

   14. Applied rewrites 88.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\left(-u1\right)}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \left(u2 \cdot u2\right)\right) \cdot -2, \sqrt{-\log \left(1 - u1\right)}, \sqrt{-\log \left(1 - u1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\left(-u1\right)}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.10999999940395355)
     (fma
      (* (* u2 u2) (* (* PI PI) (sqrt (- (- u1)))))
      -2.0
      (sqrt (+ u1 (* (* (fma 0.3333333333333333 u1 0.5) u1) u1))))
     (* t_0 (fma (* -2.0 (* u2 u2)) (* PI PI) 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))) <= 0.10999999940395355f) {
		tmp = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(-(-u1)))), -2.0f, sqrtf((u1 + ((fmaf(0.3333333333333333f, u1, 0.5f) * u1) * u1))));
	} else {
		tmp = t_0 * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 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(0.10999999940395355))
		tmp = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(-Float32(-u1))))), Float32(-2.0), sqrt(Float32(u1 + Float32(Float32(fma(Float32(0.3333333333333333), u1, Float32(0.5)) * u1) * u1))));
	else
		tmp = Float32(t_0 * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), 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))) < 0.109999999

   1. Initial program 46.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 84.6

         \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\right) \]

   8. Applied rewrites 84.6%

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lift-fma.f32 84.8

          \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   10. Applied rewrites 84.8%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\log \left(1 - u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   11. Taylor expanded in u1 around 0

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-1 \cdot u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(\frac{1}{3}, u1, \frac{1}{2}\right) \cdot u1\right) \cdot u1}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lift-neg.f32 87.5

          \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   13. Applied rewrites 87.5%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\left(-u1\right) \cdot -1}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right) \]

   if 0.109999999 < (*.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 96.2%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   4. 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. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{-\left(-u1\right)}\right), -2, \sqrt{u1 + \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.0% accurate, 0.6× speedup

Math

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

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)
     (fma
      (* (* u2 u2) (* (* PI PI) (sqrt u1)))
      -2.0
      (sqrt (- (* (- (* -0.5 u1) 1.0) u1))))
     (* t_0 (fma (* -2.0 (* u2 u2)) (* PI PI) 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 = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(u1))), -2.0f, sqrtf(-(((-0.5f * u1) - 1.0f) * u1)));
	} else {
		tmp = t_0 * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 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 = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(u1))), Float32(-2.0), sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))));
	else
		tmp = Float32(t_0 * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), 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 44.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 41.3%

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

   9. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. lower-*.f32 86.8

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

   11. Applied rewrites 86.8%

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

   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. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   4. 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. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
  (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * 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(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 94.5

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Applied rewrites 94.5%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Taylor expanded in u2 around 0

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

   1. count-2-rev N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 84.5

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \]

8. Applied rewrites 84.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)} \]
9. Add Preprocessing

Alternative 14: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.013000000268220901:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.013000000268220901)
     (fma (* (* u2 u2) (* (* PI PI) (sqrt u1))) -2.0 (sqrt (- (- u1))))
     (* t_0 (fma (* -2.0 (* u2 u2)) (* PI PI) 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))) <= 0.013000000268220901f) {
		tmp = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(u1))), -2.0f, sqrtf(-(-u1)));
	} else {
		tmp = t_0 * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 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(0.013000000268220901))
		tmp = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(u1))), Float32(-2.0), sqrt(Float32(-Float32(-u1))));
	else
		tmp = Float32(t_0 * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), 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))) < 0.0130000003

   1. Initial program 39.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f32 81.3

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

   11. Applied rewrites 81.3%

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

   if 0.0130000003 < (*.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 90.0%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   4. 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. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 83.8

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

   5. Applied rewrites 83.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.013000000268220901:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999983012676239:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.999983012676239)
   (fma (* (* u2 u2) (* (* PI PI) (sqrt u1))) -2.0 (sqrt (- (- u1))))
   (*
    (sqrt
     (fma u1 1.0 (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
    1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.999983012676239f) {
		tmp = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(u1))), -2.0f, sqrtf(-(-u1)));
	} else {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * 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(0.999983012676239))
		tmp = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(u1))), Float32(-2.0), sqrt(Float32(-Float32(-u1))));
	else
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * 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)) < 0.999983013

   1. Initial program 53.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f32 55.3

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

   11. Applied rewrites 55.3%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 54.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 95.4

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 95.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 95.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

   10. Applied rewrites 93.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999983012676239:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999983012676239:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.999983012676239)
   (fma (* (* u2 u2) (* (* PI PI) (sqrt u1))) -2.0 (sqrt (- (- u1))))
   (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.999983012676239f) {
		tmp = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(u1))), -2.0f, sqrtf(-(-u1)));
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.999983012676239))
		tmp = fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(u1))), Float32(-2.0), sqrt(Float32(-Float32(-u1))));
	else
		tmp = sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999983013

   1. Initial program 53.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f32 55.3

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

   11. Applied rewrites 55.3%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 54.9%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
   4. 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. lower-*.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. lift--.f32 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f32 N/A

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

      8. lift-fma.f32 N/A

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

      9. lift-fma.f32 N/A

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

      10. lift-*.f32 93.2

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

   8. Applied rewrites 93.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999983012676239:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{u1}\right), -2, \sqrt{-\left(-u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.1550000011920929:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.1550000011920929)
     (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
     t_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))) <= 0.1550000011920929f) {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1));
	} else {
		tmp = t_0;
	}
	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(0.1550000011920929))
		tmp = sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1));
	else
		tmp = t_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))) < 0.155000001

   1. Initial program 47.7%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
   4. 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. lower-*.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. lift--.f32 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 78.5

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \]

   8. Applied rewrites 78.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \]

   if 0.155000001 < (*.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 96.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 72.8

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 72.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 72.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. sqrt-prod 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. log-pow-rev N/A

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

      5. inv-pow N/A

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

      6. neg-log N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. lift-neg.f32 80.6

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

   10. Applied rewrites 80.6%

       \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.1550000011920929:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.9% accurate, 0.8× speedup

Math

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

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))
     t_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));
	} else {
		tmp = t_0;
	}
	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 = sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1));
	else
		tmp = t_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 44.9%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
   4. 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. lower-*.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. lift--.f32 39.2

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

   5. Applied rewrites 39.2%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 78.5

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

   8. Applied rewrites 78.5%

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

   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. Add Preprocessing

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 79.5

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied rewrites 79.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 79.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{1}, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. sqrt-prod 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. log-pow-rev N/A

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

      5. inv-pow N/A

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

      6. neg-log N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. lift-neg.f32 78.5

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

   10. Applied rewrites 78.5%

       \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1))
end

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. 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. lower-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lift-fma.f32 N/A

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

   8. lift-fma.f32 N/A

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

   9. lift-fma.f32 N/A

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

   10. lift-*.f32 76.5

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

8. Applied rewrites 76.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

9. Final simplification 76.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 20: 72.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (* (fma 0.5 u1 1.0) u1)))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1))
end

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. 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. lower-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.9

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

8. Applied rewrites 73.9%

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

9. Final simplification 73.9%

   \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 21: 65.0% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(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(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. 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. lower-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
7. Step-by-step derivation

8. Applied rewrites 66.9%

   \[\leadsto \sqrt{u1} \]

9. Final simplification 66.9%

   \[\leadsto \sqrt{u1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))