Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.2%

Time: 3.7s

Alternatives: 12

Speedup: 1.0×

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: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (- 0.25 u2) (+ PI PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(((0.25f - u2) * (((float) M_PI) + ((float) M_PI))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(Float32(0.25) - u2) * Float32(Float32(pi) + Float32(pi)))))
end

Derivation

1. Initial program 57.3%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.1

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

3. Applied rewrites 99.1%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

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

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

   7. lift-PI.f32 N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. distribute-lft-out N/A

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

   15. lower-*.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. count-2-rev N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

   18. lower-+.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

   19. lower-+.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

   20. lower-neg.f32 99.2

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

5. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + \frac{1}{4}\right) \cdot \left(\pi + \pi\right)\right)} \]

   3. lower-*.f32 99.2

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + 0.25\right) \cdot \left(\pi + \pi\right)\right)} \]

   4. lift-+.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\left(-u2\right) + \frac{1}{4}\right)} \cdot \left(\pi + \pi\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} + \left(-u2\right)\right)} \cdot \left(\pi + \pi\right)\right) \]

   6. lift-neg.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(u2\right)\right)}\right) \cdot \left(\pi + \pi\right)\right) \]

   7. sub-flip-reverse N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

   8. lower--.f32 99.2

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(0.25 - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

7. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.1

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

3. Applied rewrites 99.1%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 99.1

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

5. Applied rewrites 99.1%

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

Alternative 3: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.003700000001117587:\\ \;\;\;\;\sqrt{-t\_0} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.003700000001117587)
     (* (sqrt (- t_0)) (sin (* (- 0.25 u2) (+ PI PI))))
     (* (sqrt (fma (* u1 u1) 0.5 u1)) (sin (* (+ PI PI) (+ (- u2) 0.25)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.003700000001117587f) {
		tmp = sqrtf(-t_0) * sinf(((0.25f - u2) * (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf(fmaf((u1 * u1), 0.5f, u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * (-u2 + 0.25f)));
	}
	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(-0.003700000001117587))
		tmp = Float32(sqrt(Float32(-t_0)) * sin(Float32(Float32(Float32(0.25) - u2) * Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(sqrt(fma(Float32(u1 * u1), Float32(0.5), u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * Float32(Float32(-u2) + Float32(0.25)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0037

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + \frac{1}{4}\right) \cdot \left(\pi + \pi\right)\right)} \]

      3. lower-*.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + 0.25\right) \cdot \left(\pi + \pi\right)\right)} \]

      4. lift-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\left(-u2\right) + \frac{1}{4}\right)} \cdot \left(\pi + \pi\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} + \left(-u2\right)\right)} \cdot \left(\pi + \pi\right)\right) \]

      6. lift-neg.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(u2\right)\right)}\right) \cdot \left(\pi + \pi\right)\right) \]

      7. sub-flip-reverse N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

      8. lower--.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(0.25 - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

   7. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)} \]
   8. Step-by-step derivation

      1. lift-log1p.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(\frac{1}{4} - u2\right) \cdot \left(\pi + \pi\right)\right) \]

      2. lower-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(\frac{1}{4} - u2\right) \cdot \left(\pi + \pi\right)\right) \]

      3. lift-neg.f32 N/A

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

      4. sub-flip-reverse N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(\frac{1}{4} - u2\right) \cdot \left(\pi + \pi\right)\right) \]

      5. lower--.f32 57.3

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right) \]

   9. Applied rewrites 57.3%

      \[\leadsto \sqrt{\color{blue}{-\log \left(1 - u1\right)}} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right) \]

   if -0.0037 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. lower-unsound-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-unsound-*.f32 88.5

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

   10. Applied rewrites 88.5%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{0.5}, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.003700000001117587:\\ \;\;\;\;\sqrt{-t\_0} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.003700000001117587)
     (* (sqrt (- t_0)) (cos (* (+ PI PI) u2)))
     (* (sqrt (fma (* u1 u1) 0.5 u1)) (sin (* (+ PI PI) (+ (- u2) 0.25)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.003700000001117587f) {
		tmp = sqrtf(-t_0) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(fmaf((u1 * u1), 0.5f, u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * (-u2 + 0.25f)));
	}
	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(-0.003700000001117587))
		tmp = Float32(sqrt(Float32(-t_0)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(fma(Float32(u1 * u1), Float32(0.5), u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * Float32(Float32(-u2) + Float32(0.25)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0037

   1. Initial program 57.3%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.3

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

   3. Applied rewrites 57.3%

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

   if -0.0037 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. lower-unsound-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-unsound-*.f32 88.5

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

   10. Applied rewrites 88.5%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{0.5}, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.003700000001117587:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.003700000001117587)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* (- 0.25 u2) (+ PI PI))))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.003700000001117587f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(((0.25f - u2) * (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.003700000001117587))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(Float32(0.25) - u2) * Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.003700000001117587))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * sin(((single(0.25) - u2) * (single(pi) + single(pi))));
	else
		tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(pi) + single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0037

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + \frac{1}{4}\right) \cdot \left(\pi + \pi\right)\right)} \]

      3. lower-*.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + 0.25\right) \cdot \left(\pi + \pi\right)\right)} \]

      4. lift-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\left(-u2\right) + \frac{1}{4}\right)} \cdot \left(\pi + \pi\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} + \left(-u2\right)\right)} \cdot \left(\pi + \pi\right)\right) \]

      6. lift-neg.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(u2\right)\right)}\right) \cdot \left(\pi + \pi\right)\right) \]

      7. sub-flip-reverse N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

      8. lower--.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(0.25 - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

   7. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)} \]

   8. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right) \]

   10. Applied rewrites 88.4%

       \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right) \]

   if 0.0037 < u1

   1. Initial program 57.3%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 6: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.003700000001117587:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (cos (* (+ PI PI) u2))))
   (if (<= t_0 -0.003700000001117587)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.003700000001117587f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003700000001117587))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0037

   1. Initial program 57.3%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.3

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

   3. Applied rewrites 57.3%

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

   if -0.0037 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{\left(1 + 0.5 \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 88.4

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

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \mathsf{Rewrite=>}\left(lift-sin.f32, \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(lift-*.f32, \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f32, \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\left(-u2\right) \cdot \left(\pi + \pi\right) + \frac{1}{4} \cdot \left(\pi + \pi\right)\right)\right) \]

   10. Applied rewrites 88.3%

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

Alternative 7: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00014000000373926014:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00014000000373926014)
   (* (sqrt (- (log1p (- u1)))) (sin (* 0.5 PI)))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.00014000000373926014f) {
		tmp = sqrtf(-log1pf(-u1)) * sinf((0.5f * ((float) M_PI)));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.00014000000373926014))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(0.5) * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 1.40000004e-4

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 80.2

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

   8. Applied rewrites 80.2%

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

   if 1.40000004e-4 < u2

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{\left(1 + 0.5 \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 88.4

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

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \mathsf{Rewrite=>}\left(lift-sin.f32, \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(lift-*.f32, \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f32, \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\left(-u2\right) \cdot \left(\pi + \pi\right) + \frac{1}{4} \cdot \left(\pi + \pi\right)\right)\right) \]

   10. Applied rewrites 88.3%

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

Alternative 8: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 88.4

         \[\leadsto \sqrt{\left(1 + 0.5 \cdot u1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right) \]

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 88.4

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

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \mathsf{Rewrite=>}\left(lift-sin.f32, \sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(lift-*.f32, \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f32, \left(\left(-u2\right) + \frac{1}{4}\right)\right)\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\left(-u2\right) \cdot \left(\pi + \pi\right) + \frac{1}{4} \cdot \left(\pi + \pi\right)\right)\right) \]

   10. Applied rewrites 88.3%

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

   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 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower--.f32 49.3

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

   4. Applied rewrites 49.3%

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

Alternative 9: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.013199999928474426:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.013199999928474426)
     (* (sqrt u1) (sin (* (- 0.25 u2) (+ PI PI))))
     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))) <= 0.013199999928474426f) {
		tmp = sqrtf(u1) * sinf(((0.25f - u2) * (((float) M_PI) + ((float) M_PI))));
	} 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(0.013199999928474426))
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(0.25) - u2) * Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	tmp = single(0.0);
	if ((t_0 * cos(((single(2.0) * single(pi)) * u2))) <= single(0.013199999928474426))
		tmp = sqrt(u1) * sin(((single(0.25) - u2) * (single(pi) + single(pi))));
	else
		tmp = t_0;
	end
	tmp_2 = 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))) < 0.0132

   1. Initial program 57.3%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-PI.f32 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      18. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot \left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)\right) \]

      19. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u2\right)\right) + \frac{1}{4}\right)}\right) \]

      20. lower-neg.f32 99.2

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot \left(\color{blue}{\left(-u2\right)} + 0.25\right)\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + 0.25\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot \left(\left(-u2\right) + \frac{1}{4}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + \frac{1}{4}\right) \cdot \left(\pi + \pi\right)\right)} \]

      3. lower-*.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\left(-u2\right) + 0.25\right) \cdot \left(\pi + \pi\right)\right)} \]

      4. lift-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\left(-u2\right) + \frac{1}{4}\right)} \cdot \left(\pi + \pi\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} + \left(-u2\right)\right)} \cdot \left(\pi + \pi\right)\right) \]

      6. lift-neg.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(u2\right)\right)}\right) \cdot \left(\pi + \pi\right)\right) \]

      7. sub-flip-reverse N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\frac{1}{4} - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

      8. lower--.f32 99.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(0.25 - u2\right)} \cdot \left(\pi + \pi\right)\right) \]

   7. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right)} \]

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 76.9%

       \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(\left(0.25 - u2\right) \cdot \left(\pi + \pi\right)\right) \]

   if 0.0132 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower--.f32 49.3

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

   4. Applied rewrites 49.3%

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

Alternative 10: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.013199999928474426:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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) 0.013199999928474426) (* (sqrt u1) t_1) t_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) <= 0.013199999928474426f) {
		tmp = sqrtf(u1) * t_1;
	} else {
		tmp = t_0;
	}
	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(0.013199999928474426))
		tmp = Float32(sqrt(u1) * t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	t_1 = cos(((single(2.0) * single(pi)) * u2));
	tmp = single(0.0);
	if ((t_0 * t_1) <= single(0.013199999928474426))
		tmp = sqrt(u1) * t_1;
	else
		tmp = t_0;
	end
	tmp_2 = 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))) < 0.0132

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.9%

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

   if 0.0132 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower--.f32 49.3

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

   4. Applied rewrites 49.3%

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

Alternative 11: 49.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. lower--.f32 49.3

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

4. Applied rewrites 49.3%

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

Alternative 12: 6.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log 1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. lower--.f32 49.3

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

4. Applied rewrites 49.3%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\log 1} \]
6. Step-by-step derivation

7. Applied rewrites 6.6%

   \[\leadsto \sqrt{-\log 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(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))))