Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.8% → 99.2%

Time: 5.7s

Alternatives: 16

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: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 99.2

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

7. Applied rewrites 99.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.2

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

   4. lift-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 99.2

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

9. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right)} \]
10. 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 57.8%

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

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

Alternative 3: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0026000000070780516)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* PI (fma u2 -2.0 0.5))))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (* (fma -2.0 u2 0.5) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0026000000070780516f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf((fmaf(-2.0f, u2, 0.5f) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0026000000070780516))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(fma(Float32(-2.0), u2, Float32(0.5)) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00260000001

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.1

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

   10. Applied rewrites 88.1%

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

   if 0.00260000001 < u1

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 99.2

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

   7. Applied rewrites 99.2%

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

      1. lift-log1p.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. sub-flip-reverse N/A

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

      4. lower-log.f32 N/A

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

      5. lift--.f32 57.8

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 57.8

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

      9. lift-fma.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 57.8

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

   9. Applied rewrites 57.8%

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

Alternative 4: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0026000000070780516)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* PI (fma u2 -2.0 0.5))))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0026000000070780516f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
	} 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(0.0026000000070780516))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(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 u1 < 0.00260000001

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.1

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

   10. Applied rewrites 88.1%

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

   if 0.00260000001 < u1

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 57.8

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

   3. Applied rewrites 57.8%

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

Alternative 5: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0026000000070780516)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0026000000070780516f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * cosf(((2.0f * ((float) M_PI)) * u2));
	} 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(0.0026000000070780516))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	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(0.0026000000070780516))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * cos(((single(2.0) * single(pi)) * u2));
	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 < 0.00260000001

   1. Initial program 57.8%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.0

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

   4. Applied rewrites 88.0%

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

   if 0.00260000001 < u1

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 57.8

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

   3. Applied rewrites 57.8%

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

Alternative 6: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.08500000089406967:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \pi\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

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) 0.08500000089406967)
     (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) t_1)
     (- t_0 (* (* 2.0 (* (* (* u2 u2) PI) PI)) 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) <= 0.08500000089406967f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * t_1;
	} else {
		tmp = t_0 - ((2.0f * (((u2 * u2) * ((float) M_PI)) * ((float) M_PI))) * 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(0.08500000089406967))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * t_1);
	else
		tmp = Float32(t_0 - Float32(Float32(Float32(2.0) * Float32(Float32(Float32(u2 * u2) * Float32(pi)) * Float32(pi))) * 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(0.08500000089406967))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * t_1;
	else
		tmp = t_0 - ((single(2.0) * (((u2 * u2) * single(pi)) * single(pi))) * 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))) < 0.0850000009

   1. Initial program 57.8%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.0

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

   4. Applied rewrites 88.0%

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

   if 0.0850000009 < (*.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 57.8%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.1%

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

      1. lift-+.f32 N/A

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

      2. add-flip N/A

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

      3. lower--.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-pow.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. associate-*r* N/A

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

   6. Applied rewrites 53.1%

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

Alternative 7: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \pi, \pi \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0006000000284984708)
   (* (sqrt (- (log1p (- u1)))) (sin (* 0.5 PI)))
   (* (sqrt u1) (sin (fma -2.0 (* u2 PI) (* PI 0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0006000000284984708f) {
		tmp = sqrtf(-log1pf(-u1)) * sinf((0.5f * ((float) M_PI)));
	} else {
		tmp = sqrtf(u1) * sinf(fmaf(-2.0f, (u2 * ((float) M_PI)), (((float) M_PI) * 0.5f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0006000000284984708))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(0.5) * Float32(pi))));
	else
		tmp = Float32(sqrt(u1) * sin(fma(Float32(-2.0), Float32(u2 * Float32(pi)), Float32(Float32(pi) * Float32(0.5)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.00000028e-4

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

   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 80.4

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

   8. Applied rewrites 80.4%

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

   if 6.00000028e-4 < u2

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.4%

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

Alternative 8: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.02459999918937683:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \pi, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \pi\right), t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.02459999918937683)
     (* (sqrt u1) (sin (fma -2.0 (* u2 PI) (* PI 0.5))))
     (fma (* -2.0 (* (* (* u2 u2) PI) PI)) t_0 t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.02459999918937683f) {
		tmp = sqrtf(u1) * sinf(fmaf(-2.0f, (u2 * ((float) M_PI)), (((float) M_PI) * 0.5f)));
	} else {
		tmp = fmaf((-2.0f * (((u2 * u2) * ((float) M_PI)) * ((float) M_PI))), t_0, t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.02459999918937683))
		tmp = Float32(sqrt(u1) * sin(fma(Float32(-2.0), Float32(u2 * Float32(pi)), Float32(Float32(pi) * Float32(0.5)))));
	else
		tmp = fma(Float32(Float32(-2.0) * Float32(Float32(Float32(u2 * u2) * Float32(pi)) * Float32(pi))), t_0, t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.024599999

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.4%

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

   if 0.024599999 < (*.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 57.8%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 53.1%

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

Alternative 9: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.02459999918937683:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \pi\right), t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.02459999918937683)
     (* (sqrt u1) (sin (* PI (fma u2 -2.0 0.5))))
     (fma (* -2.0 (* (* (* u2 u2) PI) PI)) t_0 t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.02459999918937683f) {
		tmp = sqrtf(u1) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
	} else {
		tmp = fmaf((-2.0f * (((u2 * u2) * ((float) M_PI)) * ((float) M_PI))), t_0, t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.02459999918937683))
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))));
	else
		tmp = fma(Float32(Float32(-2.0) * Float32(Float32(Float32(u2 * u2) * Float32(pi)) * Float32(pi))), t_0, t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.024599999

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 76.4%

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

   if 0.024599999 < (*.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 57.8%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 53.1%

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

Alternative 10: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.02459999918937683:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \pi\right), t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 PI) u2))))
   (if (<= (* t_0 t_1) 0.02459999918937683)
     (* (sqrt u1) t_1)
     (fma (* -2.0 (* (* (* u2 u2) PI) PI)) t_0 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) <= 0.02459999918937683f) {
		tmp = sqrtf(u1) * t_1;
	} else {
		tmp = fmaf((-2.0f * (((u2 * u2) * ((float) M_PI)) * ((float) M_PI))), t_0, 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(0.02459999918937683))
		tmp = Float32(sqrt(u1) * t_1);
	else
		tmp = fma(Float32(Float32(-2.0) * Float32(Float32(Float32(u2 * u2) * Float32(pi)) * Float32(pi))), t_0, t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.024599999

   1. Initial program 57.8%

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

   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 76.3%

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

   if 0.024599999 < (*.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 57.8%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 53.1%

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

Alternative 11: 53.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t\_0 - \left(2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \pi\right)\right) \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (- t_0 (* (* 2.0 (* (* (* u2 u2) PI) PI)) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	return t_0 - ((2.0f * (((u2 * u2) * ((float) M_PI)) * ((float) M_PI))) * t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	return Float32(t_0 - Float32(Float32(Float32(2.0) * Float32(Float32(Float32(u2 * u2) * Float32(pi)) * Float32(pi))) * t_0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	tmp = t_0 - ((single(2.0) * (((u2 * u2) * single(pi)) * single(pi))) * t_0);
end

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-neg.f32 N/A

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

4. Applied rewrites 53.1%

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

   1. lift-+.f32 N/A

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

   2. add-flip N/A

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

   3. lower--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-pow.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*r* N/A

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

6. Applied rewrites 53.1%

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

Alternative 12: 53.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (fma (* -2.0 (* (* (* u2 u2) PI) PI)) t_0 t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	return fmaf((-2.0f * (((u2 * u2) * ((float) M_PI)) * ((float) M_PI))), t_0, t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	return fma(Float32(Float32(-2.0) * Float32(Float32(Float32(u2 * u2) * Float32(pi)) * Float32(pi))), t_0, t_0)
end

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-neg.f32 N/A

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

4. Applied rewrites 53.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

6. Applied rewrites 53.1%

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

Alternative 13: 53.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (fma (* -2.0 (* (* PI PI) t_0)) (* u2 u2) t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	return fmaf((-2.0f * ((((float) M_PI) * ((float) M_PI)) * t_0)), (u2 * u2), t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	return fma(Float32(Float32(-2.0) * Float32(Float32(Float32(pi) * Float32(pi)) * t_0)), Float32(u2 * u2), t_0)
end

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-neg.f32 N/A

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

4. Applied rewrites 53.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f32 N/A

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

6. Applied rewrites 53.1%

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

Alternative 14: 49.8% 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 57.8%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}} \]
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 49.8

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

4. Applied rewrites 49.8%

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

Alternative 15: 36.4% accurate, 4.7× 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(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 57.8%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 55.5

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

3. Applied rewrites 55.5%

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

4. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-log.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 48.0

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

6. Applied rewrites 48.0%

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

7. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\log \left(1 + u1\right)} \]
8. Step-by-step derivation

   1. lower-+.f32 36.4

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

9. Applied rewrites 36.4%

   \[\leadsto \sqrt{\log \left(1 + u1\right)} \]
10. Add Preprocessing

Alternative 16: 6.6% accurate, 6.0× 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(log(Float32(1.0)))
end

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 55.5

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

3. Applied rewrites 55.5%

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

4. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-log.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 48.0

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

6. Applied rewrites 48.0%

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

7. Taylor expanded in u1 around 0

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

9. Applied rewrites 6.6%

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

Reproduce

herbie shell --seed 2025150 
(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))))