Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.4%

Time: 15.2s

Alternatives: 15

Speedup: 9.6×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.1

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

4. Applied egg-rr 98.1%

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

Alternative 2: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.002199999988079071)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sqrt (* u1 (fma u1 (fma u1 0.3333333333333333 0.5) 1.0)))
    (sin (* PI (+ u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.002199999988079071f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sqrtf((u1 * fmaf(u1, fmaf(u1, 0.3333333333333333f, 0.5f), 1.0f))) * sinf((((float) M_PI) * (u2 + u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 * fma(u1, fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)))) * sin(Float32(Float32(pi) * Float32(u2 + u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0022

   1. Initial program 57.1%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.6

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 98.1

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

   7. Simplified 98.1%

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

   if 0.0022 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 91.5

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

   5. Simplified 91.5%

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

      1. *-lowering-*.f32 N/A

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

      2. sqrt-lowering-sqrt.f32 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. associate-*l* N/A

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

      7. count-2 N/A

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

      8. sin-lowering-sin.f32 N/A

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

      9. distribute-lft-out N/A

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

      10. *-lowering-*.f32 N/A

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

      11. PI-lowering-PI.f32 N/A

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

      12. +-lowering-+.f32 91.5

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

   7. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.004000000189989805)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt (fma (* u1 u1) 0.5 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.004000000189989805f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(fmaf((u1 * u1), 0.5f, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.004000000189989805))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(fma(Float32(u1 * u1), Float32(0.5), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00400000019

   1. Initial program 57.6%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 97.7

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

   7. Simplified 97.7%

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

   if 0.00400000019 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f32 89.3

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

   5. Simplified 89.3%

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. *-lowering-*.f32 89.4

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

   7. Applied egg-rr 89.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.004000000189989805)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sin (* PI (+ u2 u2))) (sqrt (* u1 (fma u1 0.5 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.004000000189989805f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf((u1 * fmaf(u1, 0.5f, 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.004000000189989805))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00400000019

   1. Initial program 57.6%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 97.7

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

   7. Simplified 97.7%

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

   if 0.00400000019 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f32 89.3

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

   5. Simplified 89.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-lowering-+.f32 N/A

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

      7. PI-lowering-PI.f32 89.3

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

   7. Applied egg-rr 89.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 93.9

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

5. Simplified 93.9%

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

6. Final simplification 93.9%

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

Alternative 6: 90.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.004000000189989805)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.004000000189989805f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.004000000189989805))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00400000019

   1. Initial program 57.6%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 97.7

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

   7. Simplified 97.7%

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

   if 0.00400000019 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 80.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ t_1 := \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}\\ \mathbf{if}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(2, \pi \cdot t\_1, -1.3333333333333333 \cdot \left(t\_1 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2))
        (t_1 (sqrt (* u1 (fma u1 (fma u1 0.3333333333333333 0.5) 1.0)))))
   (if (<= t_0 0.10000000149011612)
     (*
      u2
      (fma
       2.0
       (* PI t_1)
       (* -1.3333333333333333 (* t_1 (* (* u2 u2) (* PI (* PI PI)))))))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float t_1 = sqrtf((u1 * fmaf(u1, fmaf(u1, 0.3333333333333333f, 0.5f), 1.0f)));
	float tmp;
	if (t_0 <= 0.10000000149011612f) {
		tmp = u2 * fmaf(2.0f, (((float) M_PI) * t_1), (-1.3333333333333333f * (t_1 * ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	t_1 = sqrt(Float32(u1 * fma(u1, fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.10000000149011612))
		tmp = Float32(u2 * fma(Float32(2.0), Float32(Float32(pi) * t_1), Float32(Float32(-1.3333333333333333) * Float32(t_1 * Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.100000001

   1. Initial program 56.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 92.8%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}, -1.3333333333333333 \cdot \left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)} \]

   if 0.100000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 78.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.10000000149011612:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}, -1.3333333333333333 \cdot \left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}\\ u2 \cdot \mathsf{fma}\left(2, \pi \cdot t\_0, -1.3333333333333333 \cdot \left(t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (* u1 (fma u1 (fma u1 0.3333333333333333 0.5) 1.0)))))
   (*
    u2
    (fma
     2.0
     (* PI t_0)
     (* -1.3333333333333333 (* t_0 (* (* u2 u2) (* PI (* PI PI)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 * fmaf(u1, fmaf(u1, 0.3333333333333333f, 0.5f), 1.0f)));
	return u2 * fmaf(2.0f, (((float) M_PI) * t_0), (-1.3333333333333333f * (t_0 * ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 * fma(u1, fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0))))
	return Float32(u2 * fma(Float32(2.0), Float32(Float32(pi) * t_0), Float32(Float32(-1.3333333333333333) * Float32(t_0 * Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 92.2

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

5. Simplified 92.2%

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

6. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

8. Simplified 83.1%

   \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}, -1.3333333333333333 \cdot \left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 9: 83.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \mathbf{if}\;u1 \leq 0.0008200000156648457:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(2, \pi \cdot t\_0, -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(t\_0 \cdot \left(u2 \cdot u2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (* u1 (fma u1 0.5 1.0)))))
   (if (<= u1 0.0008200000156648457)
     (*
      u2
      (fma
       2.0
       (* PI t_0)
       (* -1.3333333333333333 (* (* PI (* PI PI)) (* t_0 (* u2 u2))))))
     (*
      (sqrt (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0)))
      (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 * fmaf(u1, 0.5f, 1.0f)));
	float tmp;
	if (u1 <= 0.0008200000156648457f) {
		tmp = u2 * fmaf(2.0f, (((float) M_PI) * t_0), (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (t_0 * (u2 * u2)))));
	} else {
		tmp = sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f))) * (2.0f * (((float) M_PI) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0008200000156648457))
		tmp = Float32(u2 * fma(Float32(2.0), Float32(Float32(pi) * t_0), Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(t_0 * Float32(u2 * u2))))));
	else
		tmp = Float32(sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 8.20000016e-4

   1. Initial program 42.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f32 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. sqrt-lowering-sqrt.f32 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 N/A

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

      12. *-lowering-*.f32 N/A

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

   8. Simplified 87.5%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}, -1.3333333333333333 \cdot \left(\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)} \]

   if 8.20000016e-4 < u1

   1. Initial program 93.1%

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

   4. Applied egg-rr 89.9%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 77.6

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

   7. Simplified 77.6%

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

   8. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 74.3

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

   10. Simplified 74.3%

       \[\leadsto \sqrt{\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.0008200000156648457:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}, -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \left(u2 \cdot u2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(\sqrt{u1}, 2 \cdot \pi, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.000699999975040555)
   (*
    (sqrt (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0)))
    (* 2.0 (* PI u2)))
   (*
    u2
    (fma
     (sqrt u1)
     (* 2.0 PI)
     (* (sqrt u1) (* (* PI PI) (* PI (* -1.3333333333333333 (* u2 u2)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.000699999975040555f) {
		tmp = sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f))) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = u2 * fmaf(sqrtf(u1), (2.0f * ((float) M_PI)), (sqrtf(u1) * ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * (-1.3333333333333333f * (u2 * u2))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.000699999975040555))
		tmp = Float32(sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(u2 * fma(sqrt(u1), Float32(Float32(2.0) * Float32(pi)), Float32(sqrt(u1) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.99999975e-4

   1. Initial program 57.6%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 53.9

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

   7. Simplified 53.9%

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

   8. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 93.8

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

   10. Simplified 93.8%

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

   if 6.99999975e-4 < u2

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 80.6%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 57.9%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(\sqrt{u1}, 2 \cdot \pi, \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(\sqrt{u1}, 2 \cdot \pi, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.000699999975040555)
   (*
    (sqrt (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0)))
    (* 2.0 (* PI u2)))
   (*
    (sqrt u1)
    (*
     u2
     (fma -1.3333333333333333 (* (* PI PI) (* PI (* u2 u2))) (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.000699999975040555f) {
		tmp = sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f))) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(-1.3333333333333333f, ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * (u2 * u2))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.000699999975040555))
		tmp = Float32(sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.99999975e-4

   1. Initial program 57.6%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 53.9

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

   7. Simplified 53.9%

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

   8. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 93.8

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

   10. Simplified 93.8%

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

   if 6.99999975e-4 < u2

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 80.6%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 57.8%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.2% accurate, 4.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f))) * (2.0f * (((float) M_PI) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)))
end

Derivation

1. Initial program 56.6%

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

4. Applied egg-rr 52.6%

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

5. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. PI-lowering-PI.f32 47.1

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

7. Simplified 47.1%

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

8. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 78.4

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

10. Simplified 78.4%

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

11. Final simplification 78.4%

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

Alternative 13: 77.0% accurate, 5.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (2.0f * (((float) M_PI) * u2)) * sqrtf((u1 * fmaf(u1, fmaf(u1, 0.3333333333333333f, 0.5f), 1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(u1 * fma(u1, fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)))))
end

Derivation

1. Initial program 56.6%

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

4. Applied egg-rr 52.6%

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

5. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. PI-lowering-PI.f32 47.1

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

7. Simplified 47.1%

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

8. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 77.0

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

10. Simplified 77.0%

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

11. Final simplification 77.0%

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

Alternative 14: 74.5% accurate, 6.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (2.0f * (((float) M_PI) * u2)) * sqrtf((u1 * fmaf(u1, 0.5f, 1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0)))))
end

Derivation

1. Initial program 56.6%

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

4. Applied egg-rr 52.6%

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

5. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. PI-lowering-PI.f32 47.1

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

7. Simplified 47.1%

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

8. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 74.4

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

10. Simplified 74.4%

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

11. Final simplification 74.4%

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

Alternative 15: 66.3% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(pi) * (u2 + u2)) * sqrt(u1);
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in u1 around 0

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

5. Simplified 78.4%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. sqrt-lowering-sqrt.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. PI-lowering-PI.f32 66.8

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

8. Simplified 66.8%

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

   1. *-commutative N/A

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

   2. count-2 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. distribute-lft-out N/A

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

   6. *-lowering-*.f32 N/A

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

   7. PI-lowering-PI.f32 N/A

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

   8. +-lowering-+.f32 N/A

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

   9. sqrt-lowering-sqrt.f32 66.8

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

10. Applied egg-rr 66.8%

    \[\leadsto \color{blue}{\left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))