Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 98.2%

Time: 5.0s

Alternatives: 18

Speedup: 4.1×

Specification

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

Math

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

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: 57.4% 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: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt (fma (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1) u1 u1))
    (sin (* (+ PI PI) u2)))
   (*
    (sqrt (- (log (- 1.0 u1))))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf(fmaf((fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1), u1, u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(fma(Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1), u1, u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 50.7%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.3%

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

   8. Applied rewrites 98.3%

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

   if 0.0399999991 < u1

   1. Initial program 97.4%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. sin-2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-sin.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cos.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lift-PI.f32 97.3

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

   3. Applied rewrites 97.3%

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

Alternative 2: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (+ PI PI) u2))))
   (if (<= u1 0.03999999910593033)
     (*
      (sqrt (fma (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1) u1 u1))
      t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf(fmaf((fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1), u1, u1)) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(fma(Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1), u1, u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 50.7%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.3%

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

   8. Applied rewrites 98.3%

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

   if 0.0399999991 < u1

   1. Initial program 97.4%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 97.4

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

   3. Applied rewrites 97.4%

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

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.03999999910593033)
     (* (sqrt (- t_0)) t_1)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0399999991

   1. Initial program 97.4%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 97.4

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

   3. Applied rewrites 97.4%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 50.6%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.3

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

   6. Applied rewrites 98.3%

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

Alternative 4: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.014999999664723873:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.014999999664723873)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (fma (* (fma 0.3333333333333333 u1 0.5) u1) u1 u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.014999999664723873f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(fmaf((fmaf(0.3333333333333333f, u1, 0.5f) * u1), u1, u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.014999999664723873))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(fma(Float32(fma(Float32(0.3333333333333333), u1, Float32(0.5)) * u1), u1, u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0149999997

   1. Initial program 96.6%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 96.6

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

   3. Applied rewrites 96.6%

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

   if -0.0149999997 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 48.4%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.4%

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 98.3%

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

Alternative 5: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.014999999664723873:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.014999999664723873)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.014999999664723873f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.014999999664723873))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0149999997

   1. Initial program 96.6%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 96.6

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

   3. Applied rewrites 96.6%

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

   if -0.0149999997 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 48.4%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.2

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

   6. Applied rewrites 98.2%

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

Alternative 6: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\left(u1 \cdot u1\right) \cdot u1}, 0.25, \sqrt{u1}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.0035000001080334187)
     (* (sqrt (- t_0)) t_1)
     (* (fma (sqrt (* (* u1 u1) u1)) 0.25 (sqrt u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.0035000001080334187f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = fmaf(sqrtf(((u1 * u1) * u1)), 0.25f, sqrtf(u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(fma(sqrt(Float32(Float32(u1 * u1) * u1)), Float32(0.25), sqrt(u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00350000011

   1. Initial program 94.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 94.7

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

   3. Applied rewrites 94.7%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 44.8%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 44.8

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

   3. Applied rewrites 44.8%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. pow3 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-sqrt.f32 97.8

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

   8. Applied rewrites 97.8%

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

Alternative 7: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.0035000001080334187)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (fma (* 0.5 u1) u1 u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.0035000001080334187f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00350000011

   1. Initial program 94.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 94.7

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

   3. Applied rewrites 94.7%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 44.8%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.4%

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 97.7%

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

Alternative 8: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;u1 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot t\_0, -1.3333333333333333, \left(\pi + \pi\right) \cdot t\_0\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= u1 0.0035000001080334187)
     (* (sqrt (fma (* 0.5 u1) u1 u1)) (sin (* (+ PI PI) u2)))
     (*
      (fma
       (* (* (* u2 u2) (* (* PI PI) PI)) t_0)
       -1.3333333333333333
       (* (+ PI PI) t_0))
      u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if (u1 <= 0.0035000001080334187f) {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = fmaf((((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))) * t_0), -1.3333333333333333f, ((((float) M_PI) + ((float) M_PI)) * t_0)) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(fma(Float32(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))) * t_0), Float32(-1.3333333333333333), Float32(Float32(Float32(pi) + Float32(pi)) * t_0)) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00350000011

   1. Initial program 44.8%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.4%

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 97.7%

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

   if 0.00350000011 < u1

   1. Initial program 94.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}, -1.3333333333333333, \left(\pi + \pi\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0035000001080334187)
     (*
      (sqrt (- t_0))
      (*
       (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
       u2))
     (* (sqrt (fma (* 0.5 u1) u1 u1)) (sin (* (+ PI PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0035000001080334187f) {
		tmp = sqrtf(-t_0) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-t_0)) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00350000011

   1. Initial program 94.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 86.1%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 44.8%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.4%

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 97.7%

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

Alternative 10: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0035000001080334187)
     (*
      (sqrt (- t_0))
      (*
       (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
       u2))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0035000001080334187f) {
		tmp = sqrtf(-t_0) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-t_0)) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00350000011

   1. Initial program 94.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 86.1%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 44.8%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   7. Applied rewrites 98.2%

      \[\leadsto \sqrt{\left(-\left(\left(-\frac{\left(\frac{1}{u1 \cdot u1} + \frac{0.5}{u1}\right) + 0.3333333333333333}{u1}\right) - 0.25\right) \cdot \left(\left(u1 \cdot u1\right) \cdot u1\right)\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.2

         \[\leadsto \sqrt{\left(-\left(\left(-\frac{\left(\frac{1}{u1 \cdot u1} + \frac{0.5}{u1}\right) + 0.3333333333333333}{u1}\right) - 0.25\right) \cdot \left(\left(u1 \cdot u1\right) \cdot u1\right)\right) \cdot u1} \cdot \sin \left(\left(\pi + \color{blue}{\pi}\right) \cdot u2\right) \]

   9. Applied rewrites 98.2%

      \[\leadsto \sqrt{\left(-\left(\left(-\frac{\left(\frac{1}{u1 \cdot u1} + \frac{0.5}{u1}\right) + 0.3333333333333333}{u1}\right) - 0.25\right) \cdot \left(\left(u1 \cdot u1\right) \cdot u1\right)\right) \cdot u1} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]

   10. Taylor expanded in u1 around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f32 97.7

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

   12. Applied rewrites 97.7%

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

Alternative 11: 88.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00033000000985339284:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00033000000985339284)
   (* (sqrt u1) (sin (* (+ PI PI) u2)))
   (*
    (sqrt (- (log (- 1.0 u1))))
    (*
     (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
     u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00033000000985339284f) {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00033000000985339284))
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 3.3000001e-4

   1. Initial program 38.8%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 91.5%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 91.5

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

   6. Applied rewrites 91.5%

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

   if 3.3000001e-4 < u1

   1. Initial program 90.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 82.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\\ \mathbf{if}\;u1 \leq 0.014800000004470348:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (*
          (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
          u2)))
   (if (<= u1 0.014800000004470348)
     (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)) t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2;
	float tmp;
	if (u1 <= 0.014800000004470348f) {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.014800000004470348))
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0148

   1. Initial program 48.4%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 89.2%

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

   if 0.0148 < u1

   1. Initial program 96.6%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 87.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\\ \mathbf{if}\;u1 \leq 0.00023999999393709004:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (*
          (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
          u2)))
   (if (<= u1 0.00023999999393709004)
     (* (sqrt u1) t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2;
	float tmp;
	if (u1 <= 0.00023999999393709004f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00023999999393709004))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 2.39999994e-4

   1. Initial program 38.0%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 84.1%

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

   if 2.39999994e-4 < u1

   1. Initial program 90.2%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 82.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00033000000985339284:\\ \;\;\;\;\sqrt{u1} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00033000000985339284)
   (*
    (sqrt u1)
    (* (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) u2))
   (* (sqrt (- (log (- 1.0 u1)))) (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00033000000985339284f) {
		tmp = sqrtf(u1) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * ((((float) M_PI) + ((float) M_PI)) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00033000000985339284))
		tmp = Float32(sqrt(u1) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 3.3000001e-4

   1. Initial program 38.8%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 83.7%

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

   if 3.3000001e-4 < u1

   1. Initial program 90.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 76.5

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

   4. Applied rewrites 76.5%

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

Alternative 15: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot u2\\ \mathbf{if}\;u1 \leq 0.014800000004470348:\\ \;\;\;\;\sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (+ PI PI) u2)))
   (if (<= u1 0.014800000004470348)
     (*
      (sqrt (- (* (- (* (- (* -0.3333333333333333 u1) 0.5) u1) 1.0) u1)))
      t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) + ((float) M_PI)) * u2;
	float tmp;
	if (u1 <= 0.014800000004470348f) {
		tmp = sqrtf(-(((((-0.3333333333333333f * u1) - 0.5f) * u1) - 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) + Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.014800000004470348))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (single(pi) + single(pi)) * u2;
	tmp = single(0.0);
	if (u1 <= single(0.014800000004470348))
		tmp = sqrt(-(((((single(-0.3333333333333333) * u1) - single(0.5)) * u1) - single(1.0)) * u1)) * t_0;
	else
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0148

   1. Initial program 48.4%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 81.5

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

   7. Applied rewrites 81.5%

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

   if 0.0148 < u1

   1. Initial program 96.6%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 16: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \left(\pi + \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (* (+ PI PI) u2)))
   (if (<= t_0 -0.0035000001080334187)
     (* (sqrt (- t_0)) t_1)
     (* t_1 (sqrt (- (* (- (* -0.5 u1) 1.0) u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = (((float) M_PI) + ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= -0.0035000001080334187f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = t_1 * sqrtf(-(((-0.5f * u1) - 1.0f) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = Float32(Float32(Float32(pi) + Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(t_1 * sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	t_1 = (single(pi) + single(pi)) * u2;
	tmp = single(0.0);
	if (t_0 <= single(-0.0035000001080334187))
		tmp = sqrt(-t_0) * t_1;
	else
		tmp = t_1 * sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00350000011

   1. Initial program 94.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 79.1

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

   4. Applied rewrites 79.1%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 44.8%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 40.8

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

   4. Applied rewrites 40.8%

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

   5. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 74.6

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

   7. Applied rewrites 74.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 74.6

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

      4. count-2-rev 74.6

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

   9. Applied rewrites 74.6%

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

   10. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. lower-*.f32 81.1

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

   12. Applied rewrites 81.1%

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

Alternative 17: 74.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\pi + \pi\right) \cdot u2\right) \cdot \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* (+ PI PI) u2) (sqrt (- (* (- (* -0.5 u1) 1.0) u1)))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(pi) + single(pi)) * u2) * sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1));
end

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 50.4

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

4. Applied rewrites 50.4%

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

5. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 66.5

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

7. Applied rewrites 66.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 66.5

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

   4. count-2-rev 66.5

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

9. Applied rewrites 66.5%

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

10. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower--.f32 N/A

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

    4. lower-*.f32 74.4

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

12. Applied rewrites 74.4%

    \[\leadsto \left(\left(\pi + \pi\right) \cdot u2\right) \cdot \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \]
13. Add Preprocessing

Alternative 18: 66.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 50.4

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

4. Applied rewrites 50.4%

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

5. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 66.5

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

7. Applied rewrites 66.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 66.5

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

   4. count-2-rev 66.5

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

9. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025111 
(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))))