Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.0% → 98.3%

Time: 5.8s

Alternatives: 11

Speedup: N/A×

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: 58.0% 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: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt{-1 \cdot \mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot \left(0.5 - u2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* -1.0 (log1p (* -1.0 u1))))
  (* 2.0 (* (sin (* PI u2)) (sin (* PI (- 0.5 u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-1.0f * log1pf((-1.0f * u1)))) * (2.0f * (sinf((((float) M_PI) * u2)) * sinf((((float) M_PI) * (0.5f - u2)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(-1.0) * log1p(Float32(Float32(-1.0) * u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * sin(Float32(Float32(pi) * Float32(Float32(0.5) - u2))))))
end

Derivation

1. Initial program 62.4%

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

   1. 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 62.3

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

4. Applied rewrites 62.3%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lift-*.f32 98.2

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

6. Applied rewrites 98.2%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 98.2

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

8. Applied rewrites 98.2%

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

9. Taylor expanded in u2 around inf

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]
10. Step-by-step derivation

    1. lower-sin.f32 N/A

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

    2. distribute-rgt-out-- N/A

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

    3. lower-*.f32 N/A

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

    4. lift-PI.f32 N/A

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

    5. lower--.f32 98.2

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot \left(0.5 - u2\right)\right)\right)\right) \]

11. Applied rewrites 98.2%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \color{blue}{\sin \left(\pi \cdot \left(0.5 - u2\right)\right)}\right)\right) \]

12. Final simplification 98.2%

    \[\leadsto \sqrt{-1 \cdot \mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot \left(0.5 - u2\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 98.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.03500000014901161:\\ \;\;\;\;\sqrt{-1 \cdot t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (* 2.0 PI) u2))))
   (if (<= t_0 -0.03500000014901161)
     (* (sqrt (* -1.0 t_0)) t_1)
     (*
      (sqrt
       (*
        -1.0
        (*
         (- (* (- (* (- (* -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(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.03500000014901161f) {
		tmp = sqrtf((-1.0f * t_0)) * t_1;
	} else {
		tmp = sqrtf((-1.0f * (((((((-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(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03500000014901161))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	t_1 = sin(((single(2.0) * single(pi)) * u2));
	tmp = single(0.0);
	if (t_0 <= single(-0.03500000014901161))
		tmp = sqrt((single(-1.0) * t_0)) * t_1;
	else
		tmp = sqrt((single(-1.0) * (((((((single(-0.25) * u1) - single(0.3333333333333333)) * u1) - single(0.5)) * u1) - single(1.0)) * u1))) * t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0350000001

   1. Initial program 97.1%

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

   if -0.0350000001 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 54.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.03500000014901161:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.04100000113248825:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(\left(\frac{1}{u1} - 1\right) \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.04100000113248825)
   (*
    (sqrt (* -1.0 (log (* (- (/ 1.0 u1) 1.0) u1))))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))
   (*
    (sqrt
     (*
      -1.0
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (logf((1.0f - u1)) <= -0.04100000113248825f) {
		tmp = sqrtf((-1.0f * logf((((1.0f / u1) - 1.0f) * u1)))) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	} else {
		tmp = sqrtf((-1.0f * (((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.04100000113248825))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(Float32(Float32(1.0) / u1) - Float32(1.0)) * u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (log((single(1.0) - u1)) <= single(-0.04100000113248825))
		tmp = sqrt((single(-1.0) * log((((single(1.0) / u1) - single(1.0)) * u1)))) * (single(2.0) * (sin((single(pi) * u2)) * cos((single(pi) * u2))));
	else
		tmp = sqrt((single(-1.0) * (((((((single(-0.25) * u1) - single(0.3333333333333333)) * u1) - single(0.5)) * u1) - single(1.0)) * u1))) * sin(((single(2.0) * single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0410000011

   1. Initial program 97.1%

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

      1. 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 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 96.9

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

   7. Applied rewrites 96.9%

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

   if -0.0410000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 55.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.04100000113248825:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(\left(\frac{1}{u1} - 1\right) \cdot u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;\log \left(1 - u1\right) \leq -0.04100000113248825:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(\left(\frac{1}{u1} - 1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 PI) u2))))
   (if (<= (log (- 1.0 u1)) -0.04100000113248825)
     (* (sqrt (* -1.0 (log (* (- (/ 1.0 u1) 1.0) u1)))) t_0)
     (*
      (sqrt
       (*
        -1.0
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (logf((1.0f - u1)) <= -0.04100000113248825f) {
		tmp = sqrtf((-1.0f * logf((((1.0f / u1) - 1.0f) * u1)))) * t_0;
	} else {
		tmp = sqrtf((-1.0f * (((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.04100000113248825))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(Float32(Float32(1.0) / u1) - Float32(1.0)) * u1)))) * t_0);
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sin(((single(2.0) * single(pi)) * u2));
	tmp = single(0.0);
	if (log((single(1.0) - u1)) <= single(-0.04100000113248825))
		tmp = sqrt((single(-1.0) * log((((single(1.0) / u1) - single(1.0)) * u1)))) * t_0;
	else
		tmp = sqrt((single(-1.0) * (((((((single(-0.25) * u1) - single(0.3333333333333333)) * u1) - single(0.5)) * u1) - single(1.0)) * u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0410000011

   1. Initial program 97.1%

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

   3. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 96.7

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

   5. Applied rewrites 96.7%

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

   if -0.0410000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 55.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.04100000113248825:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(\left(\frac{1}{u1} - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.041999999433755875:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.041999999433755875)
   (*
    (exp (* (log (log (/ 1.0 (- 1.0 u1)))) 0.5))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))
   (*
    (sqrt
     (*
      -1.0
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (logf((1.0f - u1)) <= -0.041999999433755875f) {
		tmp = expf((logf(logf((1.0f / (1.0f - u1)))) * 0.5f)) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	} else {
		tmp = sqrtf((-1.0f * (((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.041999999433755875))
		tmp = Float32(exp(Float32(log(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(0.5))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (log((single(1.0) - u1)) <= single(-0.041999999433755875))
		tmp = exp((log(log((single(1.0) / (single(1.0) - u1)))) * single(0.5))) * (single(2.0) * (sin((single(pi) * u2)) * cos((single(pi) * u2))));
	else
		tmp = sqrt((single(-1.0) * (((((((single(-0.25) * u1) - single(0.3333333333333333)) * u1) - single(0.5)) * u1) - single(1.0)) * u1))) * sin(((single(2.0) * single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0419999994

   1. Initial program 97.1%

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

      1. 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 97.0

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

   4. Applied rewrites 97.0%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-log1p.f32 N/A

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

      5. pow1/2 N/A

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

      6. neg-log N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. flip3-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. neg-log N/A

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

      14. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

      15. lower-exp.f32 N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   8. Applied rewrites 96.4%

      \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   if -0.0419999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.041999999433755875:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.041999999433755875:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333}{u1} \cdot -1 - 0.25\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.041999999433755875)
   (*
    (sqrt
     (*
      -1.0
      (*
       (-
        (*
         (/ (+ (fma (/ 1.0 u1) 0.5 (/ 1.0 (* u1 u1))) 0.3333333333333333) u1)
         -1.0)
        0.25)
       (pow u1 4.0))))
    (sin (* (* 2.0 PI) u2)))
   (*
    (exp (* (log (log (/ 1.0 (- 1.0 u1)))) 0.5))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.041999999433755875f) {
		tmp = sqrtf((-1.0f * (((((fmaf((1.0f / u1), 0.5f, (1.0f / (u1 * u1))) + 0.3333333333333333f) / u1) * -1.0f) - 0.25f) * powf(u1, 4.0f)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = expf((logf(logf((1.0f / (1.0f - u1)))) * 0.5f)) * (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(0.041999999433755875))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(fma(Float32(Float32(1.0) / u1), Float32(0.5), Float32(Float32(1.0) / Float32(u1 * u1))) + Float32(0.3333333333333333)) / u1) * Float32(-1.0)) - Float32(0.25)) * (u1 ^ Float32(4.0))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(exp(Float32(log(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(0.5))) * 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 < 0.0419999994

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in u1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 98.0%

      \[\leadsto \sqrt{-\left(\frac{\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333}{u1} \cdot -1 - 0.25\right) \cdot \color{blue}{{u1}^{4}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0419999994 < u1

   1. Initial program 97.1%

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

      1. 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 97.0

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

   4. Applied rewrites 97.0%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-log1p.f32 N/A

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

      5. pow1/2 N/A

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

      6. neg-log N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. flip3-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. neg-log N/A

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

      14. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

      15. lower-exp.f32 N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   8. Applied rewrites 96.4%

      \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.041999999433755875:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333}{u1} \cdot -1 - 0.25\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.021800000220537186:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left({u1}^{3} \cdot -1\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{1}{u1}}{u1}, 1, \frac{1}{u1} \cdot 0.5\right) + 0.3333333333333333\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.021800000220537186)
   (*
    (exp (* (log (log (/ 1.0 (- 1.0 u1)))) 0.5))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))
   (*
    (sqrt
     (*
      -1.0
      (*
       (* (pow u1 3.0) -1.0)
       (+ (fma (/ (/ 1.0 u1) u1) 1.0 (* (/ 1.0 u1) 0.5)) 0.3333333333333333))))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (logf((1.0f - u1)) <= -0.021800000220537186f) {
		tmp = expf((logf(logf((1.0f / (1.0f - u1)))) * 0.5f)) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	} else {
		tmp = sqrtf((-1.0f * ((powf(u1, 3.0f) * -1.0f) * (fmaf(((1.0f / u1) / u1), 1.0f, ((1.0f / u1) * 0.5f)) + 0.3333333333333333f)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.021800000220537186))
		tmp = Float32(exp(Float32(log(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(0.5))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32((u1 ^ Float32(3.0)) * Float32(-1.0)) * Float32(fma(Float32(Float32(Float32(1.0) / u1) / u1), Float32(1.0), Float32(Float32(Float32(1.0) / u1) * Float32(0.5))) + Float32(0.3333333333333333))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0218000002

   1. Initial program 96.9%

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

      1. 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 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-log1p.f32 N/A

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

      5. pow1/2 N/A

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

      6. neg-log N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. flip3-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. neg-log N/A

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

      14. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

      15. lower-exp.f32 N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   8. Applied rewrites 95.5%

      \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   if -0.0218000002 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-pow.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. pow2 N/A

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

      13. lower-*.f32 98.0

          \[\leadsto \sqrt{-\left({u1}^{3} \cdot -1\right) \cdot \left(\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 98.0%

      \[\leadsto \sqrt{-\left({u1}^{3} \cdot -1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-/.f32 N/A

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

      8. pow-flip N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. pow-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f32 N/A

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

      17. pow2 N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f32 N/A

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

      20. lift-/.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 N/A

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

      23. lift-/.f32 98.0

          \[\leadsto \sqrt{-\left({u1}^{3} \cdot -1\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{1}{u1}}{u1}, 1, \frac{1}{u1} \cdot 0.5\right) + 0.3333333333333333\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. Applied rewrites 98.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.021800000220537186:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left({u1}^{3} \cdot -1\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{1}{u1}}{u1}, 1, \frac{1}{u1} \cdot 0.5\right) + 0.3333333333333333\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.021800000220537186:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left({u1}^{3} \cdot -1\right) \cdot \left(\frac{-1 \cdot \mathsf{fma}\left(0.5, u1, 1\right)}{-1 \cdot \left(u1 \cdot u1\right)} + 0.3333333333333333\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.021800000220537186)
   (*
    (exp (* (log (log (/ 1.0 (- 1.0 u1)))) 0.5))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))
   (*
    (sqrt
     (*
      -1.0
      (*
       (* (pow u1 3.0) -1.0)
       (+
        (/ (* -1.0 (fma 0.5 u1 1.0)) (* -1.0 (* u1 u1)))
        0.3333333333333333))))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (logf((1.0f - u1)) <= -0.021800000220537186f) {
		tmp = expf((logf(logf((1.0f / (1.0f - u1)))) * 0.5f)) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	} else {
		tmp = sqrtf((-1.0f * ((powf(u1, 3.0f) * -1.0f) * (((-1.0f * fmaf(0.5f, u1, 1.0f)) / (-1.0f * (u1 * u1))) + 0.3333333333333333f)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.021800000220537186))
		tmp = Float32(exp(Float32(log(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(0.5))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32((u1 ^ Float32(3.0)) * Float32(-1.0)) * Float32(Float32(Float32(Float32(-1.0) * fma(Float32(0.5), u1, Float32(1.0))) / Float32(Float32(-1.0) * Float32(u1 * u1))) + Float32(0.3333333333333333))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0218000002

   1. Initial program 96.9%

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

      1. 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 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-log1p.f32 N/A

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

      5. pow1/2 N/A

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

      6. neg-log N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. flip3-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. neg-log N/A

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

      14. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

      15. lower-exp.f32 N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   8. Applied rewrites 95.5%

      \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   if -0.0218000002 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-pow.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. pow2 N/A

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

      13. lower-*.f32 98.0

          \[\leadsto \sqrt{-\left({u1}^{3} \cdot -1\right) \cdot \left(\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 98.0%

      \[\leadsto \sqrt{-\left({u1}^{3} \cdot -1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{1}{u1}, 0.5, \frac{1}{u1 \cdot u1}\right) + 0.3333333333333333\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. Taylor expanded in u1 around 0

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

       1. associate-*l/ N/A

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

       2. metadata-eval N/A

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

       3. associate-/r* N/A

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

       4. div-add N/A

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

       5. +-commutative N/A

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

       6. lower-/.f32 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f32 N/A

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

       9. pow2 N/A

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

       10. lift-*.f32 98.0

           \[\leadsto \sqrt{-\left({u1}^{3} \cdot -1\right) \cdot \left(\frac{\mathsf{fma}\left(0.5, u1, 1\right)}{u1 \cdot u1} + 0.3333333333333333\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. Applied rewrites 98.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.021800000220537186:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left({u1}^{3} \cdot -1\right) \cdot \left(\frac{-1 \cdot \mathsf{fma}\left(0.5, u1, 1\right)}{-1 \cdot \left(u1 \cdot u1\right)} + 0.3333333333333333\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(u2 \cdot \pi\right)\\ t_1 := \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\\ t_2 := \sqrt{u1} \cdot t\_1\\ t_3 := \frac{1}{\sqrt{u1}}\\ \mathbf{if}\;\log \left(1 - u1\right) \leq -0.021800000220537186:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t\_0, t\_2 \cdot t\_0\right) - \left(-1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\left(0.5 \cdot \left(-1 \cdot t\_3\right)\right) \cdot \left(\left(-1 \cdot t\_1\right) \cdot t\_0\right) - \left(-1 \cdot u1\right) \cdot \left(\left(0.3333333333333333 \cdot t\_3\right) \cdot \left(t\_1 \cdot t\_0\right) - \left(-1 \cdot \sqrt{u1}\right) \cdot \left(t\_1 \cdot \left(t\_0 \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* u2 PI)))
        (t_1 (sin (fma u2 PI (/ PI 2.0))))
        (t_2 (* (sqrt u1) t_1))
        (t_3 (/ 1.0 (sqrt u1))))
   (if (<= (log (- 1.0 u1)) -0.021800000220537186)
     (*
      (exp (* (log (log (/ 1.0 (- 1.0 u1)))) 0.5))
      (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))
     (-
      (fma t_2 t_0 (* t_2 t_0))
      (*
       (* -1.0 (* u1 u1))
       (-
        (* (* 0.5 (* -1.0 t_3)) (* (* -1.0 t_1) t_0))
        (*
         (* -1.0 u1)
         (-
          (* (* 0.3333333333333333 t_3) (* t_1 t_0))
          (*
           (* -1.0 (sqrt u1))
           (* t_1 (* t_0 (- 0.25 (* 0.0625 (/ 1.0 u1))))))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((u2 * ((float) M_PI)));
	float t_1 = sinf(fmaf(u2, ((float) M_PI), (((float) M_PI) / 2.0f)));
	float t_2 = sqrtf(u1) * t_1;
	float t_3 = 1.0f / sqrtf(u1);
	float tmp;
	if (logf((1.0f - u1)) <= -0.021800000220537186f) {
		tmp = expf((logf(logf((1.0f / (1.0f - u1)))) * 0.5f)) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	} else {
		tmp = fmaf(t_2, t_0, (t_2 * t_0)) - ((-1.0f * (u1 * u1)) * (((0.5f * (-1.0f * t_3)) * ((-1.0f * t_1) * t_0)) - ((-1.0f * u1) * (((0.3333333333333333f * t_3) * (t_1 * t_0)) - ((-1.0f * sqrtf(u1)) * (t_1 * (t_0 * (0.25f - (0.0625f * (1.0f / u1))))))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(u2 * Float32(pi)))
	t_1 = sin(fma(u2, Float32(pi), Float32(Float32(pi) / Float32(2.0))))
	t_2 = Float32(sqrt(u1) * t_1)
	t_3 = Float32(Float32(1.0) / sqrt(u1))
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.021800000220537186))
		tmp = Float32(exp(Float32(log(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(0.5))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	else
		tmp = Float32(fma(t_2, t_0, Float32(t_2 * t_0)) - Float32(Float32(Float32(-1.0) * Float32(u1 * u1)) * Float32(Float32(Float32(Float32(0.5) * Float32(Float32(-1.0) * t_3)) * Float32(Float32(Float32(-1.0) * t_1) * t_0)) - Float32(Float32(Float32(-1.0) * u1) * Float32(Float32(Float32(Float32(0.3333333333333333) * t_3) * Float32(t_1 * t_0)) - Float32(Float32(Float32(-1.0) * sqrt(u1)) * Float32(t_1 * Float32(t_0 * Float32(Float32(0.25) - Float32(Float32(0.0625) * Float32(Float32(1.0) / u1)))))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0218000002

   1. Initial program 96.9%

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

      1. 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 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-log1p.f32 N/A

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

      5. pow1/2 N/A

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

      6. neg-log N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. flip3-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. neg-log N/A

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

      14. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

      15. lower-exp.f32 N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   8. Applied rewrites 95.5%

      \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   if -0.0218000002 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.9%

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

      1. 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 52.9

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

   4. Applied rewrites 52.9%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-log1p.f32 N/A

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

      5. pow1/2 N/A

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

      6. neg-log N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. flip3-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. neg-log N/A

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

      14. pow-to-exp N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

      15. lower-exp.f32 N/A

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   8. Applied rewrites 50.3%

      \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   9. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sin \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   11. Applied rewrites 98.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right), \sin \left(u2 \cdot \pi\right), \left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-u1 \cdot u1\right) \cdot \left(\left(0.5 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) + u1 \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-\sqrt{u1}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\sin \left(u2 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.021800000220537186:\\ \;\;\;\;e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right), \sin \left(u2 \cdot \pi\right), \left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\left(0.5 \cdot \left(-1 \cdot \frac{1}{\sqrt{u1}}\right)\right) \cdot \left(\left(-1 \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-1 \cdot u1\right) \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-1 \cdot \sqrt{u1}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\sin \left(u2 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt{-1 \cdot \mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\pi \cdot \pi\right) \cdot \pi, \left(u2 \cdot u2\right) \cdot \left(\left(-0.0001984126984126984 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7} - -0.008333333333333333 \cdot {\pi}^{5}\right)\right)\right)\right) \cdot \sin \left(-1 \cdot \left(\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* -1.0 (log1p (* -1.0 u1))))
  (*
   2.0
   (*
    (*
     u2
     (+
      PI
      (*
       (* u2 u2)
       (fma
        -0.16666666666666666
        (* (* PI PI) PI)
        (*
         (* u2 u2)
         (-
          (* (* -0.0001984126984126984 (* u2 u2)) (pow PI 7.0))
          (* -0.008333333333333333 (pow PI 5.0))))))))
    (sin (+ (* -1.0 (* PI u2)) (/ PI 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-1.0f * log1pf((-1.0f * u1)))) * (2.0f * ((u2 * (((float) M_PI) + ((u2 * u2) * fmaf(-0.16666666666666666f, ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), ((u2 * u2) * (((-0.0001984126984126984f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) - (-0.008333333333333333f * powf(((float) M_PI), 5.0f)))))))) * sinf(((-1.0f * (((float) M_PI) * u2)) + (((float) M_PI) / 2.0f)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(-1.0) * log1p(Float32(Float32(-1.0) * u1)))) * Float32(Float32(2.0) * Float32(Float32(u2 * Float32(Float32(pi) + Float32(Float32(u2 * u2) * fma(Float32(-0.16666666666666666), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(u2 * u2) * Float32(Float32(Float32(Float32(-0.0001984126984126984) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) - Float32(Float32(-0.008333333333333333) * (Float32(pi) ^ Float32(5.0))))))))) * sin(Float32(Float32(Float32(-1.0) * Float32(Float32(pi) * u2)) + Float32(Float32(pi) / Float32(2.0)))))))
end

Derivation

1. Initial program 62.4%

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

   1. 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 62.3

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

4. Applied rewrites 62.3%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lift-*.f32 98.2

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

6. Applied rewrites 98.2%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 98.2

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

8. Applied rewrites 98.2%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

    2. lower-+.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{{u2}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)}\right)\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

    3. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + \color{blue}{{u2}^{2}} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

    4. lower-*.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + {u2}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)}\right)\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3}} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3}} + {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

    7. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, {u2}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

11. Applied rewrites 95.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\color{blue}{\left(u2 \cdot \left(\pi + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\pi \cdot \pi\right) \cdot \pi, \left(u2 \cdot u2\right) \cdot \left(\left(-0.0001984126984126984 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7} - -0.008333333333333333 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sin \left(\left(-\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]

12. Final simplification 95.0%

    \[\leadsto \sqrt{-1 \cdot \mathsf{log1p}\left(-1 \cdot u1\right)} \cdot \left(2 \cdot \left(\left(u2 \cdot \left(\pi + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\pi \cdot \pi\right) \cdot \pi, \left(u2 \cdot u2\right) \cdot \left(\left(-0.0001984126984126984 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7} - -0.008333333333333333 \cdot {\pi}^{5}\right)\right)\right)\right) \cdot \sin \left(-1 \cdot \left(\pi \cdot u2\right) + \frac{\pi}{2}\right)\right)\right) \]
13. Add Preprocessing

Alternative 11: 92.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\\ t_1 := \sqrt{u1} \cdot t\_0\\ t_2 := \frac{1}{\sqrt{u1}}\\ t_3 := \sin \left(u2 \cdot \pi\right)\\ \mathsf{fma}\left(t\_1, t\_3, t\_1 \cdot t\_3\right) - \left(-1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\left(0.5 \cdot \left(-1 \cdot t\_2\right)\right) \cdot \left(\left(-1 \cdot t\_0\right) \cdot t\_3\right) - \left(-1 \cdot u1\right) \cdot \left(\left(0.3333333333333333 \cdot t\_2\right) \cdot \left(t\_0 \cdot t\_3\right) - \left(-1 \cdot \sqrt{u1}\right) \cdot \left(t\_0 \cdot \left(t\_3 \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (fma u2 PI (/ PI 2.0))))
        (t_1 (* (sqrt u1) t_0))
        (t_2 (/ 1.0 (sqrt u1)))
        (t_3 (sin (* u2 PI))))
   (-
    (fma t_1 t_3 (* t_1 t_3))
    (*
     (* -1.0 (* u1 u1))
     (-
      (* (* 0.5 (* -1.0 t_2)) (* (* -1.0 t_0) t_3))
      (*
       (* -1.0 u1)
       (-
        (* (* 0.3333333333333333 t_2) (* t_0 t_3))
        (*
         (* -1.0 (sqrt u1))
         (* t_0 (* t_3 (- 0.25 (* 0.0625 (/ 1.0 u1)))))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(fmaf(u2, ((float) M_PI), (((float) M_PI) / 2.0f)));
	float t_1 = sqrtf(u1) * t_0;
	float t_2 = 1.0f / sqrtf(u1);
	float t_3 = sinf((u2 * ((float) M_PI)));
	return fmaf(t_1, t_3, (t_1 * t_3)) - ((-1.0f * (u1 * u1)) * (((0.5f * (-1.0f * t_2)) * ((-1.0f * t_0) * t_3)) - ((-1.0f * u1) * (((0.3333333333333333f * t_2) * (t_0 * t_3)) - ((-1.0f * sqrtf(u1)) * (t_0 * (t_3 * (0.25f - (0.0625f * (1.0f / u1))))))))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(fma(u2, Float32(pi), Float32(Float32(pi) / Float32(2.0))))
	t_1 = Float32(sqrt(u1) * t_0)
	t_2 = Float32(Float32(1.0) / sqrt(u1))
	t_3 = sin(Float32(u2 * Float32(pi)))
	return Float32(fma(t_1, t_3, Float32(t_1 * t_3)) - Float32(Float32(Float32(-1.0) * Float32(u1 * u1)) * Float32(Float32(Float32(Float32(0.5) * Float32(Float32(-1.0) * t_2)) * Float32(Float32(Float32(-1.0) * t_0) * t_3)) - Float32(Float32(Float32(-1.0) * u1) * Float32(Float32(Float32(Float32(0.3333333333333333) * t_2) * Float32(t_0 * t_3)) - Float32(Float32(Float32(-1.0) * sqrt(u1)) * Float32(t_0 * Float32(t_3 * Float32(Float32(0.25) - Float32(Float32(0.0625) * Float32(Float32(1.0) / u1)))))))))))
end

Derivation

1. Initial program 62.4%

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

   1. 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 62.3

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

4. Applied rewrites 62.3%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lift-*.f32 98.2

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

6. Applied rewrites 98.2%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-1 \cdot u1\right)}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log1p.f32 N/A

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

   5. pow1/2 N/A

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

   6. neg-log N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. flip3-- N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. neg-log N/A

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

   14. pow-to-exp N/A

       \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

   15. lower-exp.f32 N/A

       \[\leadsto \color{blue}{e^{\log \left(\mathsf{neg}\left(\log \left(\frac{1 - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right) \cdot \frac{1}{2}}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

8. Applied rewrites 60.0%

   \[\leadsto \color{blue}{e^{\log \log \left(\frac{1}{1 - u1}\right) \cdot 0.5}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right) \]

9. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sin \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

11. Applied rewrites 92.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right), \sin \left(u2 \cdot \pi\right), \left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-u1 \cdot u1\right) \cdot \left(\left(0.5 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) + u1 \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-\sqrt{u1}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\sin \left(u2 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

12. Final simplification 92.7%

    \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right), \sin \left(u2 \cdot \pi\right), \left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\left(0.5 \cdot \left(-1 \cdot \frac{1}{\sqrt{u1}}\right)\right) \cdot \left(\left(-1 \cdot \sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-1 \cdot u1\right) \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) - \left(-1 \cdot \sqrt{u1}\right) \cdot \left(\sin \left(\mathsf{fma}\left(u2, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\sin \left(u2 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))