Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.3%

Time: 3.8s

Alternatives: 10

Speedup: 4.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 98.3

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

5. Applied rewrites 98.3%

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

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0024999999441206455:\\ \;\;\;\;\sqrt{-t\_0} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0024999999441206455)
     (* (sqrt (- t_0)) (sin (* (+ PI PI) u2)))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* u2 (+ PI PI)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0024999999441206455f) {
		tmp = sqrtf(-t_0) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf((u2 * (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-t_0)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00249999994

   1. Initial program 57.5%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.5

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

   3. Applied rewrites 57.5%

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

   if -0.00249999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.5%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.4

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

   4. Applied rewrites 93.4%

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 88.0%

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

Alternative 3: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sqrt{-t\_0}\\ \mathbf{if}\;t\_0 \leq -0.006099999882280827:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(\pi + \pi, t\_1, \left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot t\_1\right) \cdot -1.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sqrt (- t_0))))
   (if (<= t_0 -0.006099999882280827)
     (*
      u2
      (fma
       (+ PI PI)
       t_1
       (* (* (* (* u2 u2) (* (* PI PI) PI)) t_1) -1.3333333333333333)))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* u2 (+ PI PI)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sqrtf(-t_0);
	float tmp;
	if (t_0 <= -0.006099999882280827f) {
		tmp = u2 * fmaf((((float) M_PI) + ((float) M_PI)), t_1, ((((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))) * t_1) * -1.3333333333333333f));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf((u2 * (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sqrt(Float32(-t_0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.006099999882280827))
		tmp = Float32(u2 * fma(Float32(Float32(pi) + Float32(pi)), t_1, Float32(Float32(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))) * t_1) * Float32(-1.3333333333333333))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00609999988

   1. Initial program 57.5%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 53.6%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{-\log \left(1 - u1\right)}\right), 2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 53.6

          \[\leadsto u2 \cdot \mathsf{fma}\left(\pi + \pi, \sqrt{-\log \left(1 - u1\right)}, \left({u2}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right) \cdot -1.3333333333333333\right) \]

   6. Applied rewrites 53.6%

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

   if -0.00609999988 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.5%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.4

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

   4. Applied rewrites 93.4%

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 88.0%

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

Alternative 4: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00018200000340584666:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00018200000340584666)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.00018200000340584666f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf((u2 * (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.00018200000340584666))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 1.82000003e-4

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.1

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

   6. Applied rewrites 81.1%

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

   if 1.82000003e-4 < u2

   1. Initial program 57.5%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.4

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

   4. Applied rewrites 93.4%

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 88.0%

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

Alternative 5: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0019000000320374966:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0019000000320374966)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt u1) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0019000000320374966f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0019000000320374966))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00190000003

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.1

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

   6. Applied rewrites 81.1%

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

   if 0.00190000003 < u2

   1. Initial program 57.5%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.6%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 76.6

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

   6. Applied rewrites 76.6%

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

Alternative 6: 81.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.1

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

6. Applied rewrites 81.1%

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

Alternative 7: 76.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00018000000272877514:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \frac{\sqrt{u1}}{u1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00018000000272877514)
   (* 2.0 (* u2 (* u1 (* PI (/ (sqrt u1) u1)))))
   (* 2.0 (* u2 (* PI (sqrt (- (log (- 1.0 u1)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00018000000272877514f) {
		tmp = 2.0f * (u2 * (u1 * (((float) M_PI) * (sqrtf(u1) / u1))));
	} else {
		tmp = 2.0f * (u2 * (((float) M_PI) * sqrtf(-logf((1.0f - u1)))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00018000000272877514))
		tmp = Float32(Float32(2.0) * Float32(u2 * Float32(u1 * Float32(Float32(pi) * Float32(sqrt(u1) / u1)))));
	else
		tmp = Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * sqrt(Float32(-log(Float32(Float32(1.0) - u1)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.00018000000272877514))
		tmp = single(2.0) * (u2 * (u1 * (single(pi) * (sqrt(u1) / u1))));
	else
		tmp = single(2.0) * (u2 * (single(pi) * sqrt(-log((single(1.0) - u1)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 1.80000003e-4

   1. Initial program 57.5%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in u1 around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-/.f32 66.0

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

   10. Applied rewrites 66.0%

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

   11. Taylor expanded in u1 around 0

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

       1. lower-/.f32 N/A

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

       2. lower-sqrt.f32 65.9

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

   13. Applied rewrites 65.9%

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

   if 1.80000003e-4 < u1

   1. Initial program 57.5%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

Alternative 8: 66.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \frac{\sqrt{u1}}{u1}\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(u2 * Float32(u1 * Float32(Float32(pi) * Float32(sqrt(u1) / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * (u2 * (u1 * (single(pi) * (sqrt(u1) / u1))));
end

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.0

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

7. Applied rewrites 66.0%

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

8. Taylor expanded in u1 around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 66.0

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

10. Applied rewrites 66.0%

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

11. Taylor expanded in u1 around 0

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

    1. lower-/.f32 N/A

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

    2. lower-sqrt.f32 65.9

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

13. Applied rewrites 65.9%

    \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \frac{\sqrt{u1}}{u1}\right)\right)\right) \]
14. Add Preprocessing

Alternative 9: 66.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.0

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

7. Applied rewrites 66.0%

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

8. Taylor expanded in u1 around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 66.0

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

10. Applied rewrites 66.0%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f32 66.0

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

12. Applied rewrites 66.0%

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

Alternative 10: 65.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.0

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

7. Applied rewrites 66.0%

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

Reproduce

herbie shell --seed 2025142 
(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))))