Beckmann Sample, near normal, slope_y

Percentage Accurate: 56.7% → 98.4%

Time: 4.1s

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: 56.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

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 56.7%

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

   1. lift-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 98.4

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

3. Applied rewrites 98.4%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 98.4

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

5. Applied rewrites 98.4%

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

Alternative 2: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.003599999938160181:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(0.5 \cdot u1\right)\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (+ PI PI) u2))))
   (if (<= u1 0.003599999938160181)
     (* (sqrt (fma u1 1.0 (* u1 (* 0.5 u1)))) t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.003599999938160181f) {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (0.5f * u1)))) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.003599999938160181))
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(Float32(0.5) * u1)))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00359999994

   1. Initial program 56.7%

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

      1. lift-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 98.4

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

   3. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.3

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

   8. Applied rewrites 88.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 88.3

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

   10. Applied rewrites 88.3%

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

   if 0.00359999994 < u1

   1. Initial program 56.7%

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

      1. lift-*.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 56.7

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

   3. Applied rewrites 56.7%

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

Alternative 3: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.003599999938160181:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (+ PI PI) u2))))
   (if (<= u1 0.003599999938160181)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.003599999938160181f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.003599999938160181))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00359999994

   1. Initial program 56.7%

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

      1. lift-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 98.4

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

   3. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.3

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

   8. Applied rewrites 88.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 88.3

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 88.3

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

   10. Applied rewrites 88.3%

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

   if 0.00359999994 < u1

   1. Initial program 56.7%

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

      1. lift-*.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 56.7

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

   3. Applied rewrites 56.7%

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

Alternative 4: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sqrt{-t\_0}\\ \mathbf{if}\;t\_0 \leq -0.003599999938160181:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(\pi + \pi, t\_1, \left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot t\_1\right) \cdot -1.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sqrt (- t_0))))
   (if (<= t_0 -0.003599999938160181)
     (*
      u2
      (fma
       (+ PI PI)
       t_1
       (* (* (* (* u2 u2) (* (* PI PI) PI)) t_1) -1.3333333333333333)))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sqrtf(-t_0);
	float tmp;
	if (t_0 <= -0.003599999938160181f) {
		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(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	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(-0.003599999938160181))
		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(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00359999994

   1. Initial program 56.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   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 52.9%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{-\log \left(1 - u1\right)}\right), 2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)} \]
   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 52.9

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

   6. Applied rewrites 52.9%

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

   if -0.00359999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 56.7%

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

      1. lift-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 98.4

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

   3. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.3

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

   8. Applied rewrites 88.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 88.3

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 88.3

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

   10. Applied rewrites 88.3%

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

Alternative 5: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.003599999938160181:\\ \;\;\;\;\sqrt{-t\_0} \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right)\right) \cdot u2 + u2 \cdot \left(\pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.003599999938160181)
     (*
      (sqrt (- t_0))
      (+
       (* (* (* u2 u2) (* (* (* PI PI) PI) -1.3333333333333333)) u2)
       (* u2 (+ PI PI))))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.003599999938160181f) {
		tmp = sqrtf(-t_0) * ((((u2 * u2) * (((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * -1.3333333333333333f)) * u2) + (u2 * (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003599999938160181))
		tmp = Float32(sqrt(Float32(-t_0)) * Float32(Float32(Float32(Float32(u2 * u2) * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(-1.3333333333333333))) * u2) + Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00359999994

   1. Initial program 56.7%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-pow.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 52.9

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

   4. Applied rewrites 52.9%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-*.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 52.9

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

      9. lift-pow.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 52.9

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

      12. lift-pow.f32 N/A

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

      13. unpow3 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 52.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 52.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot -1.3333333333333333\right) \cdot u2 + \color{blue}{u2 \cdot \left(\pi + \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 52.9

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

   8. Applied rewrites 52.9%

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

   if -0.00359999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 56.7%

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

      1. lift-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 98.4

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

   3. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.3

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

   8. Applied rewrites 88.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 88.3

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 88.3

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

   10. Applied rewrites 88.3%

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

Alternative 6: 90.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0026000000070780516)
   (* (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 <= 0.0026000000070780516f) {
		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(0.0026000000070780516))
		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 < 0.00260000001

   1. Initial program 56.7%

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

      1. lift-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 98.4

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

   3. Applied rewrites 98.4%

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

   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 81.5

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

   6. Applied rewrites 81.5%

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

   if 0.00260000001 < u2

   1. Initial program 56.7%

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

      1. lift-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 98.4

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

   3. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 77.0%

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

Alternative 7: 81.5% 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 56.7%

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

   1. lift-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 98.4

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

3. Applied rewrites 98.4%

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

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 81.5

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

6. Applied rewrites 81.5%

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

Alternative 8: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \leq 0.01600000075995922:\\ \;\;\;\;\left(\left(\frac{\pi}{\sqrt{u1}} \cdot u1\right) \cdot u2\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= t_0 0.01600000075995922)
     (* (* (* (/ PI (sqrt u1)) u1) u2) 2.0)
     (* 2.0 (* u2 (* PI t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if (t_0 <= 0.01600000075995922f) {
		tmp = (((((float) M_PI) / sqrtf(u1)) * u1) * u2) * 2.0f;
	} else {
		tmp = 2.0f * (u2 * (((float) M_PI) * t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.01600000075995922))
		tmp = Float32(Float32(Float32(Float32(Float32(pi) / sqrt(u1)) * u1) * u2) * Float32(2.0));
	else
		tmp = Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	tmp = single(0.0);
	if (t_0 <= single(0.01600000075995922))
		tmp = (((single(pi) / sqrt(u1)) * u1) * u2) * single(2.0);
	else
		tmp = single(2.0) * (u2 * (single(pi) * t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) < 0.0160000008

   1. Initial program 56.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   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 49.9

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

   4. Applied rewrites 49.9%

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

   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 66.5

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

   7. Applied rewrites 66.5%

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

   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 66.5

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

   10. Applied rewrites 66.5%

       \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right)\right) \]
   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 66.5

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

   12. Applied rewrites 66.5%

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

   if 0.0160000008 < (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))))

   1. Initial program 56.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   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 49.9

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

   4. Applied rewrites 49.9%

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

Alternative 9: 66.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
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 49.9

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

4. Applied rewrites 49.9%

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

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 66.5

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

7. Applied rewrites 66.5%

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

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 66.5

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

10. Applied rewrites 66.5%

    \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right)\right) \]
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 66.5

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

12. Applied rewrites 66.5%

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

Alternative 10: 66.5% 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 56.7%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
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 49.9

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

4. Applied rewrites 49.9%

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

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 66.5

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

7. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025149 
(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))))