Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.9% → 98.1%

Time: 12.2s

Alternatives: 17

Speedup: 8.9×

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: 57.9% 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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9639999866485596:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 PI) u2))))
   (if (<= (- 1.0 u1) 0.9639999866485596)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (*
      t_0
      (sqrt
       (*
        (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0)
        (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if ((1.0f - u1) <= 0.9639999866485596f) {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	} else {
		tmp = t_0 * sqrtf((fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) * -u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9639999866485596))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	else
		tmp = Float32(t_0 * sqrt(Float32(fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)) * Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.963999987

   1. Initial program 97.8%

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

   if 0.963999987 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 49.4%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9639999866485596:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 94.2

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

5. Simplified 94.2%

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

6. Final simplification 94.2%

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

Alternative 3: 93.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9929999709129333:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, -0.5, -1\right) \cdot \left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= (- 1.0 u1) 0.9929999709129333)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (* (sin t_0) (sqrt (* (fma u1 -0.5 -1.0) (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if ((1.0f - u1) <= 0.9929999709129333f) {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf((fmaf(u1, -0.5f, -1.0f) * -u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9929999709129333))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(fma(u1, Float32(-0.5), Float32(-1.0)) * Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.992999971

   1. Initial program 96.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 80.2

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

   5. Simplified 80.2%

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

   if 0.992999971 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 46.9%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 96.9

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

   5. Simplified 96.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9929999709129333:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, -0.5, -1\right) \cdot \left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 92.7

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

5. Simplified 92.7%

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

6. Final simplification 92.7%

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

Alternative 5: 90.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.2150000035762787:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \left(\frac{0.3333333333333333}{u1} + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.2150000035762787)
   (*
    u2
    (*
     (sqrt
      (+
       (/ 1.0 (* u1 (* u1 u1)))
       (+ 0.25 (+ (/ 0.3333333333333333 u1) (/ 0.5 (* u1 u1))))))
     (*
      (* u1 u1)
      (fma -1.3333333333333333 (* (* PI PI) (* PI (* u2 u2))) (* 2.0 PI)))))
   (* (sqrt u1) (sin (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.2150000035762787f) {
		tmp = u2 * (sqrtf(((1.0f / (u1 * (u1 * u1))) + (0.25f + ((0.3333333333333333f / u1) + (0.5f / (u1 * u1)))))) * ((u1 * u1) * fmaf(-1.3333333333333333f, ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * (u2 * u2))), (2.0f * ((float) M_PI)))));
	} else {
		tmp = sqrtf(u1) * sinf((2.0f * (((float) M_PI) * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.2150000035762787))
		tmp = Float32(u2 * Float32(sqrt(Float32(Float32(Float32(1.0) / Float32(u1 * Float32(u1 * u1))) + Float32(Float32(0.25) + Float32(Float32(Float32(0.3333333333333333) / u1) + Float32(Float32(0.5) / Float32(u1 * u1)))))) * Float32(Float32(u1 * u1) * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(Float32(2.0) * Float32(pi))))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.215000004

   1. Initial program 55.9%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in u1 around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f32 N/A

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

      9. lower-/.f32 N/A

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

   8. Simplified 94.4%

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

   9. Taylor expanded in u1 around 0

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

       1. lower-/.f32 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f32 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f32 94.4

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

   11. Simplified 94.4%

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

   12. Taylor expanded in u2 around 0

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

   14. Simplified 93.6%

       \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \left(\frac{0.3333333333333333}{u1} + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\right)} \]

   if 0.215000004 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.4%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in u1 around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f32 N/A

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

      9. lower-/.f32 N/A

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

   8. Simplified 91.8%

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

   9. Taylor expanded in u1 around 0

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

       1. lower-*.f32 N/A

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

       2. lower-sqrt.f32 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-sin.f32 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-PI.f32 78.1

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

   11. Simplified 78.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.2150000035762787:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \left(\frac{0.3333333333333333}{u1} + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \left(\frac{0.3333333333333333}{u1} + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (sqrt
    (+
     (/ 1.0 (* u1 (* u1 u1)))
     (+ 0.25 (+ (/ 0.3333333333333333 u1) (/ 0.5 (* u1 u1))))))
   (*
    (* u1 u1)
    (fma -1.3333333333333333 (* (* PI PI) (* PI (* u2 u2))) (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf(((1.0f / (u1 * (u1 * u1))) + (0.25f + ((0.3333333333333333f / u1) + (0.5f / (u1 * u1)))))) * ((u1 * u1) * fmaf(-1.3333333333333333f, ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * (u2 * u2))), (2.0f * ((float) M_PI)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(Float32(Float32(1.0) / Float32(u1 * Float32(u1 * u1))) + Float32(Float32(0.25) + Float32(Float32(Float32(0.3333333333333333) / u1) + Float32(Float32(0.5) / Float32(u1 * u1)))))) * Float32(Float32(u1 * u1) * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(Float32(2.0) * Float32(pi))))))
end

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 94.2

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

5. Simplified 94.2%

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

6. Taylor expanded in u1 around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-/.f32 N/A

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

8. Simplified 94.0%

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

9. Taylor expanded in u1 around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. cube-mult N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f32 94.0

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

11. Simplified 94.0%

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

12. Taylor expanded in u2 around 0

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

14. Simplified 85.1%

    \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \left(\frac{0.3333333333333333}{u1} + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)\right)} \]
15. Add Preprocessing

Alternative 7: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{0.3333333333333333}{u1} + \left(\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (sqrt
    (+
     (/ 0.3333333333333333 u1)
     (+ (/ 1.0 (* u1 (* u1 u1))) (+ 0.25 (/ 0.5 (* u1 u1))))))
   (*
    (* u1 u1)
    (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf(((0.3333333333333333f / u1) + ((1.0f / (u1 * (u1 * u1))) + (0.25f + (0.5f / (u1 * u1)))))) * ((u1 * u1) * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(Float32(Float32(0.3333333333333333) / u1) + Float32(Float32(Float32(1.0) / Float32(u1 * Float32(u1 * u1))) + Float32(Float32(0.25) + Float32(Float32(0.5) / Float32(u1 * u1)))))) * Float32(Float32(u1 * u1) * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0))))))
end

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 94.2

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

5. Simplified 94.2%

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

6. Taylor expanded in u1 around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-/.f32 N/A

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

8. Simplified 94.0%

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

9. Taylor expanded in u2 around 0

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

11. Simplified 85.1%

    \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{0.3333333333333333}{u1} + \left(\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(\frac{0.5}{u1 \cdot u1} + 0.25\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\right)} \]

12. Final simplification 85.1%

    \[\leadsto u2 \cdot \left(\sqrt{\frac{0.3333333333333333}{u1} + \left(\frac{1}{u1 \cdot \left(u1 \cdot u1\right)} + \left(0.25 + \frac{0.5}{u1 \cdot u1}\right)\right)} \cdot \left(\left(u1 \cdot u1\right) \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 80.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right) \cdot \mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (fma -1.3333333333333333 (* (* PI PI) (* PI (* u2 u2))) (* 2.0 PI))
   (fma 0.25 (sqrt (* u1 (* u1 u1))) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (fmaf(-1.3333333333333333f, ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * (u2 * u2))), (2.0f * ((float) M_PI))) * fmaf(0.25f, sqrtf((u1 * (u1 * u1))), sqrtf(u1)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(Float32(2.0) * Float32(pi))) * fma(Float32(0.25), sqrt(Float32(u1 * Float32(u1 * u1))), sqrt(u1))))
end

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 94.2

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

5. Simplified 94.2%

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

6. Taylor expanded in u1 around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-/.f32 N/A

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

8. Simplified 94.0%

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

9. Taylor expanded in u1 around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-out N/A

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

    4. lower-*.f32 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lower-sin.f32 N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. lower-*.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. lower-PI.f32 N/A

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

    13. +-commutative N/A

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

    14. lower-fma.f32 N/A

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

11. Simplified 89.7%

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

12. Taylor expanded in u2 around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. +-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. +-commutative N/A

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

14. Simplified 81.6%

    \[\leadsto \color{blue}{u2 \cdot \left(\mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right)\right)} \]

15. Final simplification 81.6%

    \[\leadsto u2 \cdot \left(\mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 2 \cdot \pi\right) \cdot \mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right)\right) \]
16. Add Preprocessing

Alternative 9: 81.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (*
      (sqrt
       (*
        (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0)
        (- u1)))
      t_0)
     (*
      (sqrt u1)
      (* u2 (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf((fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) * -u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * (u2 * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)) * Float32(-u1))) * t_0);
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 54.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 93.8

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

   8. Simplified 93.8%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.5%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 77.1

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

   5. Simplified 77.1%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow3 N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

   8. Simplified 59.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right) \cdot \left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (* (fma u1 (fma u1 -0.3333333333333333 -0.5) -1.0) (- u1))) t_0)
     (*
      (sqrt u1)
      (* u2 (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf((fmaf(u1, fmaf(u1, -0.3333333333333333f, -0.5f), -1.0f) * -u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * (u2 * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(fma(u1, fma(u1, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0)) * Float32(-u1))) * t_0);
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 54.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 92.3

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

   8. Simplified 92.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.5%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 77.1

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

   5. Simplified 77.1%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow3 N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

   8. Simplified 59.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right) \cdot \left(-u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right) \cdot \left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (* (fma u1 (fma u1 -0.3333333333333333 -0.5) -1.0) (- u1))) t_0)
     (*
      u2
      (*
       (sqrt u1)
       (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf((fmaf(u1, fmaf(u1, -0.3333333333333333f, -0.5f), -1.0f) * -u1)) * t_0;
	} else {
		tmp = u2 * (sqrtf(u1) * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(fma(u1, fma(u1, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0)) * Float32(-u1))) * t_0);
	else
		tmp = Float32(u2 * Float32(sqrt(u1) * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 54.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 92.3

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

   8. Simplified 92.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.5%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 77.1

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

   5. Simplified 77.1%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 59.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right) \cdot \left(-u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.5% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 48.8

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

5. Simplified 48.8%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 76.9

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

8. Simplified 76.9%

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

9. Final simplification 76.9%

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

Alternative 13: 73.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 48.8

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

5. Simplified 48.8%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 74.9

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

8. Simplified 74.9%

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

9. Final simplification 74.9%

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

Alternative 14: 66.0% accurate, 8.9× 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 55.6%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 78.6

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

5. Simplified 78.6%

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

6. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-PI.f32 68.2

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

8. Simplified 68.2%

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

9. Final simplification 68.2%

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

Alternative 15: 21.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot u2\right) \cdot \mathsf{fma}\left(u1 + 0.6666666666666666, u1, 0.7777777777777778\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* PI u2) (fma (+ u1 0.6666666666666666) u1 0.7777777777777778)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((float) M_PI) * u2) * fmaf((u1 + 0.6666666666666666f), u1, 0.7777777777777778f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(pi) * u2) * fma(Float32(u1 + Float32(0.6666666666666666)), u1, Float32(0.7777777777777778)))
end

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 48.8

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

5. Simplified 48.8%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 77.9

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

8. Simplified 77.9%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{{u1}^{2} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \color{blue}{{u1}^{2} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    4. associate-+r+ N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    5. associate-*r/ N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \color{blue}{\frac{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{{u1}^{2}}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \frac{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{u1 \cdot u1}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    7. times-frac N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \color{blue}{\frac{\frac{7}{9}}{u1} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    8. distribute-rgt-out N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\color{blue}{\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} \cdot \left(\frac{2}{3} + \frac{\frac{7}{9}}{u1}\right)} + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    9. lower-fma.f32 N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    10. lower-/.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    11. lower-*.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{u2 \cdot \mathsf{PI}\left(\right)}}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    12. lower-PI.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    13. lower-+.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \color{blue}{\frac{2}{3} + \frac{\frac{7}{9}}{u1}}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    14. lower-/.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \color{blue}{\frac{\frac{7}{9}}{u1}}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    15. lower-*.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right) \]

    16. lower-PI.f32 20.7

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \pi}{u1}, 0.6666666666666666 + \frac{0.7777777777777778}{u1}, u2 \cdot \color{blue}{\pi}\right) \]

11. Simplified 20.7%

    \[\leadsto \color{blue}{\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \pi}{u1}, 0.6666666666666666 + \frac{0.7777777777777778}{u1}, u2 \cdot \pi\right)} \]

12. Taylor expanded in u1 around 0

    \[\leadsto \color{blue}{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + u1 \cdot \left(\frac{2}{3} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + u1 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{u1 \cdot \left(\frac{2}{3} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + u1 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + u1 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u1} + \frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) \]

    3. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{2}{3} + u1\right)\right)} \cdot u1 + \frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{2}{3} + u1\right) \cdot u1\right)} + \frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{2}{3} + u1\right) \cdot u1\right) + \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{7}{9}} \]

    6. distribute-lft-out N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{2}{3} + u1\right) \cdot u1 + \frac{7}{9}\right)} \]

    7. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{2}{3} + u1\right) \cdot u1 + \frac{7}{9}\right)} \]

    8. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(\frac{2}{3} + u1\right) \cdot u1 + \frac{7}{9}\right) \]

    9. lower-PI.f32 N/A

       \[\leadsto \left(u2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\frac{2}{3} + u1\right) \cdot u1 + \frac{7}{9}\right) \]

    10. lower-fma.f32 N/A

        \[\leadsto \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{3} + u1, u1, \frac{7}{9}\right)} \]

    11. +-commutative N/A

        \[\leadsto \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\color{blue}{u1 + \frac{2}{3}}, u1, \frac{7}{9}\right) \]

    12. lower-+.f32 20.7

        \[\leadsto \left(u2 \cdot \pi\right) \cdot \mathsf{fma}\left(\color{blue}{u1 + 0.6666666666666666}, u1, 0.7777777777777778\right) \]

14. Simplified 20.7%

    \[\leadsto \color{blue}{\left(u2 \cdot \pi\right) \cdot \mathsf{fma}\left(u1 + 0.6666666666666666, u1, 0.7777777777777778\right)} \]

15. Final simplification 20.7%

    \[\leadsto \left(\pi \cdot u2\right) \cdot \mathsf{fma}\left(u1 + 0.6666666666666666, u1, 0.7777777777777778\right) \]
16. Add Preprocessing

Alternative 16: 20.9% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot u2\right) \cdot \mathsf{fma}\left(u1, 0.6666666666666666, 0.7777777777777778\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* PI u2) (fma u1 0.6666666666666666 0.7777777777777778)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((float) M_PI) * u2) * fmaf(u1, 0.6666666666666666f, 0.7777777777777778f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(pi) * u2) * fma(u1, Float32(0.6666666666666666), Float32(0.7777777777777778)))
end

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 48.8

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

5. Simplified 48.8%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 77.9

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

8. Simplified 77.9%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{{u1}^{2} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \color{blue}{{u1}^{2} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    4. associate-+r+ N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    5. associate-*r/ N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \color{blue}{\frac{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{{u1}^{2}}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \frac{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{u1 \cdot u1}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    7. times-frac N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \color{blue}{\frac{\frac{7}{9}}{u1} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    8. distribute-rgt-out N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\color{blue}{\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} \cdot \left(\frac{2}{3} + \frac{\frac{7}{9}}{u1}\right)} + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    9. lower-fma.f32 N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    10. lower-/.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    11. lower-*.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{u2 \cdot \mathsf{PI}\left(\right)}}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    12. lower-PI.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    13. lower-+.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \color{blue}{\frac{2}{3} + \frac{\frac{7}{9}}{u1}}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    14. lower-/.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \color{blue}{\frac{\frac{7}{9}}{u1}}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    15. lower-*.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right) \]

    16. lower-PI.f32 20.7

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \pi}{u1}, 0.6666666666666666 + \frac{0.7777777777777778}{u1}, u2 \cdot \color{blue}{\pi}\right) \]

11. Simplified 20.7%

    \[\leadsto \color{blue}{\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \pi}{u1}, 0.6666666666666666 + \frac{0.7777777777777778}{u1}, u2 \cdot \pi\right)} \]

12. Taylor expanded in u1 around 0

    \[\leadsto \color{blue}{\frac{2}{3} \cdot \left(u1 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)} \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot u1\right) \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)} + \frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) \]

    2. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{2}{3} \cdot u1 + \frac{7}{9}\right)} \]

    3. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{2}{3} \cdot u1 + \frac{7}{9}\right)} \]

    4. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{2}{3} \cdot u1 + \frac{7}{9}\right) \]

    5. lower-PI.f32 N/A

       \[\leadsto \left(u2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{2}{3} \cdot u1 + \frac{7}{9}\right) \]

    6. *-commutative N/A

       \[\leadsto \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{u1 \cdot \frac{2}{3}} + \frac{7}{9}\right) \]

    7. lower-fma.f32 20.4

       \[\leadsto \left(u2 \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(u1, 0.6666666666666666, 0.7777777777777778\right)} \]

14. Simplified 20.4%

    \[\leadsto \color{blue}{\left(u2 \cdot \pi\right) \cdot \mathsf{fma}\left(u1, 0.6666666666666666, 0.7777777777777778\right)} \]

15. Final simplification 20.4%

    \[\leadsto \left(\pi \cdot u2\right) \cdot \mathsf{fma}\left(u1, 0.6666666666666666, 0.7777777777777778\right) \]
16. Add Preprocessing

Alternative 17: 20.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\pi \cdot 0.7777777777777778\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (* PI 0.7777777777777778)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (((float) M_PI) * 0.7777777777777778f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(pi) * Float32(0.7777777777777778)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(pi) * single(0.7777777777777778));
end

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 48.8

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

5. Simplified 48.8%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 77.9

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

8. Simplified 77.9%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{{u1}^{2} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \color{blue}{{u1}^{2} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \left(\frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}} + u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

    4. associate-+r+ N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \frac{7}{9} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{{u1}^{2}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    5. associate-*r/ N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \color{blue}{\frac{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{{u1}^{2}}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \frac{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{u1 \cdot u1}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    7. times-frac N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\left(\frac{2}{3} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} + \color{blue}{\frac{\frac{7}{9}}{u1} \cdot \frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}}\right) + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    8. distribute-rgt-out N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \left(\color{blue}{\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1} \cdot \left(\frac{2}{3} + \frac{\frac{7}{9}}{u1}\right)} + u2 \cdot \mathsf{PI}\left(\right)\right) \]

    9. lower-fma.f32 N/A

       \[\leadsto \left(u1 \cdot u1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right)} \]

    10. lower-/.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    11. lower-*.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{u2 \cdot \mathsf{PI}\left(\right)}}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    12. lower-PI.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    13. lower-+.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \color{blue}{\frac{2}{3} + \frac{\frac{7}{9}}{u1}}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    14. lower-/.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \color{blue}{\frac{\frac{7}{9}}{u1}}, u2 \cdot \mathsf{PI}\left(\right)\right) \]

    15. lower-*.f32 N/A

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \mathsf{PI}\left(\right)}{u1}, \frac{2}{3} + \frac{\frac{7}{9}}{u1}, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right) \]

    16. lower-PI.f32 20.7

        \[\leadsto \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \pi}{u1}, 0.6666666666666666 + \frac{0.7777777777777778}{u1}, u2 \cdot \color{blue}{\pi}\right) \]

11. Simplified 20.7%

    \[\leadsto \color{blue}{\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{u2 \cdot \pi}{u1}, 0.6666666666666666 + \frac{0.7777777777777778}{u1}, u2 \cdot \pi\right)} \]

12. Taylor expanded in u1 around 0

    \[\leadsto \color{blue}{\frac{7}{9} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{7}{9}} \]

    2. associate-*l* N/A

       \[\leadsto \color{blue}{u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{7}{9}\right)} \]

    3. lower-*.f32 N/A

       \[\leadsto \color{blue}{u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{7}{9}\right)} \]

    4. lower-*.f32 N/A

       \[\leadsto u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{7}{9}\right)} \]

    5. lower-PI.f32 20.3

       \[\leadsto u2 \cdot \left(\color{blue}{\pi} \cdot 0.7777777777777778\right) \]

14. Simplified 20.3%

    \[\leadsto \color{blue}{u2 \cdot \left(\pi \cdot 0.7777777777777778\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))