Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.1% → 98.7%

Time: 5.5s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-1 \cdot \left(\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.05000000074505806)
   (*
    (sqrt
     (*
      -1.0
      (*
       (/
        (- (pow (* (- (* (* -0.25 u1) u1) 0.5) u1) 3.0) 1.0)
        (+
         1.0
         (*
          u1
          (- (* u1 (- (* 0.08333333333333333 u1) 0.08333333333333333)) 0.5))))
       u1)))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (* -1.0 (log (- 1.0 u1)))) (sin (fma (* PI u2) 2.0 (/ PI 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.05000000074505806f) {
		tmp = sqrtf((-1.0f * (((powf(((((-0.25f * u1) * u1) - 0.5f) * u1), 3.0f) - 1.0f) / (1.0f + (u1 * ((u1 * ((0.08333333333333333f * u1) - 0.08333333333333333f)) - 0.5f)))) * u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf((-1.0f * logf((1.0f - u1)))) * sinf(fmaf((((float) M_PI) * u2), 2.0f, (((float) M_PI) / 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32((Float32(Float32(Float32(Float32(Float32(-0.25) * u1) * u1) - Float32(0.5)) * u1) ^ Float32(3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(u1 * Float32(Float32(u1 * Float32(Float32(Float32(0.08333333333333333) * u1) - Float32(0.08333333333333333))) - Float32(0.5))))) * u1))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(1.0) - u1)))) * sin(fma(Float32(Float32(pi) * u2), Float32(2.0), Float32(Float32(pi) / Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0500000007

   1. Initial program 53.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. flip3-- N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 98.9%

      \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{\mathsf{fma}\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1, \left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1, 1 + \left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right) \cdot 1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-*.f32 99.0

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. Applied rewrites 99.0%

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around inf

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   12. Step-by-step derivation

       1. lift-*.f32 99.0

          \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. Applied rewrites 99.0%

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0500000007 < u1

   1. Initial program 97.8%

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

      1. lift-cos.f32 N/A

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

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

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lift-PI.f32 97.9

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

   4. Applied rewrites 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-1 \cdot \left(\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-1 \cdot \left(\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= u1 0.05000000074505806)
     (*
      (sqrt
       (*
        -1.0
        (*
         (/
          (- (pow (* (- (* (* -0.25 u1) u1) 0.5) u1) 3.0) 1.0)
          (+
           1.0
           (*
            u1
            (-
             (* u1 (- (* 0.08333333333333333 u1) 0.08333333333333333))
             0.5))))
         u1)))
      t_0)
     (* (sqrt (* -1.0 (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.05000000074505806f) {
		tmp = sqrtf((-1.0f * (((powf(((((-0.25f * u1) * u1) - 0.5f) * u1), 3.0f) - 1.0f) / (1.0f + (u1 * ((u1 * ((0.08333333333333333f * u1) - 0.08333333333333333f)) - 0.5f)))) * u1))) * t_0;
	} else {
		tmp = sqrtf((-1.0f * logf((1.0f - u1)))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32((Float32(Float32(Float32(Float32(Float32(-0.25) * u1) * u1) - Float32(0.5)) * u1) ^ Float32(3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(u1 * Float32(Float32(u1 * Float32(Float32(Float32(0.08333333333333333) * u1) - Float32(0.08333333333333333))) - Float32(0.5))))) * u1))) * t_0);
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos(((single(2.0) * single(pi)) * u2));
	tmp = single(0.0);
	if (u1 <= single(0.05000000074505806))
		tmp = sqrt((single(-1.0) * ((((((((single(-0.25) * u1) * u1) - single(0.5)) * u1) ^ single(3.0)) - single(1.0)) / (single(1.0) + (u1 * ((u1 * ((single(0.08333333333333333) * u1) - single(0.08333333333333333))) - single(0.5))))) * u1))) * t_0;
	else
		tmp = sqrt((single(-1.0) * log((single(1.0) - u1)))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0500000007

   1. Initial program 53.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. flip3-- N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 98.9%

      \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{\mathsf{fma}\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1, \left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1, 1 + \left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right) \cdot 1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-*.f32 99.0

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. Applied rewrites 99.0%

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around inf

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   12. Step-by-step derivation

       1. lift-*.f32 99.0

          \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. Applied rewrites 99.0%

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0500000007 < u1

   1. Initial program 97.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-1 \cdot \left(\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-1 \cdot \left(\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.054999999701976776)
   (*
    (sqrt
     (*
      -1.0
      (*
       (/
        (- (pow (* (- (* (* -0.25 u1) u1) 0.5) u1) 3.0) 1.0)
        (+
         1.0
         (*
          u1
          (- (* u1 (- (* 0.08333333333333333 u1) 0.08333333333333333)) 0.5))))
       u1)))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (sin (fma (* PI u2) 2.0 (/ PI 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.054999999701976776f) {
		tmp = sqrtf((-1.0f * (((powf(((((-0.25f * u1) * u1) - 0.5f) * u1), 3.0f) - 1.0f) / (1.0f + (u1 * ((u1 * ((0.08333333333333333f * u1) - 0.08333333333333333f)) - 0.5f)))) * u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * sinf(fmaf((((float) M_PI) * u2), 2.0f, (((float) M_PI) / 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32((Float32(Float32(Float32(Float32(Float32(-0.25) * u1) * u1) - Float32(0.5)) * u1) ^ Float32(3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(u1 * Float32(Float32(u1 * Float32(Float32(Float32(0.08333333333333333) * u1) - Float32(0.08333333333333333))) - Float32(0.5))))) * u1))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * sin(fma(Float32(Float32(pi) * u2), Float32(2.0), Float32(Float32(pi) / Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0549999997

   1. Initial program 53.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. flip3-- N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 98.8%

      \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{\mathsf{fma}\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1, \left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1, 1 + \left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right) \cdot 1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-*.f32 98.9

         \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. Applied rewrites 98.9%

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around inf

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(\frac{-1}{4} \cdot u1\right) \cdot u1 - \frac{1}{2}\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{12} \cdot u1 - \frac{1}{12}\right) - \frac{1}{2}\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   12. Step-by-step derivation

       1. lift-*.f32 99.0

          \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   13. Applied rewrites 99.0%

       \[\leadsto \sqrt{-\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0549999997 < u1

   1. Initial program 98.0%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.8

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-1 \cdot \left(\frac{{\left(\left(\left(-0.25 \cdot u1\right) \cdot u1 - 0.5\right) \cdot u1\right)}^{3} - 1}{1 + u1 \cdot \left(u1 \cdot \left(0.08333333333333333 \cdot u1 - 0.08333333333333333\right) - 0.5\right)} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.042100001126527786:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.042100001126527786)
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (sin (fma (* PI u2) 2.0 (/ PI 2.0))))
   (*
    (sqrt
     (*
      -1.0
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (cos (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (logf((1.0f - u1)) <= -0.042100001126527786f) {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * sinf(fmaf((((float) M_PI) * u2), 2.0f, (((float) M_PI) / 2.0f)));
	} else {
		tmp = sqrtf((-1.0f * (((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.042100001126527786))
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * sin(fma(Float32(Float32(pi) * u2), Float32(2.0), Float32(Float32(pi) / Float32(2.0)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0421000011

   1. Initial program 97.7%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.6

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 97.6%

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

   if -0.0421000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 53.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.042100001126527786:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.05000000074505806:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{\frac{0.5 + \frac{1}{u1}}{u1} + 0.3333333333333333}{u1} \cdot -1 - 0.25\right) \cdot {u1}^{4}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.05000000074505806)
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (sin (fma (* PI u2) 2.0 (/ PI 2.0))))
   (*
    (sqrt
     (*
      -1.0
      (*
       (-
        (* (/ (+ (/ (+ 0.5 (/ 1.0 u1)) u1) 0.3333333333333333) u1) -1.0)
        0.25)
       (pow u1 4.0))))
    (cos (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (logf((1.0f - u1)) <= -0.05000000074505806f) {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * sinf(fmaf((((float) M_PI) * u2), 2.0f, (((float) M_PI) / 2.0f)));
	} else {
		tmp = sqrtf((-1.0f * (((((((0.5f + (1.0f / u1)) / u1) + 0.3333333333333333f) / u1) * -1.0f) - 0.25f) * powf(u1, 4.0f)))) * cosf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.05000000074505806))
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * sin(fma(Float32(Float32(pi) * u2), Float32(2.0), Float32(Float32(pi) / Float32(2.0)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(1.0) / u1)) / u1) + Float32(0.3333333333333333)) / u1) * Float32(-1.0)) - Float32(0.25)) * (u1 ^ Float32(4.0))))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0500000007

   1. Initial program 97.8%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.7

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 97.7%

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

   if -0.0500000007 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 53.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in u1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in u1 around inf

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

       1. lower-/.f32 N/A

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

       2. lower-+.f32 N/A

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

       3. lift-/.f32 98.8

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

   11. Applied rewrites 98.8%

       \[\leadsto \sqrt{-\left(\frac{\frac{0.5 + \frac{1}{u1}}{u1} + 0.3333333333333333}{u1} \cdot -1 - 0.25\right) \cdot {u1}^{4}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.05000000074505806:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{\frac{0.5 + \frac{1}{u1}}{u1} + 0.3333333333333333}{u1} \cdot -1 - 0.25\right) \cdot {u1}^{4}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\\ t_1 := \frac{1}{\sqrt{u1}} \cdot t\_0\\ \mathbf{if}\;\log \left(1 - u1\right) \leq -0.02800000086426735:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, t\_0, \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot t\_1 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, t\_1, 0.5 \cdot \left(\sqrt{u1} \cdot \left(t\_0 \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (fma 0.5 PI (* 2.0 (* u2 PI)))))
        (t_1 (* (/ 1.0 (sqrt u1)) t_0)))
   (if (<= (log (- 1.0 u1)) -0.02800000086426735)
     (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (sin (fma (* PI u2) 2.0 (/ PI 2.0))))
     (fma
      (sqrt u1)
      t_0
      (*
       (* u1 u1)
       (-
        (* 0.25 t_1)
        (*
         (* -1.0 u1)
         (fma
          0.16666666666666666
          t_1
          (* 0.5 (* (sqrt u1) (* t_0 (- 0.25 (* 0.0625 (/ 1.0 u1))))))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(fmaf(0.5f, ((float) M_PI), (2.0f * (u2 * ((float) M_PI)))));
	float t_1 = (1.0f / sqrtf(u1)) * t_0;
	float tmp;
	if (logf((1.0f - u1)) <= -0.02800000086426735f) {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * sinf(fmaf((((float) M_PI) * u2), 2.0f, (((float) M_PI) / 2.0f)));
	} else {
		tmp = fmaf(sqrtf(u1), t_0, ((u1 * u1) * ((0.25f * t_1) - ((-1.0f * u1) * fmaf(0.16666666666666666f, t_1, (0.5f * (sqrtf(u1) * (t_0 * (0.25f - (0.0625f * (1.0f / u1)))))))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(fma(Float32(0.5), Float32(pi), Float32(Float32(2.0) * Float32(u2 * Float32(pi)))))
	t_1 = Float32(Float32(Float32(1.0) / sqrt(u1)) * t_0)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.02800000086426735))
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * sin(fma(Float32(Float32(pi) * u2), Float32(2.0), Float32(Float32(pi) / Float32(2.0)))));
	else
		tmp = fma(sqrt(u1), t_0, Float32(Float32(u1 * u1) * Float32(Float32(Float32(0.25) * t_1) - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(0.16666666666666666), t_1, Float32(Float32(0.5) * Float32(sqrt(u1) * Float32(t_0 * Float32(Float32(0.25) - Float32(Float32(0.0625) * Float32(Float32(1.0) / u1)))))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0280000009

   1. Initial program 97.4%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.0

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 97.2%

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

   if -0.0280000009 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.5%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 49.7

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.25 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right)\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) \cdot \sqrt{u1}\right)} \]

   8. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(2 \cdot \left(\sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(u2 \cdot \left(-2 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(-1 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right) + \color{blue}{\left(\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]

   10. Applied rewrites 90.2%

       \[\leadsto \mathsf{fma}\left(u2, \color{blue}{\mathsf{fma}\left(2, \sqrt{u1} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \mathsf{fma}\left(u2, -2 \cdot \left(\sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - \left(-u1 \cdot u1\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(-1, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right), -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.3333333333333333, \frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \sqrt{u1} \cdot \left(\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right)}, \sqrt{u1} \cdot \sin \left(0.5 \cdot \pi\right) - \left(-u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

   11. Taylor expanded in u2 around inf

       \[\leadsto \sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{{u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)} \]

   13. Applied rewrites 98.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{u1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)}, \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.02800000086426735:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\\ t_1 := \frac{1}{\sqrt{u1}} \cdot t\_0\\ \mathbf{if}\;\log \left(1 - u1\right) \leq -0.02800000086426735:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{\left(\frac{1}{u1} - 1\right) \cdot u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, t\_0, \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot t\_1 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, t\_1, 0.5 \cdot \left(\sqrt{u1} \cdot \left(t\_0 \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (fma 0.5 PI (* 2.0 (* u2 PI)))))
        (t_1 (* (/ 1.0 (sqrt u1)) t_0)))
   (if (<= (log (- 1.0 u1)) -0.02800000086426735)
     (*
      (sqrt (log (/ 1.0 (* (- (/ 1.0 u1) 1.0) u1))))
      (sin (fma (* PI u2) 2.0 (/ PI 2.0))))
     (fma
      (sqrt u1)
      t_0
      (*
       (* u1 u1)
       (-
        (* 0.25 t_1)
        (*
         (* -1.0 u1)
         (fma
          0.16666666666666666
          t_1
          (* 0.5 (* (sqrt u1) (* t_0 (- 0.25 (* 0.0625 (/ 1.0 u1))))))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(fmaf(0.5f, ((float) M_PI), (2.0f * (u2 * ((float) M_PI)))));
	float t_1 = (1.0f / sqrtf(u1)) * t_0;
	float tmp;
	if (logf((1.0f - u1)) <= -0.02800000086426735f) {
		tmp = sqrtf(logf((1.0f / (((1.0f / u1) - 1.0f) * u1)))) * sinf(fmaf((((float) M_PI) * u2), 2.0f, (((float) M_PI) / 2.0f)));
	} else {
		tmp = fmaf(sqrtf(u1), t_0, ((u1 * u1) * ((0.25f * t_1) - ((-1.0f * u1) * fmaf(0.16666666666666666f, t_1, (0.5f * (sqrtf(u1) * (t_0 * (0.25f - (0.0625f * (1.0f / u1)))))))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(fma(Float32(0.5), Float32(pi), Float32(Float32(2.0) * Float32(u2 * Float32(pi)))))
	t_1 = Float32(Float32(Float32(1.0) / sqrt(u1)) * t_0)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.02800000086426735))
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / u1) - Float32(1.0)) * u1)))) * sin(fma(Float32(Float32(pi) * u2), Float32(2.0), Float32(Float32(pi) / Float32(2.0)))));
	else
		tmp = fma(sqrt(u1), t_0, Float32(Float32(u1 * u1) * Float32(Float32(Float32(0.25) * t_1) - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(0.16666666666666666), t_1, Float32(Float32(0.5) * Float32(sqrt(u1) * Float32(t_0 * Float32(Float32(0.25) - Float32(Float32(0.0625) * Float32(Float32(1.0) / u1)))))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0280000009

   1. Initial program 97.4%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.0

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 96.6

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

   7. Applied rewrites 96.6%

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

   if -0.0280000009 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.5%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 49.7

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

      8. lift-cos.f32 N/A

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

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

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

      10. lower-sin.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.25 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right)\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) \cdot \sqrt{u1}\right)} \]

   8. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \left(2 \cdot \left(\sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(u2 \cdot \left(-2 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(-1 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right) + \color{blue}{\left(\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]

   10. Applied rewrites 90.2%

       \[\leadsto \mathsf{fma}\left(u2, \color{blue}{\mathsf{fma}\left(2, \sqrt{u1} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \mathsf{fma}\left(u2, -2 \cdot \left(\sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - \left(-u1 \cdot u1\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(-1, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right), -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.3333333333333333, \frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \sqrt{u1} \cdot \left(\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right)}, \sqrt{u1} \cdot \sin \left(0.5 \cdot \pi\right) - \left(-u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

   11. Taylor expanded in u2 around inf

       \[\leadsto \sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{{u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)} \]

   13. Applied rewrites 98.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{u1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)}, \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.02800000086426735:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{\left(\frac{1}{u1} - 1\right) \cdot u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\\ t_1 := \frac{1}{\sqrt{u1}} \cdot t\_0\\ \mathsf{fma}\left(\sqrt{u1}, t\_0, \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot t\_1 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, t\_1, 0.5 \cdot \left(\sqrt{u1} \cdot \left(t\_0 \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (fma 0.5 PI (* 2.0 (* u2 PI)))))
        (t_1 (* (/ 1.0 (sqrt u1)) t_0)))
   (fma
    (sqrt u1)
    t_0
    (*
     (* u1 u1)
     (-
      (* 0.25 t_1)
      (*
       (* -1.0 u1)
       (fma
        0.16666666666666666
        t_1
        (* 0.5 (* (sqrt u1) (* t_0 (- 0.25 (* 0.0625 (/ 1.0 u1)))))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(fmaf(0.5f, ((float) M_PI), (2.0f * (u2 * ((float) M_PI)))));
	float t_1 = (1.0f / sqrtf(u1)) * t_0;
	return fmaf(sqrtf(u1), t_0, ((u1 * u1) * ((0.25f * t_1) - ((-1.0f * u1) * fmaf(0.16666666666666666f, t_1, (0.5f * (sqrtf(u1) * (t_0 * (0.25f - (0.0625f * (1.0f / u1)))))))))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(fma(Float32(0.5), Float32(pi), Float32(Float32(2.0) * Float32(u2 * Float32(pi)))))
	t_1 = Float32(Float32(Float32(1.0) / sqrt(u1)) * t_0)
	return fma(sqrt(u1), t_0, Float32(Float32(u1 * u1) * Float32(Float32(Float32(0.25) * t_1) - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(0.16666666666666666), t_1, Float32(Float32(0.5) * Float32(sqrt(u1) * Float32(t_0 * Float32(Float32(0.25) - Float32(Float32(0.0625) * Float32(Float32(1.0) / u1)))))))))))
end

Derivation

1. Initial program 60.7%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. neg-log N/A

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

   5. lower-log.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift--.f32 58.4

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

   8. lift-cos.f32 N/A

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

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

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

   10. lower-sin.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 N/A

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

4. Applied rewrites 58.4%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

7. Applied rewrites 92.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.25 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right)\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) \cdot \sqrt{u1}\right)} \]

8. Taylor expanded in u2 around 0

   \[\leadsto u2 \cdot \left(2 \cdot \left(\sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(u2 \cdot \left(-2 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(-1 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right) + \color{blue}{\left(\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]

10. Applied rewrites 84.6%

    \[\leadsto \mathsf{fma}\left(u2, \color{blue}{\mathsf{fma}\left(2, \sqrt{u1} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \mathsf{fma}\left(u2, -2 \cdot \left(\sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - \left(-u1 \cdot u1\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(-1, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right), -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.3333333333333333, \frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \sqrt{u1} \cdot \left(\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right)}, \sqrt{u1} \cdot \sin \left(0.5 \cdot \pi\right) - \left(-u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

11. Taylor expanded in u2 around inf

    \[\leadsto \sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{{u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)} \]

13. Applied rewrites 92.6%

    \[\leadsto \mathsf{fma}\left(\sqrt{u1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)}, \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

14. Final simplification 92.6%

    \[\leadsto \mathsf{fma}\left(\sqrt{u1}, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 9: 83.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := 0.25 - 0.0625 \cdot \frac{1}{u1}\\ t_2 := \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\\ t_3 := \sin \left(0.5 \cdot \pi\right)\\ t_4 := \left(\pi \cdot \pi\right) \cdot t\_3\\ t_5 := t\_0 \cdot t\_4\\ t_6 := t\_0 \cdot t\_3\\ t_7 := t\_3 \cdot t\_1\\ t_8 := \pi \cdot t\_2\\ t_9 := t\_0 \cdot t\_8\\ t_10 := \left(-1 \cdot u1\right) \cdot u1\\ \mathsf{fma}\left(u2, \mathsf{fma}\left(2, \sqrt{u1} \cdot t\_8, \mathsf{fma}\left(u2, -2 \cdot \left(\sqrt{u1} \cdot t\_4\right) - t\_10 \cdot \left(-0.5 \cdot t\_5 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(-1, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_7\right), -0.3333333333333333 \cdot t\_5\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.5 \cdot t\_9 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.3333333333333333, t\_9, \sqrt{u1} \cdot \left(\pi \cdot \left(t\_2 \cdot t\_1\right)\right)\right)\right)\right)\right), \sqrt{u1} \cdot t\_3 - t\_10 \cdot \left(0.25 \cdot t\_6 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, t\_6, 0.5 \cdot \left(\sqrt{u1} \cdot t\_7\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1)))
        (t_1 (- 0.25 (* 0.0625 (/ 1.0 u1))))
        (t_2 (sin (fma 0.5 PI (/ PI 2.0))))
        (t_3 (sin (* 0.5 PI)))
        (t_4 (* (* PI PI) t_3))
        (t_5 (* t_0 t_4))
        (t_6 (* t_0 t_3))
        (t_7 (* t_3 t_1))
        (t_8 (* PI t_2))
        (t_9 (* t_0 t_8))
        (t_10 (* (* -1.0 u1) u1)))
   (fma
    u2
    (fma
     2.0
     (* (sqrt u1) t_8)
     (fma
      u2
      (-
       (* -2.0 (* (sqrt u1) t_4))
       (*
        t_10
        (-
         (* -0.5 t_5)
         (*
          (* -1.0 u1)
          (fma
           -1.0
           (* (sqrt u1) (* (* PI PI) t_7))
           (* -0.3333333333333333 t_5))))))
      (*
       (* u1 u1)
       (-
        (* 0.5 t_9)
        (*
         (* -1.0 u1)
         (fma 0.3333333333333333 t_9 (* (sqrt u1) (* PI (* t_2 t_1)))))))))
    (-
     (* (sqrt u1) t_3)
     (*
      t_10
      (-
       (* 0.25 t_6)
       (*
        (* -1.0 u1)
        (fma 0.16666666666666666 t_6 (* 0.5 (* (sqrt u1) t_7))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f / sqrtf(u1);
	float t_1 = 0.25f - (0.0625f * (1.0f / u1));
	float t_2 = sinf(fmaf(0.5f, ((float) M_PI), (((float) M_PI) / 2.0f)));
	float t_3 = sinf((0.5f * ((float) M_PI)));
	float t_4 = (((float) M_PI) * ((float) M_PI)) * t_3;
	float t_5 = t_0 * t_4;
	float t_6 = t_0 * t_3;
	float t_7 = t_3 * t_1;
	float t_8 = ((float) M_PI) * t_2;
	float t_9 = t_0 * t_8;
	float t_10 = (-1.0f * u1) * u1;
	return fmaf(u2, fmaf(2.0f, (sqrtf(u1) * t_8), fmaf(u2, ((-2.0f * (sqrtf(u1) * t_4)) - (t_10 * ((-0.5f * t_5) - ((-1.0f * u1) * fmaf(-1.0f, (sqrtf(u1) * ((((float) M_PI) * ((float) M_PI)) * t_7)), (-0.3333333333333333f * t_5)))))), ((u1 * u1) * ((0.5f * t_9) - ((-1.0f * u1) * fmaf(0.3333333333333333f, t_9, (sqrtf(u1) * (((float) M_PI) * (t_2 * t_1))))))))), ((sqrtf(u1) * t_3) - (t_10 * ((0.25f * t_6) - ((-1.0f * u1) * fmaf(0.16666666666666666f, t_6, (0.5f * (sqrtf(u1) * t_7))))))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = Float32(Float32(0.25) - Float32(Float32(0.0625) * Float32(Float32(1.0) / u1)))
	t_2 = sin(fma(Float32(0.5), Float32(pi), Float32(Float32(pi) / Float32(2.0))))
	t_3 = sin(Float32(Float32(0.5) * Float32(pi)))
	t_4 = Float32(Float32(Float32(pi) * Float32(pi)) * t_3)
	t_5 = Float32(t_0 * t_4)
	t_6 = Float32(t_0 * t_3)
	t_7 = Float32(t_3 * t_1)
	t_8 = Float32(Float32(pi) * t_2)
	t_9 = Float32(t_0 * t_8)
	t_10 = Float32(Float32(Float32(-1.0) * u1) * u1)
	return fma(u2, fma(Float32(2.0), Float32(sqrt(u1) * t_8), fma(u2, Float32(Float32(Float32(-2.0) * Float32(sqrt(u1) * t_4)) - Float32(t_10 * Float32(Float32(Float32(-0.5) * t_5) - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(-1.0), Float32(sqrt(u1) * Float32(Float32(Float32(pi) * Float32(pi)) * t_7)), Float32(Float32(-0.3333333333333333) * t_5)))))), Float32(Float32(u1 * u1) * Float32(Float32(Float32(0.5) * t_9) - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(0.3333333333333333), t_9, Float32(sqrt(u1) * Float32(Float32(pi) * Float32(t_2 * t_1))))))))), Float32(Float32(sqrt(u1) * t_3) - Float32(t_10 * Float32(Float32(Float32(0.25) * t_6) - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(0.16666666666666666), t_6, Float32(Float32(0.5) * Float32(sqrt(u1) * t_7))))))))
end

Derivation

1. Initial program 60.7%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. neg-log N/A

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

   5. lower-log.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift--.f32 58.4

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

   8. lift-cos.f32 N/A

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

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

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

   10. lower-sin.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 N/A

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

4. Applied rewrites 58.4%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

7. Applied rewrites 92.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.25 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right)\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \pi \cdot 0.5\right)\right) \cdot \sqrt{u1}\right)} \]

8. Taylor expanded in u2 around 0

   \[\leadsto u2 \cdot \left(2 \cdot \left(\sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(u2 \cdot \left(-2 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(-1 \cdot \left(\sqrt{u1} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \frac{-1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right) + \color{blue}{\left(\sqrt{u1} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) - -1 \cdot \left(u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]

10. Applied rewrites 84.6%

    \[\leadsto \mathsf{fma}\left(u2, \color{blue}{\mathsf{fma}\left(2, \sqrt{u1} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \mathsf{fma}\left(u2, -2 \cdot \left(\sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - \left(-u1 \cdot u1\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(-1, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right), -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.3333333333333333, \frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \sqrt{u1} \cdot \left(\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)\right)}, \sqrt{u1} \cdot \sin \left(0.5 \cdot \pi\right) - \left(-u1 \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right)\right) - -1 \cdot \left(u1 \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

11. Final simplification 84.6%

    \[\leadsto \mathsf{fma}\left(u2, \mathsf{fma}\left(2, \sqrt{u1} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \mathsf{fma}\left(u2, -2 \cdot \left(\sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - \left(\left(-1 \cdot u1\right) \cdot u1\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(-1, \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right), -0.3333333333333333 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right), \left(u1 \cdot u1\right) \cdot \left(0.5 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.3333333333333333, \frac{1}{\sqrt{u1}} \cdot \left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right), \sqrt{u1} \cdot \left(\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right), \sqrt{u1} \cdot \sin \left(0.5 \cdot \pi\right) - \left(\left(-1 \cdot u1\right) \cdot u1\right) \cdot \left(0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{1}{\sqrt{u1}} \cdot \sin \left(0.5 \cdot \pi\right), 0.5 \cdot \left(\sqrt{u1} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(0.25 - 0.0625 \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))