Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 98.5%

Time: 17.5s

Alternatives: 10

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: 57.4% 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: 98.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03799999877810478)
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       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 <= 0.03799999877810478f) {
		tmp = sqrtf(-(((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * 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(0.03799999877810478))
		tmp = Float32(sqrt(Float32(-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)));
	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 < 0.0379999988

   1. Initial program 48.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.0379999988 < u1

   1. Initial program 98.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-cos.f32 N/A

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

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

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lift-PI.f32 98.4

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

   6. Applied rewrites 98.4%

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

Alternative 2: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right)}{{u1}^{3}} + 0.25\right) \cdot \left(-{u1}^{4}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03799999877810478)
   (*
    (sqrt
     (-
      (*
       (+ (/ (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) (pow u1 3.0)) 0.25)
       (- (pow u1 4.0)))))
    (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 <= 0.03799999877810478f) {
		tmp = sqrtf(-(((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) / powf(u1, 3.0f)) + 0.25f) * -powf(u1, 4.0f))) * 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(0.03799999877810478))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) / (u1 ^ Float32(3.0))) + Float32(0.25)) * Float32(-(u1 ^ Float32(4.0)))))) * 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 < 0.0379999988

   1. Initial program 48.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   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 98.6%

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

   9. Taylor expanded in u1 around 0

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

       1. lower-/.f32 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. lift-pow.f32 98.6

          \[\leadsto \sqrt{-\left(\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right)}{{u1}^{3}}\right) - 0.25\right) \cdot {u1}^{4}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   11. Applied rewrites 98.6%

       \[\leadsto \sqrt{-\left(\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right)}{{u1}^{3}}\right) - 0.25\right) \cdot {u1}^{4}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0379999988 < u1

   1. Initial program 98.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-cos.f32 N/A

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

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

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lift-PI.f32 98.4

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

   6. Applied rewrites 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right)}{{u1}^{3}} + 0.25\right) \cdot \left(-{u1}^{4}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot t\_0, t\_1, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot t\_1, \left(0.16666666666666666 \cdot t\_0\right) \cdot t\_1\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, -\pi, \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot u2, 2, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1))) (t_1 (cos (* u2 (* PI 2.0)))))
   (if (<= u1 0.029999999329447746)
     (fma
      (fma
       (* 0.25 t_0)
       t_1
       (*
        (fma
         (* 0.5 (sqrt u1))
         (* (- 0.25 (/ 0.0625 u1)) t_1)
         (* (* 0.16666666666666666 t_0) t_1))
        u1))
      (* u1 u1)
      (* (sin (fma (* 2.0 u2) (- PI) (/ PI 2.0))) (sqrt u1)))
     (*
      (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 t_0 = 1.0f / sqrtf(u1);
	float t_1 = cosf((u2 * (((float) M_PI) * 2.0f)));
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = fmaf(fmaf((0.25f * t_0), t_1, (fmaf((0.5f * sqrtf(u1)), ((0.25f - (0.0625f / u1)) * t_1), ((0.16666666666666666f * t_0) * t_1)) * u1)), (u1 * u1), (sinf(fmaf((2.0f * u2), -((float) M_PI), (((float) M_PI) / 2.0f))) * sqrtf(u1)));
	} 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)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = fma(fma(Float32(Float32(0.25) * t_0), t_1, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * t_1), Float32(Float32(Float32(0.16666666666666666) * t_0) * t_1)) * u1)), Float32(u1 * u1), Float32(sin(fma(Float32(Float32(2.0) * u2), Float32(-Float32(pi)), Float32(Float32(pi) / Float32(2.0)))) * sqrt(u1)));
	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 < 0.0299999993

   1. Initial program 47.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot \sqrt{u1}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      4. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      6. lower-neg.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      15. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      16. lower-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      17. lift-PI.f32 98.7

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

   9. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]
   10. Step-by-step derivation

       1. lift-+.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       2. lift-neg.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \pi\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       3. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       4. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(2 \cdot u2\right) \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       6. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, \mathsf{neg}\left(\mathsf{PI}\left(\right)\right), \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right) \]

       7. lower-neg.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, -\mathsf{PI}\left(\right), \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right) \]

       8. lift-PI.f32 98.7

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

   11. Applied rewrites 98.7%

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

   if 0.0299999993 < u1

   1. Initial program 97.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 97.9

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

   4. Applied rewrites 97.9%

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

      1. lift-cos.f32 N/A

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

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

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lift-PI.f32 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 4: 98.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot t\_0, t\_1, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot t\_1, \left(0.16666666666666666 \cdot t\_0\right) \cdot t\_1\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, -\pi, \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1))) (t_1 (cos (* u2 (* PI 2.0)))))
   (if (<= u1 0.029999999329447746)
     (fma
      (fma
       (* 0.25 t_0)
       t_1
       (*
        (fma
         (* 0.5 (sqrt u1))
         (* (- 0.25 (/ 0.0625 u1)) t_1)
         (* (* 0.16666666666666666 t_0) t_1))
        u1))
      (* u1 u1)
      (* (sin (fma (* 2.0 u2) (- PI) (/ PI 2.0))) (sqrt u1)))
     (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f / sqrtf(u1);
	float t_1 = cosf((u2 * (((float) M_PI) * 2.0f)));
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = fmaf(fmaf((0.25f * t_0), t_1, (fmaf((0.5f * sqrtf(u1)), ((0.25f - (0.0625f / u1)) * t_1), ((0.16666666666666666f * t_0) * t_1)) * u1)), (u1 * u1), (sinf(fmaf((2.0f * u2), -((float) M_PI), (((float) M_PI) / 2.0f))) * sqrtf(u1)));
	} else {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * cosf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = fma(fma(Float32(Float32(0.25) * t_0), t_1, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * t_1), Float32(Float32(Float32(0.16666666666666666) * t_0) * t_1)) * u1)), Float32(u1 * u1), Float32(sin(fma(Float32(Float32(2.0) * u2), Float32(-Float32(pi)), Float32(Float32(pi) / Float32(2.0)))) * sqrt(u1)));
	else
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(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 u1 < 0.0299999993

   1. Initial program 47.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot \sqrt{u1}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      4. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      6. lower-neg.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      15. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      16. lower-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

      17. lift-PI.f32 98.7

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

   9. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]
   10. Step-by-step derivation

       1. lift-+.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       2. lift-neg.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \pi\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       3. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       4. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(2 \cdot u2\right) \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

       6. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, \mathsf{neg}\left(\mathsf{PI}\left(\right)\right), \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right) \]

       7. lower-neg.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, -\mathsf{PI}\left(\right), \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right) \]

       8. lift-PI.f32 98.7

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

   11. Applied rewrites 98.7%

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

   if 0.0299999993 < u1

   1. Initial program 97.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 97.9

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

   4. Applied rewrites 97.9%

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

Alternative 5: 93.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 92.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right)} \]
8. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot \sqrt{u1}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   6. lower-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   15. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   16. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   17. lift-PI.f32 92.4

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

9. Applied rewrites 92.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]
10. Step-by-step derivation

    1. lift-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

    2. lift-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \pi\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

    3. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

    4. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

    5. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(2 \cdot u2\right) \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) + \frac{\pi}{2}\right) \cdot \sqrt{u1}\right) \]

    6. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, \mathsf{neg}\left(\mathsf{PI}\left(\right)\right), \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right) \]

    7. lower-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\mathsf{fma}\left(2 \cdot u2, -\mathsf{PI}\left(\right), \frac{\pi}{2}\right)\right) \cdot \sqrt{u1}\right) \]

    8. lift-PI.f32 92.4

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

11. Applied rewrites 92.4%

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

Alternative 6: 93.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 92.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right)} \]
8. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot \sqrt{u1}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   6. lower-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   15. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   16. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   17. lift-PI.f32 92.4

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

9. Applied rewrites 92.4%

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

10. Final simplification 92.4%

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

Alternative 7: 93.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 92.2%

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

Alternative 8: 92.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ t_1 := \frac{1}{\sqrt{u1}}\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_1 \cdot \left(-t\_0\right)\\ {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, t\_3, 0.16666666666666666 \cdot t\_2\right), \frac{\mathsf{fma}\left(-1, -{u1}^{-1.5} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.25 \cdot t\_2\right)}{-u1}\right)}{u1}, -0.125 \cdot t\_3\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 2.0 (* u2 PI))))
        (t_1 (/ 1.0 (sqrt u1)))
        (t_2 (* t_1 t_0))
        (t_3 (* t_1 (- t_0))))
   (*
    (pow u1 4.0)
    (fma
     -1.0
     (/
      (fma
       -1.0
       (fma 0.03125 t_3 (* 0.16666666666666666 t_2))
       (/
        (fma
         -1.0
         (- (* (pow u1 -1.5) (sin (fma 0.5 PI (* -2.0 (* u2 PI))))))
         (* 0.25 t_2))
        (- u1)))
      u1)
     (* -0.125 t_3)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((2.0f * (u2 * ((float) M_PI))));
	float t_1 = 1.0f / sqrtf(u1);
	float t_2 = t_1 * t_0;
	float t_3 = t_1 * -t_0;
	return powf(u1, 4.0f) * fmaf(-1.0f, (fmaf(-1.0f, fmaf(0.03125f, t_3, (0.16666666666666666f * t_2)), (fmaf(-1.0f, -(powf(u1, -1.5f) * sinf(fmaf(0.5f, ((float) M_PI), (-2.0f * (u2 * ((float) M_PI)))))), (0.25f * t_2)) / -u1)) / u1), (-0.125f * t_3));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
	t_1 = Float32(Float32(1.0) / sqrt(u1))
	t_2 = Float32(t_1 * t_0)
	t_3 = Float32(t_1 * Float32(-t_0))
	return Float32((u1 ^ Float32(4.0)) * fma(Float32(-1.0), Float32(fma(Float32(-1.0), fma(Float32(0.03125), t_3, Float32(Float32(0.16666666666666666) * t_2)), Float32(fma(Float32(-1.0), Float32(-Float32((u1 ^ Float32(-1.5)) * sin(fma(Float32(0.5), Float32(pi), Float32(Float32(-2.0) * Float32(u2 * Float32(pi))))))), Float32(Float32(0.25) * t_2)) / Float32(-u1))) / u1), Float32(Float32(-0.125) * t_3)))
end

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 92.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right)} \]
8. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \cos \left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot \sqrt{u1}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\mathsf{neg}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   6. lower-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\pi \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   15. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   16. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \left(\frac{1}{4} - \frac{\frac{1}{16}}{u1}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(-\left(2 \cdot u2\right) \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{u1}\right) \]

   17. lift-PI.f32 92.4

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

9. Applied rewrites 92.4%

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

10. Taylor expanded in u1 around -inf

    \[\leadsto {u1}^{4} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\frac{1}{32} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + -1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{1}{{u1}^{3}}} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{u1}}{u1} + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

12. Applied rewrites 91.7%

    \[\leadsto {u1}^{4} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, \frac{1}{\sqrt{u1}} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.16666666666666666 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -1 \cdot \frac{\mathsf{fma}\left(-1, {u1}^{-1.5} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, -2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot -1\right), 0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)}{u1}\right)}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right)} \]

13. Final simplification 91.7%

    \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, \frac{1}{\sqrt{u1}} \cdot \left(-\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.16666666666666666 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), \frac{\mathsf{fma}\left(-1, -{u1}^{-1.5} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)}{-u1}\right)}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(-\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 9: 90.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\\ \mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, -t\_0, \frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, t\_0, \sqrt{{u1}^{-3}} \cdot t\_0\right)}{-u1}\right) \cdot \left(-{u1}^{3}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 u2) PI))))
   (*
    (fma
     (/ 0.16666666666666666 (sqrt u1))
     (- t_0)
     (/ (fma (/ 0.25 (sqrt u1)) t_0 (* (sqrt (pow u1 -3.0)) t_0)) (- u1)))
    (- (pow u1 3.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * u2) * ((float) M_PI)));
	return fmaf((0.16666666666666666f / sqrtf(u1)), -t_0, (fmaf((0.25f / sqrtf(u1)), t_0, (sqrtf(powf(u1, -3.0f)) * t_0)) / -u1)) * -powf(u1, 3.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * u2) * Float32(pi)))
	return Float32(fma(Float32(Float32(0.16666666666666666) / sqrt(u1)), Float32(-t_0), Float32(fma(Float32(Float32(0.25) / sqrt(u1)), t_0, Float32(sqrt((u1 ^ Float32(-3.0))) * t_0)) / Float32(-u1))) * Float32(-(u1 ^ Float32(3.0))))
end

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 89.8%

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

8. Taylor expanded in u1 around -inf

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

10. Applied rewrites 89.3%

    \[\leadsto -\mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot {u1}^{3} \]

11. Final simplification 89.3%

    \[\leadsto \mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, -\cos \left(\left(2 \cdot u2\right) \cdot \pi\right), \frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), \sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right)}{-u1}\right) \cdot \left(-{u1}^{3}\right) \]
12. Add Preprocessing

Alternative 10: 83.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\\ \mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, -t\_0, \frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, t\_0, \sqrt{{u1}^{-3}} \cdot t\_0\right)}{-u1}\right) \cdot \left(-e^{\log u1 \cdot 3}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 u2) PI))))
   (*
    (fma
     (/ 0.16666666666666666 (sqrt u1))
     (- t_0)
     (/ (fma (/ 0.25 (sqrt u1)) t_0 (* (sqrt (pow u1 -3.0)) t_0)) (- u1)))
    (- (exp (* (log u1) 3.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * u2) * ((float) M_PI)));
	return fmaf((0.16666666666666666f / sqrtf(u1)), -t_0, (fmaf((0.25f / sqrtf(u1)), t_0, (sqrtf(powf(u1, -3.0f)) * t_0)) / -u1)) * -expf((logf(u1) * 3.0f));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * u2) * Float32(pi)))
	return Float32(fma(Float32(Float32(0.16666666666666666) / sqrt(u1)), Float32(-t_0), Float32(fma(Float32(Float32(0.25) / sqrt(u1)), t_0, Float32(sqrt((u1 ^ Float32(-3.0))) * t_0)) / Float32(-u1))) * Float32(-exp(Float32(log(u1) * Float32(3.0)))))
end

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 89.8%

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

8. Taylor expanded in u1 around -inf

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

10. Applied rewrites 89.3%

    \[\leadsto -\mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot {u1}^{3} \]
11. Step-by-step derivation

    1. lift-pow.f32 N/A

       \[\leadsto -\mathsf{fma}\left(\frac{\frac{1}{6}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{\frac{1}{4}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot {u1}^{3} \]

    2. pow-to-exp N/A

       \[\leadsto -\mathsf{fma}\left(\frac{\frac{1}{6}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{\frac{1}{4}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot e^{\log u1 \cdot 3} \]

    3. lift-log.f32 N/A

       \[\leadsto -\mathsf{fma}\left(\frac{\frac{1}{6}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{\frac{1}{4}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot e^{\log u1 \cdot 3} \]

    4. lift-*.f32 N/A

       \[\leadsto -\mathsf{fma}\left(\frac{\frac{1}{6}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{\frac{1}{4}}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot e^{\log u1 \cdot 3} \]

    5. lift-exp.f32 82.3

       \[\leadsto -\mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot e^{\log u1 \cdot 3} \]

12. Applied rewrites 82.3%

    \[\leadsto -\mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right) \cdot -1, -\frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), -\left(\sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right) \cdot -1\right)}{u1}\right) \cdot e^{\log u1 \cdot 3} \]

13. Final simplification 82.3%

    \[\leadsto \mathsf{fma}\left(\frac{0.16666666666666666}{\sqrt{u1}}, -\cos \left(\left(2 \cdot u2\right) \cdot \pi\right), \frac{\mathsf{fma}\left(\frac{0.25}{\sqrt{u1}}, \cos \left(\left(2 \cdot u2\right) \cdot \pi\right), \sqrt{{u1}^{-3}} \cdot \cos \left(\left(2 \cdot u2\right) \cdot \pi\right)\right)}{-u1}\right) \cdot \left(-e^{\log u1 \cdot 3}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))