normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 13.3s

Alternatives: 5

Speedup: 1.5×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{0 - \log u1}, \left(\sqrt{2} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right) \cdot 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (- 0.0 (log u1)))
  (* (* (sqrt 2.0) (cos (* PI (+ u2 u2)))) 0.16666666666666666)
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((0.0 - log(u1))), ((sqrt(2.0) * cos((((double) M_PI) * (u2 + u2)))) * 0.16666666666666666), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(0.0 - log(u1))), Float64(Float64(sqrt(2.0) * cos(Float64(pi * Float64(u2 + u2)))) * 0.16666666666666666), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(0.0 - N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow1/2 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \color{blue}{0.16666666666666666}, 0.5\right) \]

4. Applied egg-rr 99.5%

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

   1. pow1/2 N/A

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

   2. pow-to-exp N/A

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

   3. rem-log-exp N/A

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

   4. pow-to-exp N/A

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

   5. pow1/2 N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. pow1/2 N/A

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

   8. log-pow N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow1 N/A

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

   11. metadata-eval N/A

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

   12. sqrt-pow2 N/A

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

   13. log-lowering-log.f64 N/A

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

   14. sqrt-prod N/A

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

   15. unpow-prod-down N/A

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

   16. sqrt-pow2 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. *-lowering-*.f64 N/A

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

   20. pow2 N/A

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

   21. rem-square-sqrt N/A

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

   22. log-lowering-log.f64 99.0

       \[\leadsto \mathsf{fma}\left(e^{0.5 \cdot \log \left(-2 \cdot \color{blue}{\log u1}\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.16666666666666666, 0.5\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{e^{0.5 \cdot \log \left(-2 \cdot \log u1\right)}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.16666666666666666, 0.5\right) \]

7. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. log-rec N/A

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{\color{blue}{0 - \log u1}} \cdot \sqrt{2}\right) \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{6}, \frac{1}{2}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{\color{blue}{0 - \log u1}} \cdot \sqrt{2}\right) \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{6}, \frac{1}{2}\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{0 - \color{blue}{\log u1}} \cdot \sqrt{2}\right) \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{6}, \frac{1}{2}\right) \]

   7. sqrt-lowering-sqrt.f64 99.5

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{0 - \log u1} \cdot \color{blue}{\sqrt{2}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.16666666666666666, 0.5\right) \]

9. Simplified 99.5%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt{0 - \log u1} \cdot \sqrt{2}\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.16666666666666666, 0.5\right) \]
10. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \color{blue}{\left(\sqrt{0 - \log u1} \cdot \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right)\right)} \cdot \frac{1}{6} + \frac{1}{2} \]

    2. associate-*l* N/A

       \[\leadsto \color{blue}{\sqrt{0 - \log u1} \cdot \left(\left(\sqrt{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{0 - \log u1}, \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}, \frac{1}{2}\right)} \]

11. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{0 - \log u1}, \left(\sqrt{2} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right) \cdot 0.16666666666666666, 0.5\right)} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* PI (+ u2 u2))) (sqrt (* (log u1) -2.0)))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((cos((((double) M_PI) * (u2 + u2))) * sqrt((log(u1) * -2.0))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(pi * Float64(u2 + u2))) * sqrt(Float64(log(u1) * -2.0))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. unpow1/2 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. metadata-eval N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666, \cos \left(2 \cdot \left(\color{blue}{\pi} \cdot u2\right)\right), 0.5\right) \]

4. Applied egg-rr 99.5%

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

   1. *-commutative N/A

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

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.16666666666666666, 0.5\right)} \]

7. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \]
8. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}, \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* 0.16666666666666666 (sqrt (* (log u1) -2.0)))
  (cos (* PI (+ u2 u2)))
  0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * sqrt((log(u1) * -2.0))), cos((((double) M_PI) * (u2 + u2))), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * sqrt(Float64(log(u1) * -2.0))), cos(Float64(pi * Float64(u2 + u2))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. unpow1/2 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. metadata-eval N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666, \cos \left(2 \cdot \left(\color{blue}{\pi} \cdot u2\right)\right), 0.5\right) \]

4. Applied egg-rr 99.5%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. log-lowering-log.f64 N/A

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

   6. cos-2 N/A

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

   7. cos-sum N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-lowering-*.f64 N/A

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

   11. PI-lowering-PI.f64 N/A

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

   12. +-lowering-+.f64 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666, \cos \left(\pi \cdot \color{blue}{\left(u2 + u2\right)}\right), 0.5\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666, \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.5\right)} \]

7. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}, \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.5\right) \]
8. Add Preprocessing

Alternative 4: 98.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right), 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (* (log u1) -2.0)) (fma (* u2 u2) (* -2.0 (* PI PI)) 1.0))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((sqrt((log(u1) * -2.0)) * fma((u2 * u2), (-2.0 * (((double) M_PI) * ((double) M_PI))), 1.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(log(u1) * -2.0)) * fma(Float64(u2 * u2), Float64(-2.0 * Float64(pi * pi)), 1.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(u2 * u2), $MachinePrecision] * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow1/2 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \color{blue}{0.16666666666666666}, 0.5\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.16666666666666666, 0.5\right)} \]

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. rem-square-sqrt N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-commutative N/A

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

   13. rem-square-sqrt N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 N/A

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

   18. PI-lowering-PI.f64 N/A

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

   19. PI-lowering-PI.f64 N/A

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

   20. unpow2 N/A

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

   21. rem-square-sqrt 98.2

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right), 0.16666666666666666, 0.5\right) \]
9. Add Preprocessing

Alternative 5: 98.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -2.0)) 0.16666666666666666 0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -2.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -2.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{\log u1} \cdot \left(\sqrt{-2} \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\log u1} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} + \frac{1}{2} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right)} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   7. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   10. sqrt-lowering-sqrt.f64 0.0

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2}} \cdot 0.16666666666666666, 0.5\right) \]

5. Simplified 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \sqrt{-2} \cdot 0.16666666666666666, 0.5\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} \cdot \sqrt{\log u1} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   4. sqrt-prod N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}} + \frac{1}{2} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \frac{1}{6}, \frac{1}{2}\right)} \]

7. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666, 0.5\right)} \]

8. Final simplification 97.8%

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))