normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 4.2s

Alternatives: 3

Speedup: N/A×

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: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

Alternative 2: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-2 \cdot \log u1}\\ \mathsf{fma}\left(\frac{0.5}{t\_0}, t\_0, t\_0 \cdot 0.16666666666666666\right) \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (sqrt (* -2.0 (log u1)))))
   (fma (/ 0.5 t_0) t_0 (* t_0 0.16666666666666666))))

C

double code(double u1, double u2) {
	double t_0 = sqrt((-2.0 * log(u1)));
	return fma((0.5 / t_0), t_0, (t_0 * 0.16666666666666666));
}

Julia

function code(u1, u2)
	t_0 = sqrt(Float64(-2.0 * log(u1)))
	return fma(Float64(0.5 / t_0), t_0, Float64(t_0 * 0.16666666666666666))
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.5 / t$95$0), $MachinePrecision] * t$95$0 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 97.9

      \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

      \[\leadsto \frac{1}{2} + e^{\log \left(-2 \cdot \log u1\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + e^{\log \left(-2 \cdot \log u1\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{2} + e^{\log \left(\log u1 \cdot -2\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + e^{\log \left(\log u1 \cdot -2\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} \]

   9. pow-to-exp N/A

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

   10. pow1/2 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-sqrt.f64 N/A

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

   14. pow1/2 N/A

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

   15. metadata-eval N/A

       \[\leadsto \frac{1}{2} + \frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \]

   16. pow-flip N/A

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

   17. lift-pow.f64 N/A

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

   18. mult-flip-rev N/A

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

   19. lower-/.f64 97.9

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\color{blue}{{\left(\log u1 \cdot -2\right)}^{-0.5}}} \]

   20. lift-pow.f64 N/A

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

   21. metadata-eval N/A

       \[\leadsto \frac{1}{2} + \frac{\frac{1}{6}}{{\left(\log u1 \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

   22. pow-neg N/A

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

   23. lower-unsound-pow.f32 N/A

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

   24. lower-pow.f32 N/A

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

   25. pow1/2 N/A

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

   26. lift-sqrt.f64 N/A

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

6. Applied rewrites 97.8%

   \[\leadsto 0.5 + \frac{0.16666666666666666}{\color{blue}{\frac{1}{\sqrt{-2 \cdot \log u1}}}} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. mult-flip N/A

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

   9. lift-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. sub-flip N/A

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

   12. lift--.f64 N/A

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

   13. mult-flip N/A

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

   14. lift-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 97.9%

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

Alternative 3: 97.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 97.9

      \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-to-exp N/A

      \[\leadsto e^{\log \left(-2 \cdot \log u1\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} + \frac{1}{2} \]

   8. lift-*.f64 N/A

      \[\leadsto e^{\log \left(-2 \cdot \log u1\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} + \frac{1}{2} \]

   9. *-commutative N/A

      \[\leadsto e^{\log \left(\log u1 \cdot -2\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} + \frac{1}{2} \]

   10. lift-*.f64 N/A

       \[\leadsto e^{\log \left(\log u1 \cdot -2\right) \cdot \frac{1}{2}} \cdot \frac{1}{6} + \frac{1}{2} \]

   11. pow-to-exp N/A

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

   12. pow1/2 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lower-fma.f64 97.9

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

6. Applied rewrites 97.9%

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

Reproduce

herbie shell --seed 2025154 
(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))