normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 4.0s

Alternatives: 4

Speedup: 1.3×

Specification

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

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Sin[N[(N[(2.0 * u2 + 0.5), $MachinePrecision] * Pi), $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-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. cos-2 N/A

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

   6. lower--.f64 N/A

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-PI.f64 N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. count-2-rev N/A

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

   13. associate-*l* N/A

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

   14. distribute-rgt-in N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.4

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

   19. lift-fma.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(\cos \left(6.283185307179586 \cdot u2\right) \cdot \sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* 6.283185307179586 u2)) (sqrt (* (log u1) -2.0)))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((cos((6.283185307179586 * u2)) * sqrt((log(u1) * -2.0))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(6.283185307179586 * u2)) * sqrt(Float64(log(u1) * -2.0))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(6.283185307179586 * u2), $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-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. cos-2 N/A

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

   6. lower--.f64 N/A

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

6. Evaluated real constant 99.4%

   \[\leadsto \mathsf{fma}\left(\cos \left(\color{blue}{\frac{884279719003555}{140737488355328}} \cdot u2\right) \cdot \sqrt{\log u1 \cdot -2}, \frac{1}{6}, \frac{1}{2}\right) \]
7. Add Preprocessing

Alternative 3: 98.2% accurate, 3.3× speedup

Math

\[\mathsf{fma}\left(\sqrt{\frac{1}{\frac{1}{-2 \cdot \log u1}}}, 0.16666666666666666, 0.5\right) \]

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(1.0 / N[(1.0 / N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]), $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 98.2

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

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. 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. lift-/.f64 98.2

      \[\leadsto 0.5 + \frac{1}{6} \cdot \sqrt{\color{blue}{-2 \cdot \log u1}} \]

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. metadata-eval N/A

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

   6. pow-flip N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-/.f64 98.1

       \[\leadsto 0.5 + \frac{1}{6} \cdot \frac{1}{\color{blue}{{\left(\log u1 \cdot -2\right)}^{-0.5}}} \]

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. pow-flip N/A

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

   17. metadata-eval N/A

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

   18. pow1/2 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. lift-sqrt.f64 N/A

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

6. Applied rewrites 98.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. remove-double-div N/A

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

   14. lift-/.f64 N/A

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

   15. remove-double-div N/A

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

   16. lift-/.f64 N/A

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

   17. frac-times N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

8. Applied rewrites 98.2%

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

Alternative 4: 98.2% accurate, 4.6× speedup

Math

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

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 99.4%

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

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 98.2

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

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. 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. lift-/.f64 98.2

      \[\leadsto 0.5 + \frac{1}{6} \cdot \sqrt{\color{blue}{-2 \cdot \log u1}} \]

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. metadata-eval N/A

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

   6. pow-flip N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-/.f64 98.1

       \[\leadsto 0.5 + \frac{1}{6} \cdot \frac{1}{\color{blue}{{\left(\log u1 \cdot -2\right)}^{-0.5}}} \]

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. pow-flip N/A

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

   17. metadata-eval N/A

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

   18. pow1/2 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. lift-sqrt.f64 N/A

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

6. Applied rewrites 98.2%

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

Reproduce

herbie shell --seed 2025170 
(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))