normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 4.7s

Alternatives: 5

Speedup: 1.3×

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666\right) \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, 0.5\right)\right)\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (* (sqrt (* (log u1) -2.0)) 0.16666666666666666)
   (sin (* (pow PI 0.6666666666666666) (* (cbrt PI) (fma 2.0 u2 0.5)))))
  0.5))

C

double code(double u1, double u2) {
	return ((sqrt((log(u1) * -2.0)) * 0.16666666666666666) * sin((pow(((double) M_PI), 0.6666666666666666) * (cbrt(((double) M_PI)) * fma(2.0, u2, 0.5))))) + 0.5;
}

Julia

function code(u1, u2)
	return Float64(Float64(Float64(sqrt(Float64(log(u1) * -2.0)) * 0.16666666666666666) * sin(Float64((pi ^ 0.6666666666666666) * Float64(cbrt(pi) * fma(2.0, u2, 0.5))))) + 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[N[(N[Power[Pi, 0.6666666666666666], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(2.0 * u2 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 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. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-PI.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-out N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.4

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

3. Applied rewrites 99.4%

   \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(u2, 2, 0.5\right)\right)} + 0.5 \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\left(\sqrt[3]{\color{blue}{\pi}}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(u2, 2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   8. pow-cbrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(u2, 2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   9. lower-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(u2, 2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(u2, 2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(u2, 2, \frac{1}{2}\right)\right)}\right) + \frac{1}{2} \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \mathsf{fma}\left(u2, 2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   13. lower-cbrt.f64 99.4

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \mathsf{fma}\left(u2, 2, 0.5\right)\right)\right) + 0.5 \]

   14. lift-fma.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(u2 \cdot 2 + \frac{1}{2}\right)}\right)\right) + \frac{1}{2} \]

   15. *-commutative N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\color{blue}{2 \cdot u2} + \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   16. lower-fma.f64 99.4

       \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\mathsf{fma}\left(2, u2, 0.5\right)}\right)\right) + 0.5 \]

5. Applied rewrites 99.4%

   \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sin \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, 0.5\right)\right)\right)} + 0.5 \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right)} \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   3. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\frac{1}{2}} \cdot \frac{1}{6}\right)} \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}} \cdot \frac{1}{6}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   6. unpow1/2 N/A

      \[\leadsto \left(\color{blue}{\sqrt{-2 \cdot \log u1}} \cdot \frac{1}{6}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{-2 \cdot \log u1}} \cdot \frac{1}{6}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{-2 \cdot \log u1}} \cdot \frac{1}{6}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   9. *-commutative N/A

      \[\leadsto \left(\sqrt{\color{blue}{\log u1 \cdot -2}} \cdot \frac{1}{6}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\color{blue}{\log u1 \cdot -2}} \cdot \frac{1}{6}\right) \cdot \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, \frac{1}{2}\right)\right)\right) + \frac{1}{2} \]

   11. lift-*.f64 99.4

       \[\leadsto \color{blue}{\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666\right)} \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, 0.5\right)\right)\right) + 0.5 \]

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666\right)} \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \mathsf{fma}\left(2, u2, 0.5\right)\right)\right) + 0.5 \]
8. Add Preprocessing

Alternative 2: 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 3: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

Alternative 4: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -0.3333333333333333\right) \cdot u2, u2, 0.16666666666666666\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (* (log u1) -2.0))
  (fma (* (* (* PI PI) -0.3333333333333333) u2) u2 0.16666666666666666)
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -2.0)), fma((((((double) M_PI) * ((double) M_PI)) * -0.3333333333333333) * u2), u2, 0.16666666666666666), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -2.0)), fma(Float64(Float64(Float64(pi * pi) * -0.3333333333333333) * u2), u2, 0.16666666666666666), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * u2), $MachinePrecision] * u2 + 0.16666666666666666), $MachinePrecision] + 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

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} + \frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-PI.f64 98.9

      \[\leadsto \mathsf{fma}\left(0.16666666666666666 + -0.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{2}\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \]

6. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{0.16666666666666666 + -0.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{2}\right)}, \sqrt{\log u1 \cdot -2}, 0.5\right) \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666 + -0.3333333333333333 \cdot \left({u2}^{2} \cdot {\pi}^{2}\right), 0.5\right)} \]

8. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot -0.3333333333333333\right) \cdot u2, u2, 0.16666666666666666\right), 0.5\right)} \]
9. Add Preprocessing

Alternative 5: 98.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in u2 around 0

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

6. Applied rewrites 98.3%

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

Reproduce

herbie shell --seed 2025147 
(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))