normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 2.9s

Alternatives: 6

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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \left({\left({\log u1}^{2}\right)}^{0.25} \cdot {4}^{0.25}\right), 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* u2 (* PI 2.0))) (* (pow (pow (log u1) 2.0) 0.25) (pow 4.0 0.25)))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((cos((u2 * (((double) M_PI) * 2.0))) * (pow(pow(log(u1), 2.0), 0.25) * pow(4.0, 0.25))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(u2 * Float64(pi * 2.0))) * Float64(((log(u1) ^ 2.0) ^ 0.25) * (4.0 ^ 0.25))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(u2 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Power[N[Log[u1], $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision] * N[Power[4.0, 0.25], $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%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. swap-sqr N/A

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

   9. unpow-prod-down N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \left({\left({\log u1}^{2}\right)}^{0.25} \cdot {\color{blue}{4}}^{0.25}\right), 0.16666666666666666, 0.5\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \color{blue}{\left({\left({\log u1}^{2}\right)}^{0.25} \cdot {4}^{0.25}\right)}, 0.16666666666666666, 0.5\right) \]
6. Add Preprocessing

Alternative 2: 99.4% accurate, 1.5× 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%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.5× 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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 4: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(u2 * u2), $MachinePrecision] * -2.0 + 1.0), $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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 5: 98.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[(u2 * u2), $MachinePrecision] * -0.3333333333333333 + 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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 6: 98.2% 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 99.4%

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

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in u2 around 0

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

   1. sqrt-prod N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 98.2

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

8. Applied rewrites 98.2%

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

Reproduce

herbie shell --seed 2025106 
(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))