normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 10.2s

Alternatives: 4

Speedup: 1.0×

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (* (sqrt (* -0.05555555555555555 (log u1))) (cos (* (* 2.0 PI) u2))) 0.5))

C

double code(double u1, double u2) {
	return (sqrt((-0.05555555555555555 * log(u1))) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (Math.sqrt((-0.05555555555555555 * Math.log(u1))) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (math.sqrt((-0.05555555555555555 * math.log(u1))) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(sqrt(Float64(-0.05555555555555555 * log(u1))) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (sqrt((-0.05555555555555555 * log(u1))) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $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. pow1/2 99.4%

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

   2. add-sqr-sqrt 99.1%

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

   3. sqrt-unprod 99.4%

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

   4. *-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. swap-sqr 99.5%

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

   7. add-sqr-sqrt 99.7%

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

   8. add-log-exp 55.2%

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

   9. *-commutative 55.2%

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

   10. exp-to-pow 55.2%

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

   11. metadata-eval 55.2%

       \[\leadsto \sqrt{\log \left({u1}^{-2}\right) \cdot \left(\color{blue}{0.16666666666666666} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   12. metadata-eval 55.2%

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

   13. metadata-eval 55.2%

       \[\leadsto \sqrt{\log \left({u1}^{-2}\right) \cdot \color{blue}{0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

3. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{\sqrt{\log \left({u1}^{-2}\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
4. Step-by-step derivation

   1. *-commutative 55.2%

      \[\leadsto \sqrt{\color{blue}{0.027777777777777776 \cdot \log \left({u1}^{-2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. log-pow 99.7%

      \[\leadsto \sqrt{0.027777777777777776 \cdot \color{blue}{\left(-2 \cdot \log u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. associate-*r* 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(0.027777777777777776 \cdot -2\right) \cdot \log u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   4. metadata-eval 99.7%

      \[\leadsto \sqrt{\color{blue}{-0.05555555555555555} \cdot \log u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{-0.05555555555555555 \cdot \log u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Final simplification 99.7%

   \[\leadsto \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

Alternative 2: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0.5 + {\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ 0.5 (pow (* (pow (log u1) 2.0) 0.0030864197530864196) 0.25)))

C

double code(double u1, double u2) {
	return 0.5 + pow((pow(log(u1), 2.0) * 0.0030864197530864196), 0.25);
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + (((log(u1) ** 2.0d0) * 0.0030864197530864196d0) ** 0.25d0)
end function

Java

public static double code(double u1, double u2) {
	return 0.5 + Math.pow((Math.pow(Math.log(u1), 2.0) * 0.0030864197530864196), 0.25);
}

Python

def code(u1, u2):
	return 0.5 + math.pow((math.pow(math.log(u1), 2.0) * 0.0030864197530864196), 0.25)

Julia

function code(u1, u2)
	return Float64(0.5 + (Float64((log(u1) ^ 2.0) * 0.0030864197530864196) ^ 0.25))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (((log(u1) ^ 2.0) * 0.0030864197530864196) ^ 0.25);
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Power[N[(N[Power[N[Log[u1], $MachinePrecision], 2.0], $MachinePrecision] * 0.0030864197530864196), $MachinePrecision], 0.25], $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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. unpow1/2 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u2 around 0 98.6%

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

   1. fma-udef 98.6%

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

   2. log-pow 54.7%

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

   3. *-commutative 54.7%

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

   4. +-commutative 54.7%

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

   5. add-sqr-sqrt 54.6%

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

   6. sqrt-unprod 54.7%

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

   7. *-commutative 54.7%

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

   8. *-commutative 54.7%

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

   9. swap-sqr 54.7%

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

   10. add-sqr-sqrt 54.8%

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

   11. pow-to-exp 54.8%

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

   12. add-log-exp 98.8%

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

   13. metadata-eval 98.8%

       \[\leadsto 0.5 + \sqrt{\left(\log u1 \cdot -2\right) \cdot \color{blue}{0.027777777777777776}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{0.5 + \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}} \]
7. Step-by-step derivation

   1. associate-*l* 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \]

   2. metadata-eval 98.8%

      \[\leadsto 0.5 + \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \]

   3. *-commutative 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{-0.05555555555555555 \cdot \log u1}} \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{0.5 + \sqrt{-0.05555555555555555 \cdot \log u1}} \]
9. Step-by-step derivation

   1. add-cbrt-cube 98.3%

      \[\leadsto 0.5 + \color{blue}{\sqrt[3]{\left(\sqrt{-0.05555555555555555 \cdot \log u1} \cdot \sqrt{-0.05555555555555555 \cdot \log u1}\right) \cdot \sqrt{-0.05555555555555555 \cdot \log u1}}} \]

   2. add-sqr-sqrt 98.4%

      \[\leadsto 0.5 + \sqrt[3]{\color{blue}{\left(-0.05555555555555555 \cdot \log u1\right)} \cdot \sqrt{-0.05555555555555555 \cdot \log u1}} \]

   3. pow1 98.4%

      \[\leadsto 0.5 + \sqrt[3]{\color{blue}{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1}} \cdot \sqrt{-0.05555555555555555 \cdot \log u1}} \]

   4. pow1/2 98.4%

      \[\leadsto 0.5 + \sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1} \cdot \color{blue}{{\left(-0.05555555555555555 \cdot \log u1\right)}^{0.5}}} \]

   5. pow-prod-up 98.5%

      \[\leadsto 0.5 + \sqrt[3]{\color{blue}{{\left(-0.05555555555555555 \cdot \log u1\right)}^{\left(1 + 0.5\right)}}} \]

   6. metadata-eval 98.5%

      \[\leadsto 0.5 + \sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{\color{blue}{1.5}}} \]

10. Applied egg-rr 98.5%

    \[\leadsto 0.5 + \color{blue}{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}}} \]
11. Step-by-step derivation

    1. pow1/3 98.3%

       \[\leadsto 0.5 + \color{blue}{{\left({\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 98.8%

       \[\leadsto 0.5 + \color{blue}{{\left(-0.05555555555555555 \cdot \log u1\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 98.8%

       \[\leadsto 0.5 + {\left(-0.05555555555555555 \cdot \log u1\right)}^{\color{blue}{0.5}} \]

    4. metadata-eval 98.8%

       \[\leadsto 0.5 + {\left(-0.05555555555555555 \cdot \log u1\right)}^{\color{blue}{\left(2 \cdot 0.25\right)}} \]

    5. pow-sqr 98.4%

       \[\leadsto 0.5 + \color{blue}{{\left(-0.05555555555555555 \cdot \log u1\right)}^{0.25} \cdot {\left(-0.05555555555555555 \cdot \log u1\right)}^{0.25}} \]

    6. pow-prod-down 98.8%

       \[\leadsto 0.5 + \color{blue}{{\left(\left(-0.05555555555555555 \cdot \log u1\right) \cdot \left(-0.05555555555555555 \cdot \log u1\right)\right)}^{0.25}} \]

    7. *-commutative 98.8%

       \[\leadsto 0.5 + {\left(\color{blue}{\left(\log u1 \cdot -0.05555555555555555\right)} \cdot \left(-0.05555555555555555 \cdot \log u1\right)\right)}^{0.25} \]

    8. *-commutative 98.8%

       \[\leadsto 0.5 + {\left(\left(\log u1 \cdot -0.05555555555555555\right) \cdot \color{blue}{\left(\log u1 \cdot -0.05555555555555555\right)}\right)}^{0.25} \]

    9. swap-sqr 98.9%

       \[\leadsto 0.5 + {\color{blue}{\left(\left(\log u1 \cdot \log u1\right) \cdot \left(-0.05555555555555555 \cdot -0.05555555555555555\right)\right)}}^{0.25} \]

    10. pow2 98.9%

        \[\leadsto 0.5 + {\left(\color{blue}{{\log u1}^{2}} \cdot \left(-0.05555555555555555 \cdot -0.05555555555555555\right)\right)}^{0.25} \]

    11. metadata-eval 98.9%

        \[\leadsto 0.5 + {\left({\log u1}^{2} \cdot \color{blue}{0.0030864197530864196}\right)}^{0.25} \]

12. Applied egg-rr 98.9%

    \[\leadsto 0.5 + \color{blue}{{\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25}} \]

13. Final simplification 98.9%

    \[\leadsto 0.5 + {\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25} \]

Alternative 3: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0.5 + \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ 0.5 (sqrt (log (pow u1 -0.05555555555555555)))))

C

double code(double u1, double u2) {
	return 0.5 + sqrt(log(pow(u1, -0.05555555555555555)));
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + sqrt(log((u1 ** (-0.05555555555555555d0))))
end function

Java

public static double code(double u1, double u2) {
	return 0.5 + Math.sqrt(Math.log(Math.pow(u1, -0.05555555555555555)));
}

Python

def code(u1, u2):
	return 0.5 + math.sqrt(math.log(math.pow(u1, -0.05555555555555555)))

Julia

function code(u1, u2)
	return Float64(0.5 + sqrt(log((u1 ^ -0.05555555555555555))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + sqrt(log((u1 ^ -0.05555555555555555)));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Sqrt[N[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. unpow1/2 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u2 around 0 98.6%

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

   1. fma-udef 98.6%

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

   2. log-pow 54.7%

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

   3. *-commutative 54.7%

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

   4. +-commutative 54.7%

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

   5. add-sqr-sqrt 54.6%

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

   6. sqrt-unprod 54.7%

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

   7. *-commutative 54.7%

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

   8. *-commutative 54.7%

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

   9. swap-sqr 54.7%

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

   10. add-sqr-sqrt 54.8%

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

   11. pow-to-exp 54.8%

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

   12. add-log-exp 98.8%

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

   13. metadata-eval 98.8%

       \[\leadsto 0.5 + \sqrt{\left(\log u1 \cdot -2\right) \cdot \color{blue}{0.027777777777777776}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{0.5 + \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}} \]
7. Step-by-step derivation

   1. associate-*l* 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \]

   2. metadata-eval 98.8%

      \[\leadsto 0.5 + \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \]

   3. *-commutative 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{-0.05555555555555555 \cdot \log u1}} \]

   4. log-pow 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{\log \left({u1}^{-0.05555555555555555}\right)}} \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{0.5 + \sqrt{\log \left({u1}^{-0.05555555555555555}\right)}} \]

9. Final simplification 98.8%

   \[\leadsto 0.5 + \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \]

Alternative 4: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-0.05555555555555555 \cdot \log u1} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (* -0.05555555555555555 (log u1))) 0.5))

C

double code(double u1, double u2) {
	return sqrt((-0.05555555555555555 * log(u1))) + 0.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt(((-0.05555555555555555d0) * log(u1))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((-0.05555555555555555 * Math.log(u1))) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt((-0.05555555555555555 * math.log(u1))) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(-0.05555555555555555 * log(u1))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((-0.05555555555555555 * log(u1))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $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. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. unpow1/2 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u2 around 0 98.6%

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

   1. fma-udef 98.6%

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

   2. log-pow 54.7%

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

   3. *-commutative 54.7%

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

   4. +-commutative 54.7%

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

   5. add-sqr-sqrt 54.6%

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

   6. sqrt-unprod 54.7%

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

   7. *-commutative 54.7%

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

   8. *-commutative 54.7%

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

   9. swap-sqr 54.7%

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

   10. add-sqr-sqrt 54.8%

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

   11. pow-to-exp 54.8%

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

   12. add-log-exp 98.8%

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

   13. metadata-eval 98.8%

       \[\leadsto 0.5 + \sqrt{\left(\log u1 \cdot -2\right) \cdot \color{blue}{0.027777777777777776}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{0.5 + \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}} \]
7. Step-by-step derivation

   1. associate-*l* 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \]

   2. metadata-eval 98.8%

      \[\leadsto 0.5 + \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \]

   3. *-commutative 98.8%

      \[\leadsto 0.5 + \sqrt{\color{blue}{-0.05555555555555555 \cdot \log u1}} \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{0.5 + \sqrt{-0.05555555555555555 \cdot \log u1}} \]

9. Final simplification 98.8%

   \[\leadsto \sqrt{-0.05555555555555555 \cdot \log u1} + 0.5 \]

Reproduce

herbie shell --seed 2023318 
(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))