normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 2.9s

Alternatives: 3

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, 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. Taylor expanded in u1 around 0

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

Alternative 2: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\log u1 \cdot -2}\\ \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_0\right), -0.3333333333333333, t\_0 \cdot 0.16666666666666666\right) + 0.5 \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (sqrt (* (log u1) -2.0))))
   (+
    (fma
     (* (* u2 u2) (* (* PI PI) t_0))
     -0.3333333333333333
     (* t_0 0.16666666666666666))
    0.5)))

C

double code(double u1, double u2) {
	double t_0 = sqrt((log(u1) * -2.0));
	return fma(((u2 * u2) * ((((double) M_PI) * ((double) M_PI)) * t_0)), -0.3333333333333333, (t_0 * 0.16666666666666666)) + 0.5;
}

Julia

function code(u1, u2)
	t_0 = sqrt(Float64(log(u1) * -2.0))
	return Float64(fma(Float64(Float64(u2 * u2) * Float64(Float64(pi * pi) * t_0)), -0.3333333333333333, Float64(t_0 * 0.16666666666666666)) + 0.5)
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(u2 * u2), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[(t$95$0 * 0.16666666666666666), $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. Taylor expanded in u2 around 0

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 98.7%

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

Alternative 3: 98.2% accurate, 4.6× 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. Taylor expanded in u2 around 0

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. unpow1/2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. metadata-eval 98.2

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

4. Applied rewrites 98.2%

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

Reproduce

herbie shell --seed 2025132 
(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))