normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 4.3s

Alternatives: 9

Speedup: 1.4×

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: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[(N[Sin[N[(N[(-2.0 * u2), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $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. Add Preprocessing

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-sin.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[Cos[N[(N[(-2.0 * u2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-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. Add Preprocessing

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. associate-*l* N/A

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

6. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

Alternative 6: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(N[(Pi * u2), $MachinePrecision] * -2.0), $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. Add Preprocessing

4. Applied rewrites 99.4%

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

Alternative 7: 98.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.3

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

9. Applied rewrites 98.3%

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

    1. lift-pow.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lift-PI.f64 98.3

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

11. Applied rewrites 98.3%

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

Alternative 8: 98.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-sqrt.f64 97.6

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

7. Applied rewrites 97.6%

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

Alternative 9: 98.1% 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. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. 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 97.5

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

5. Applied rewrites 97.5%

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

Reproduce

herbie shell --seed 2025064 
(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))