(/ (+ (* (/ (exp (* -3333333333333333/10000000000000000 (/ z1 z0))) (* z0 PI)) 1/8) (/ 1/8 (* (exp (/ z1 z0)) (* z0 PI)))) z1)

Percentage Accurate: 99.6% → 99.6%

Time: 3.4s

Alternatives: 16

Speedup: 1.1×

• could not determine a ground truth

  1. z1 = 177203621513954.22
  2. z0 = -1.357925752705053e-290

Specification

Math

\[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (+
  (* (/ (exp (* -0.3333333333333333 (/ z1 z0))) (* z0 PI)) 0.125)
  (/ 0.125 (* (exp (/ z1 z0)) (* z0 PI))))
 z1))

C

double code(double z1, double z0) {
	return (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * ((double) M_PI))) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * ((double) M_PI))))) / z1;
}

Java

public static double code(double z1, double z0) {
	return (((Math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * Math.PI)) * 0.125) + (0.125 / (Math.exp((z1 / z0)) * (z0 * Math.PI)))) / z1;
}

Python

def code(z1, z0):
	return (((math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * math.pi)) * 0.125) + (0.125 / (math.exp((z1 / z0)) * (z0 * math.pi)))) / z1

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(exp(Float64(-0.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * pi)) * 0.125) + Float64(0.125 / Float64(exp(Float64(z1 / z0)) * Float64(z0 * pi)))) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * pi)) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * pi)))) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 / N[(N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision] * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (+
  (* (/ (exp (* -0.3333333333333333 (/ z1 z0))) (* z0 PI)) 0.125)
  (/ 0.125 (* (exp (/ z1 z0)) (* z0 PI))))
 z1))

C

double code(double z1, double z0) {
	return (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * ((double) M_PI))) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * ((double) M_PI))))) / z1;
}

Java

public static double code(double z1, double z0) {
	return (((Math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * Math.PI)) * 0.125) + (0.125 / (Math.exp((z1 / z0)) * (z0 * Math.PI)))) / z1;
}

Python

def code(z1, z0):
	return (((math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * math.pi)) * 0.125) + (0.125 / (math.exp((z1 / z0)) * (z0 * math.pi)))) / z1

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(exp(Float64(-0.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * pi)) * 0.125) + Float64(0.125 / Float64(exp(Float64(z1 / z0)) * Float64(z0 * pi)))) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * pi)) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * pi)))) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 / N[(N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision] * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\frac{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (/
  (+
   (/ (* 0.125 (exp (* (/ z1 z0) -0.3333333333333333))) z0)
   (/ 0.125 (* (exp (/ z1 z0)) z0)))
  PI)
 z1))

C

double code(double z1, double z0) {
	return ((((0.125 * exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (exp((z1 / z0)) * z0))) / ((double) M_PI)) / z1;
}

Java

public static double code(double z1, double z0) {
	return ((((0.125 * Math.exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (Math.exp((z1 / z0)) * z0))) / Math.PI) / z1;
}

Python

def code(z1, z0):
	return ((((0.125 * math.exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (math.exp((z1 / z0)) * z0))) / math.pi) / z1

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(Float64(0.125 * exp(Float64(Float64(z1 / z0) * -0.3333333333333333))) / z0) + Float64(0.125 / Float64(exp(Float64(z1 / z0)) * z0))) / pi) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = ((((0.125 * exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (exp((z1 / z0)) * z0))) / pi) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[(0.125 * N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision] + N[(0.125 / N[(N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / z1), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{\color{blue}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot z0\right) \cdot \pi}}}{z1} \]

   11. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \color{blue}{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\frac{\frac{\frac{\left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right) \cdot 0.125}{z0}}{z1}}{\pi} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (/
  (/
   (*
    (+ (exp (/ (- z1) z0)) (exp (* -0.3333333333333333 (/ z1 z0))))
    0.125)
   z0)
  z1)
 PI))

C

double code(double z1, double z0) {
	return ((((exp((-z1 / z0)) + exp((-0.3333333333333333 * (z1 / z0)))) * 0.125) / z0) / z1) / ((double) M_PI);
}

Java

public static double code(double z1, double z0) {
	return ((((Math.exp((-z1 / z0)) + Math.exp((-0.3333333333333333 * (z1 / z0)))) * 0.125) / z0) / z1) / Math.PI;
}

Python

def code(z1, z0):
	return ((((math.exp((-z1 / z0)) + math.exp((-0.3333333333333333 * (z1 / z0)))) * 0.125) / z0) / z1) / math.pi

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(Float64(exp(Float64(Float64(-z1) / z0)) + exp(Float64(-0.3333333333333333 * Float64(z1 / z0)))) * 0.125) / z0) / z1) / pi)
end

MATLAB

function tmp = code(z1, z0)
	tmp = ((((exp((-z1 / z0)) + exp((-0.3333333333333333 * (z1 / z0)))) * 0.125) / z0) / z1) / pi;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[(N[Exp[N[((-z1) / z0), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] / z0), $MachinePrecision] / z1), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{\color{blue}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot z0\right) \cdot \pi}}}{z1} \]

   11. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \color{blue}{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}{z1}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi \cdot z1}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\color{blue}{z1 \cdot \pi}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{z1}}{\pi}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{z1}}{\pi}} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right) \cdot 0.125}{z0}}{z1}}{\pi}} \]
6. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

\[\frac{\frac{e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{\pi \cdot z0}}{z1 \cdot 8} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (/
  (+ (exp (/ (- z1) z0)) (exp (* -0.3333333333333333 (/ z1 z0))))
  (* PI z0))
 (* z1 8.0)))

C

double code(double z1, double z0) {
	return ((exp((-z1 / z0)) + exp((-0.3333333333333333 * (z1 / z0)))) / (((double) M_PI) * z0)) / (z1 * 8.0);
}

Java

public static double code(double z1, double z0) {
	return ((Math.exp((-z1 / z0)) + Math.exp((-0.3333333333333333 * (z1 / z0)))) / (Math.PI * z0)) / (z1 * 8.0);
}

Python

def code(z1, z0):
	return ((math.exp((-z1 / z0)) + math.exp((-0.3333333333333333 * (z1 / z0)))) / (math.pi * z0)) / (z1 * 8.0)

Julia

function code(z1, z0)
	return Float64(Float64(Float64(exp(Float64(Float64(-z1) / z0)) + exp(Float64(-0.3333333333333333 * Float64(z1 / z0)))) / Float64(pi * z0)) / Float64(z1 * 8.0))
end

MATLAB

function tmp = code(z1, z0)
	tmp = ((exp((-z1 / z0)) + exp((-0.3333333333333333 * (z1 / z0)))) / (pi * z0)) / (z1 * 8.0);
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[Exp[N[((-z1) / z0), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] / N[(z1 * 8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{\color{blue}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot z0\right) \cdot \pi}}}{z1} \]

   11. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \color{blue}{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{\pi \cdot z0}}{z1 \cdot 8}} \]
6. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\frac{0.125 \cdot \frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} + e^{-\frac{z1}{z0}}}{\pi \cdot z0}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (*
  0.125
  (/
   (+ (exp (* (/ z1 z0) -0.3333333333333333)) (exp (- (/ z1 z0))))
   (* PI z0)))
 z1))

C

double code(double z1, double z0) {
	return (0.125 * ((exp(((z1 / z0) * -0.3333333333333333)) + exp(-(z1 / z0))) / (((double) M_PI) * z0))) / z1;
}

Java

public static double code(double z1, double z0) {
	return (0.125 * ((Math.exp(((z1 / z0) * -0.3333333333333333)) + Math.exp(-(z1 / z0))) / (Math.PI * z0))) / z1;
}

Python

def code(z1, z0):
	return (0.125 * ((math.exp(((z1 / z0) * -0.3333333333333333)) + math.exp(-(z1 / z0))) / (math.pi * z0))) / z1

Julia

function code(z1, z0)
	return Float64(Float64(0.125 * Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) + exp(Float64(-Float64(z1 / z0)))) / Float64(pi * z0))) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.125 * ((exp(((z1 / z0) * -0.3333333333333333)) + exp(-(z1 / z0))) / (pi * z0))) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(0.125 * N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   5. mult-flip N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   6. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}\right)}}{z1} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}\right)}}{z1} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} + \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}\right)}{z1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \frac{1}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}\right)}{z1} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \color{blue}{\frac{\frac{1}{e^{\frac{z1}{z0}}}}{z0 \cdot \pi}}\right)}{z1} \]

   11. lift-exp.f64 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \frac{\frac{1}{\color{blue}{e^{\frac{z1}{z0}}}}}{z0 \cdot \pi}\right)}{z1} \]

   12. rec-exp N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} + \frac{\color{blue}{e^{\mathsf{neg}\left(\frac{z1}{z0}\right)}}}{z0 \cdot \pi}\right)}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{0.125 \cdot \frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} + e^{-\frac{z1}{z0}}}{\pi \cdot z0}}}{z1} \]
4. Add Preprocessing

Alternative 5: 96.0% accurate, 1.1× speedup

Math

\[\frac{0.125 \cdot \left(e^{\frac{z1}{z0} \cdot -0.3333333333333333} + e^{\frac{-z1}{z0}}\right)}{\left(z0 \cdot z1\right) \cdot \pi} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (*
  0.125
  (+ (exp (* (/ z1 z0) -0.3333333333333333)) (exp (/ (- z1) z0))))
 (* (* z0 z1) PI)))

C

double code(double z1, double z0) {
	return (0.125 * (exp(((z1 / z0) * -0.3333333333333333)) + exp((-z1 / z0)))) / ((z0 * z1) * ((double) M_PI));
}

Java

public static double code(double z1, double z0) {
	return (0.125 * (Math.exp(((z1 / z0) * -0.3333333333333333)) + Math.exp((-z1 / z0)))) / ((z0 * z1) * Math.PI);
}

Python

def code(z1, z0):
	return (0.125 * (math.exp(((z1 / z0) * -0.3333333333333333)) + math.exp((-z1 / z0)))) / ((z0 * z1) * math.pi)

Julia

function code(z1, z0)
	return Float64(Float64(0.125 * Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) + exp(Float64(Float64(-z1) / z0)))) / Float64(Float64(z0 * z1) * pi))
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.125 * (exp(((z1 / z0) * -0.3333333333333333)) + exp((-z1 / z0)))) / ((z0 * z1) * pi);
end

Wolfram

code[z1_, z0_] := N[(N[(0.125 * N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] + N[Exp[N[((-z1) / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(z0 * z1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{\color{blue}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot z0\right) \cdot \pi}}}{z1} \]

   11. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \color{blue}{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}{z1}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi \cdot z1}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\color{blue}{z1 \cdot \pi}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{z1}}{\pi}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{z1}}{\pi}} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right) \cdot 0.125}{z0}}{z1}}{\pi}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{z0}}{z1}}{\pi}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{z0}}{z1}}}{\pi} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{z0}}}{z1}}{\pi} \]

   4. associate-/l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{z0 \cdot z1}}}{\pi} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{\color{blue}{z0 \cdot z1}}}{\pi} \]

   6. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{\left(z0 \cdot z1\right) \cdot \pi}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(e^{\frac{-z1}{z0}} + e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}\right) \cdot \frac{1}{8}}{\color{blue}{\left(z0 \cdot z1\right) \cdot \pi}} \]

   8. lower-/.f64 96.0%

      \[\leadsto \color{blue}{\frac{\left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right) \cdot 0.125}{\left(z0 \cdot z1\right) \cdot \pi}} \]

7. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{z1}{z0} \cdot -0.3333333333333333} + e^{\frac{-z1}{z0}}\right)}{\left(z0 \cdot z1\right) \cdot \pi}} \]
8. Add Preprocessing

Alternative 6: 90.6% accurate, 1.3× speedup

Math

\[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\left(1 + \frac{\left(\left(\pi \cdot \frac{z1}{z0}\right) \cdot 0.5 + \pi\right) \cdot z1}{\pi \cdot z0}\right) \cdot \left(\pi \cdot z0\right)}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (+
  (* (/ (exp (* -0.3333333333333333 (/ z1 z0))) (* z0 PI)) 0.125)
  (/
   0.125
   (*
    (+ 1.0 (/ (* (+ (* (* PI (/ z1 z0)) 0.5) PI) z1) (* PI z0)))
    (* PI z0))))
 z1))

C

double code(double z1, double z0) {
	return (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * ((double) M_PI))) * 0.125) + (0.125 / ((1.0 + (((((((double) M_PI) * (z1 / z0)) * 0.5) + ((double) M_PI)) * z1) / (((double) M_PI) * z0))) * (((double) M_PI) * z0)))) / z1;
}

Java

public static double code(double z1, double z0) {
	return (((Math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * Math.PI)) * 0.125) + (0.125 / ((1.0 + (((((Math.PI * (z1 / z0)) * 0.5) + Math.PI) * z1) / (Math.PI * z0))) * (Math.PI * z0)))) / z1;
}

Python

def code(z1, z0):
	return (((math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * math.pi)) * 0.125) + (0.125 / ((1.0 + (((((math.pi * (z1 / z0)) * 0.5) + math.pi) * z1) / (math.pi * z0))) * (math.pi * z0)))) / z1

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(exp(Float64(-0.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * pi)) * 0.125) + Float64(0.125 / Float64(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(pi * Float64(z1 / z0)) * 0.5) + pi) * z1) / Float64(pi * z0))) * Float64(pi * z0)))) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * pi)) * 0.125) + (0.125 / ((1.0 + (((((pi * (z1 / z0)) * 0.5) + pi) * z1) / (pi * z0))) * (pi * z0)))) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 / N[(N[(1.0 + N[(N[(N[(N[(N[(Pi * N[(z1 / z0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + Pi), $MachinePrecision] * z1), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

   6. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)}}{z1} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   10. lower-PI.f64 87.8%

       \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}{z1} \]

4. Applied rewrites 87.8%

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + \color{blue}{z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]

   2. sum-to-mult N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{\left(1 + \frac{z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}{z0 \cdot \pi}\right) \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

   3. lower-unsound-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{\left(1 + \frac{z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}{z0 \cdot \pi}\right) \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

6. Applied rewrites 90.6%

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\left(1 + \frac{\left(\left(\pi \cdot \frac{z1}{z0}\right) \cdot 0.5 + \pi\right) \cdot z1}{\pi \cdot z0}\right) \cdot \color{blue}{\left(\pi \cdot z0\right)}}}{z1} \]
7. Add Preprocessing

Alternative 7: 90.2% accurate, 1.4× speedup

Math

\[\frac{\frac{\frac{0.125}{\left(z1 \cdot \left(\frac{0.5}{z0} \cdot z1 - -1\right)\right) \cdot \pi + \pi \cdot z0} \cdot \left(\pi \cdot z0\right) + e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125}{\pi \cdot z0}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (/
  (+
   (*
    (/ 0.125 (+ (* (* z1 (- (* (/ 0.5 z0) z1) -1.0)) PI) (* PI z0)))
    (* PI z0))
   (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125))
  (* PI z0))
 z1))

C

double code(double z1, double z0) {
	return ((((0.125 / (((z1 * (((0.5 / z0) * z1) - -1.0)) * ((double) M_PI)) + (((double) M_PI) * z0))) * (((double) M_PI) * z0)) + (exp(((z1 / z0) * -0.3333333333333333)) * 0.125)) / (((double) M_PI) * z0)) / z1;
}

Java

public static double code(double z1, double z0) {
	return ((((0.125 / (((z1 * (((0.5 / z0) * z1) - -1.0)) * Math.PI) + (Math.PI * z0))) * (Math.PI * z0)) + (Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125)) / (Math.PI * z0)) / z1;
}

Python

def code(z1, z0):
	return ((((0.125 / (((z1 * (((0.5 / z0) * z1) - -1.0)) * math.pi) + (math.pi * z0))) * (math.pi * z0)) + (math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125)) / (math.pi * z0)) / z1

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(Float64(0.125 / Float64(Float64(Float64(z1 * Float64(Float64(Float64(0.5 / z0) * z1) - -1.0)) * pi) + Float64(pi * z0))) * Float64(pi * z0)) + Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125)) / Float64(pi * z0)) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = ((((0.125 / (((z1 * (((0.5 / z0) * z1) - -1.0)) * pi) + (pi * z0))) * (pi * z0)) + (exp(((z1 / z0) * -0.3333333333333333)) * 0.125)) / (pi * z0)) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[(0.125 / N[(N[(N[(z1 * N[(N[(N[(0.5 / z0), $MachinePrecision] * z1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] + N[(Pi * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * z0), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

   6. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)}}{z1} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

   10. lower-PI.f64 87.8%

       \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}{z1} \]

4. Applied rewrites 87.8%

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \pi}{z0}}\right)}}{z1} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{z1 \cdot \pi}{z0} \cdot \color{blue}{\frac{1}{2}}\right)}}{z1} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{z1 \cdot \pi}{z0} \cdot \frac{1}{2}\right)}}{z1} \]

   4. mult-flip N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\left(z1 \cdot \pi\right) \cdot \frac{1}{z0}\right) \cdot \frac{1}{2}\right)}}{z1} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(z1 \cdot \pi\right) \cdot \color{blue}{\left(\frac{1}{z0} \cdot \frac{1}{2}\right)}\right)}}{z1} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(z1 \cdot \pi\right) \cdot \color{blue}{\left(\frac{1}{z0} \cdot \frac{1}{2}\right)}\right)}}{z1} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(z1 \cdot \pi\right) \cdot \left(\color{blue}{\frac{1}{z0}} \cdot \frac{1}{2}\right)\right)}}{z1} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \left(\color{blue}{\frac{1}{z0}} \cdot \frac{1}{2}\right)\right)}}{z1} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \left(\color{blue}{\frac{1}{z0}} \cdot \frac{1}{2}\right)\right)}}{z1} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \left(\frac{1}{z0} \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{z1} \]

   11. lower-/.f64 87.8%

       \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \left(\frac{1}{z0} \cdot 0.5\right)\right)}}{z1} \]

6. Applied rewrites 87.8%

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \color{blue}{\left(\frac{1}{z0} \cdot 0.5\right)}\right)}}{z1} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \color{blue}{\left(\frac{1}{z0} \cdot \frac{1}{2}\right)}\right)}}{z1} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\pi \cdot z1\right) \cdot \left(\color{blue}{\frac{1}{z0}} \cdot \frac{1}{2}\right)\right)}}{z1} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \pi \cdot \color{blue}{\left(z1 \cdot \left(\frac{1}{z0} \cdot \frac{1}{2}\right)\right)}\right)}}{z1} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(z1 \cdot \left(\frac{1}{z0} \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\pi}\right)}}{z1} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(z1 \cdot \left(\frac{1}{z0} \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\pi}\right)}}{z1} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\left(\frac{1}{z0} \cdot \frac{1}{2}\right) \cdot z1\right) \cdot \pi\right)}}{z1} \]

   7. lower-*.f64 87.8%

      \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\left(\frac{1}{z0} \cdot 0.5\right) \cdot z1\right) \cdot \pi\right)}}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\left(\frac{1}{z0} \cdot \frac{1}{2}\right) \cdot z1\right) \cdot \pi\right)}}{z1} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\left(\frac{1}{z0} \cdot \frac{1}{2}\right) \cdot z1\right) \cdot \pi\right)}}{z1} \]

   10. associate-*l/ N/A

       \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\frac{1 \cdot \frac{1}{2}}{z0} \cdot z1\right) \cdot \pi\right)}}{z1} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\frac{\frac{1}{2}}{z0} \cdot z1\right) \cdot \pi\right)}}{z1} \]

   12. lower-/.f64 87.8%

       \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\frac{0.5}{z0} \cdot z1\right) \cdot \pi\right)}}{z1} \]

8. Applied rewrites 87.8%

   \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + \left(\frac{0.5}{z0} \cdot z1\right) \cdot \color{blue}{\pi}\right)}}{z1} \]

10. Applied rewrites 90.2%

    \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{\left(z1 \cdot \left(\frac{0.5}{z0} \cdot z1 - -1\right)\right) \cdot \pi + \pi \cdot z0} \cdot \left(\pi \cdot z0\right) + e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125}{\pi \cdot z0}}}{z1} \]
11. Add Preprocessing

Alternative 8: 87.8% accurate, 1.4× speedup

Math

\[\frac{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{z0 + z1 \cdot \left(1 + 0.5 \cdot \frac{z1}{z0}\right)}}{\pi}}{z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 (/
  (+
   (/ (* 0.125 (exp (* (/ z1 z0) -0.3333333333333333))) z0)
   (/ 0.125 (+ z0 (* z1 (+ 1.0 (* 0.5 (/ z1 z0)))))))
  PI)
 z1))

C

double code(double z1, double z0) {
	return ((((0.125 * exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (z0 + (z1 * (1.0 + (0.5 * (z1 / z0))))))) / ((double) M_PI)) / z1;
}

Java

public static double code(double z1, double z0) {
	return ((((0.125 * Math.exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (z0 + (z1 * (1.0 + (0.5 * (z1 / z0))))))) / Math.PI) / z1;
}

Python

def code(z1, z0):
	return ((((0.125 * math.exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (z0 + (z1 * (1.0 + (0.5 * (z1 / z0))))))) / math.pi) / z1

Julia

function code(z1, z0)
	return Float64(Float64(Float64(Float64(Float64(0.125 * exp(Float64(Float64(z1 / z0) * -0.3333333333333333))) / z0) + Float64(0.125 / Float64(z0 + Float64(z1 * Float64(1.0 + Float64(0.5 * Float64(z1 / z0))))))) / pi) / z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = ((((0.125 * exp(((z1 / z0) * -0.3333333333333333))) / z0) + (0.125 / (z0 + (z1 * (1.0 + (0.5 * (z1 / z0))))))) / pi) / z1;
end

Wolfram

code[z1_, z0_] := N[(N[(N[(N[(N[(0.125 * N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision] + N[(0.125 / N[(z0 + N[(z1 * N[(1.0 + N[(0.5 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / z1), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{\color{blue}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot z0\right) \cdot \pi}}}{z1} \]

   11. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \color{blue}{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \frac{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{\color{blue}{z0 + z1 \cdot \left(1 + \frac{1}{2} \cdot \frac{z1}{z0}\right)}}}{\pi}}{z1} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{z0 + \color{blue}{z1 \cdot \left(1 + \frac{1}{2} \cdot \frac{z1}{z0}\right)}}}{\pi}}{z1} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{z0 + z1 \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot \frac{z1}{z0}\right)}}}{\pi}}{z1} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{z0 + z1 \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot \frac{z1}{z0}}\right)}}{\pi}}{z1} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{8} \cdot e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}}}{z0} + \frac{\frac{1}{8}}{z0 + z1 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\frac{z1}{z0}}\right)}}{\pi}}{z1} \]

   5. lower-/.f64 87.8%

      \[\leadsto \frac{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{z0 + z1 \cdot \left(1 + 0.5 \cdot \frac{z1}{\color{blue}{z0}}\right)}}{\pi}}{z1} \]

6. Applied rewrites 87.8%

   \[\leadsto \frac{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{\color{blue}{z0 + z1 \cdot \left(1 + 0.5 \cdot \frac{z1}{z0}\right)}}}{\pi}}{z1} \]
7. Add Preprocessing

Alternative 9: 87.1% accurate, 3.2× speedup

Math

\[\frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(-0.2222222222222222 \cdot \frac{\pi}{z0} + 0.1388888888888889 \cdot \frac{\pi}{z0}\right)\right) + 2.6666666666666665 \cdot \pi\right)\right)} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 1.0
 (*
  z1
  (+
   (* 4.0 (* z0 PI))
   (*
    z1
    (+
     (*
      -8.0
      (*
       z1
       (+
        (* -0.2222222222222222 (/ PI z0))
        (* 0.1388888888888889 (/ PI z0)))))
     (* 2.6666666666666665 PI)))))))

C

double code(double z1, double z0) {
	return 1.0 / (z1 * ((4.0 * (z0 * ((double) M_PI))) + (z1 * ((-8.0 * (z1 * ((-0.2222222222222222 * (((double) M_PI) / z0)) + (0.1388888888888889 * (((double) M_PI) / z0))))) + (2.6666666666666665 * ((double) M_PI))))));
}

Java

public static double code(double z1, double z0) {
	return 1.0 / (z1 * ((4.0 * (z0 * Math.PI)) + (z1 * ((-8.0 * (z1 * ((-0.2222222222222222 * (Math.PI / z0)) + (0.1388888888888889 * (Math.PI / z0))))) + (2.6666666666666665 * Math.PI)))));
}

Python

def code(z1, z0):
	return 1.0 / (z1 * ((4.0 * (z0 * math.pi)) + (z1 * ((-8.0 * (z1 * ((-0.2222222222222222 * (math.pi / z0)) + (0.1388888888888889 * (math.pi / z0))))) + (2.6666666666666665 * math.pi)))))

Julia

function code(z1, z0)
	return Float64(1.0 / Float64(z1 * Float64(Float64(4.0 * Float64(z0 * pi)) + Float64(z1 * Float64(Float64(-8.0 * Float64(z1 * Float64(Float64(-0.2222222222222222 * Float64(pi / z0)) + Float64(0.1388888888888889 * Float64(pi / z0))))) + Float64(2.6666666666666665 * pi))))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.0 / (z1 * ((4.0 * (z0 * pi)) + (z1 * ((-8.0 * (z1 * ((-0.2222222222222222 * (pi / z0)) + (0.1388888888888889 * (pi / z0))))) + (2.6666666666666665 * pi)))));
end

Wolfram

code[z1_, z0_] := N[(1.0 / N[(z1 * N[(N[(4.0 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] + N[(z1 * N[(N[(-8.0 * N[(z1 * N[(N[(-0.2222222222222222 * N[(Pi / z0), $MachinePrecision]), $MachinePrecision] + N[(0.1388888888888889 * N[(Pi / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.6666666666666665 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   4. lower-unsound-/.f64 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{z1}{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}} \]

   8. mult-flip N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}}}} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{z1}{0.125 \cdot \frac{e^{-\frac{z1}{z0}} + e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{\pi \cdot z0}}}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \frac{1}{\color{blue}{z1 \cdot \left(4 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\pi}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\pi}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \pi\right)\right)}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \color{blue}{\left(4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z1} \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \color{blue}{\left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(4 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(\frac{-177777777777777768888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0} + \frac{111111111111111108888888888888889}{800000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{z0}\right)\right) + \color{blue}{\frac{13333333333333333}{5000000000000000} \cdot \mathsf{PI}\left(\right)}\right)\right)} \]

6. Applied rewrites 87.1%

   \[\leadsto \frac{1}{\color{blue}{z1 \cdot \left(4 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(-8 \cdot \left(z1 \cdot \left(-0.2222222222222222 \cdot \frac{\pi}{z0} + 0.1388888888888889 \cdot \frac{\pi}{z0}\right)\right) + 2.6666666666666665 \cdot \pi\right)\right)}} \]
7. Add Preprocessing

Alternative 10: 85.2% accurate, 4.1× speedup

Math

\[\frac{1}{z0 \cdot \left(-2 \cdot \frac{z1 \cdot \left(\pi \cdot \left(-1 \cdot z1 + -0.3333333333333333 \cdot z1\right)\right)}{z0} + 4 \cdot \left(z1 \cdot \pi\right)\right)} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 1.0
 (*
  z0
  (+
   (*
    -2.0
    (/ (* z1 (* PI (+ (* -1.0 z1) (* -0.3333333333333333 z1)))) z0))
   (* 4.0 (* z1 PI))))))

C

double code(double z1, double z0) {
	return 1.0 / (z0 * ((-2.0 * ((z1 * (((double) M_PI) * ((-1.0 * z1) + (-0.3333333333333333 * z1)))) / z0)) + (4.0 * (z1 * ((double) M_PI)))));
}

Java

public static double code(double z1, double z0) {
	return 1.0 / (z0 * ((-2.0 * ((z1 * (Math.PI * ((-1.0 * z1) + (-0.3333333333333333 * z1)))) / z0)) + (4.0 * (z1 * Math.PI))));
}

Python

def code(z1, z0):
	return 1.0 / (z0 * ((-2.0 * ((z1 * (math.pi * ((-1.0 * z1) + (-0.3333333333333333 * z1)))) / z0)) + (4.0 * (z1 * math.pi))))

Julia

function code(z1, z0)
	return Float64(1.0 / Float64(z0 * Float64(Float64(-2.0 * Float64(Float64(z1 * Float64(pi * Float64(Float64(-1.0 * z1) + Float64(-0.3333333333333333 * z1)))) / z0)) + Float64(4.0 * Float64(z1 * pi)))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.0 / (z0 * ((-2.0 * ((z1 * (pi * ((-1.0 * z1) + (-0.3333333333333333 * z1)))) / z0)) + (4.0 * (z1 * pi))));
end

Wolfram

code[z1_, z0_] := N[(1.0 / N[(z0 * N[(N[(-2.0 * N[(N[(z1 * N[(Pi * N[(N[(-1.0 * z1), $MachinePrecision] + N[(-0.3333333333333333 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   4. lower-unsound-/.f64 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{z1}{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}} \]

   8. mult-flip N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}}}} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{z1}{0.125 \cdot \frac{e^{-\frac{z1}{z0}} + e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{\pi \cdot z0}}}} \]

4. Taylor expanded in z0 around inf

   \[\leadsto \frac{1}{\color{blue}{z0 \cdot \left(-2 \cdot \frac{z1 \cdot \left(\pi \cdot \left(-1 \cdot z1 + \frac{-3333333333333333}{10000000000000000} \cdot z1\right)\right)}{z0} + 4 \cdot \left(z1 \cdot \pi\right)\right)}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(-2 \cdot \frac{z1 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot z1 + \frac{-3333333333333333}{10000000000000000} \cdot z1\right)\right)}{z0} + 4 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{z0 \cdot \left(-2 \cdot \frac{z1 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot z1 + \frac{-3333333333333333}{10000000000000000} \cdot z1\right)\right)}{z0} + \color{blue}{4 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

6. Applied rewrites 85.2%

   \[\leadsto \frac{1}{\color{blue}{z0 \cdot \left(-2 \cdot \frac{z1 \cdot \left(\pi \cdot \left(-1 \cdot z1 + -0.3333333333333333 \cdot z1\right)\right)}{z0} + 4 \cdot \left(z1 \cdot \pi\right)\right)}} \]
7. Add Preprocessing

Alternative 11: 81.0% accurate, 7.1× speedup

Math

\[\frac{1}{z1 \cdot \left(2.6666666666666665 \cdot \left(z1 \cdot \pi\right) + 4 \cdot \left(z0 \cdot \pi\right)\right)} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 1.0 (* z1 (+ (* 2.6666666666666665 (* z1 PI)) (* 4.0 (* z0 PI))))))

C

double code(double z1, double z0) {
	return 1.0 / (z1 * ((2.6666666666666665 * (z1 * ((double) M_PI))) + (4.0 * (z0 * ((double) M_PI)))));
}

Java

public static double code(double z1, double z0) {
	return 1.0 / (z1 * ((2.6666666666666665 * (z1 * Math.PI)) + (4.0 * (z0 * Math.PI))));
}

Python

def code(z1, z0):
	return 1.0 / (z1 * ((2.6666666666666665 * (z1 * math.pi)) + (4.0 * (z0 * math.pi))))

Julia

function code(z1, z0)
	return Float64(1.0 / Float64(z1 * Float64(Float64(2.6666666666666665 * Float64(z1 * pi)) + Float64(4.0 * Float64(z0 * pi)))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.0 / (z1 * ((2.6666666666666665 * (z1 * pi)) + (4.0 * (z0 * pi))));
end

Wolfram

code[z1_, z0_] := N[(1.0 / N[(z1 * N[(N[(2.6666666666666665 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   4. lower-unsound-/.f64 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{z1}{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}} \]

   8. mult-flip N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}} + \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\frac{1}{8} \cdot \frac{1}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}}}} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{z1}{0.125 \cdot \frac{e^{-\frac{z1}{z0}} + e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{\pi \cdot z0}}}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \frac{1}{\color{blue}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \pi\right) + 4 \cdot \left(z0 \cdot \pi\right)\right)}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \color{blue}{\left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right) + 4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right) + 4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \pi\right) + 4 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \pi\right) + 4 \cdot \color{blue}{\left(z0 \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{z1 \cdot \left(\frac{13333333333333333}{5000000000000000} \cdot \left(z1 \cdot \pi\right) + 4 \cdot \left(z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)} \]

   8. lower-PI.f64 81.0%

      \[\leadsto \frac{1}{z1 \cdot \left(2.6666666666666665 \cdot \left(z1 \cdot \pi\right) + 4 \cdot \left(z0 \cdot \pi\right)\right)} \]

6. Applied rewrites 81.0%

   \[\leadsto \frac{1}{\color{blue}{z1 \cdot \left(2.6666666666666665 \cdot \left(z1 \cdot \pi\right) + 4 \cdot \left(z0 \cdot \pi\right)\right)}} \]
7. Add Preprocessing

Alternative 12: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\frac{0.25}{z1}}{\pi \cdot z0}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.25 \cdot z1}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.25}{z0}}{\pi}}{z1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (/
         (+
          (*
           (/ (exp (* -0.3333333333333333 (/ z1 z0))) (* z0 PI))
           0.125)
          (/ 0.125 (* (exp (/ z1 z0)) (* z0 PI))))
         z1)))
  (if (<= t_0 -1e-318)
    (/ (/ 0.25 z1) (* PI z0))
    (if (<= t_0 0.0)
      (/ (* 0.25 z1) (* (* (* PI z1) z0) z1))
      (/ (/ (/ 0.25 z0) PI) z1)))))

C

double code(double z1, double z0) {
	double t_0 = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * ((double) M_PI))) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * ((double) M_PI))))) / z1;
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (0.25 / z1) / (((double) M_PI) * z0);
	} else if (t_0 <= 0.0) {
		tmp = (0.25 * z1) / (((((double) M_PI) * z1) * z0) * z1);
	} else {
		tmp = ((0.25 / z0) / ((double) M_PI)) / z1;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (((Math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * Math.PI)) * 0.125) + (0.125 / (Math.exp((z1 / z0)) * (z0 * Math.PI)))) / z1;
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (0.25 / z1) / (Math.PI * z0);
	} else if (t_0 <= 0.0) {
		tmp = (0.25 * z1) / (((Math.PI * z1) * z0) * z1);
	} else {
		tmp = ((0.25 / z0) / Math.PI) / z1;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = (((math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * math.pi)) * 0.125) + (0.125 / (math.exp((z1 / z0)) * (z0 * math.pi)))) / z1
	tmp = 0
	if t_0 <= -1e-318:
		tmp = (0.25 / z1) / (math.pi * z0)
	elif t_0 <= 0.0:
		tmp = (0.25 * z1) / (((math.pi * z1) * z0) * z1)
	else:
		tmp = ((0.25 / z0) / math.pi) / z1
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(Float64(Float64(exp(Float64(-0.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * pi)) * 0.125) + Float64(0.125 / Float64(exp(Float64(z1 / z0)) * Float64(z0 * pi)))) / z1)
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(0.25 / z1) / Float64(pi * z0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(0.25 * z1) / Float64(Float64(Float64(pi * z1) * z0) * z1));
	else
		tmp = Float64(Float64(Float64(0.25 / z0) / pi) / z1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * pi)) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * pi)))) / z1;
	tmp = 0.0;
	if (t_0 <= -1e-318)
		tmp = (0.25 / z1) / (pi * z0);
	elseif (t_0 <= 0.0)
		tmp = (0.25 * z1) / (((pi * z1) * z0) * z1);
	else
		tmp = ((0.25 / z0) / pi) / z1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[(N[(N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 / N[(N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision] * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(0.25 / z1), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(0.25 * z1), $MachinePrecision] / N[(N[(N[(Pi * z1), $MachinePrecision] * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 / z0), $MachinePrecision] / Pi), $MachinePrecision] / z1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 #s(literal -3333333333333333/10000000000000000 binary64) (/.f64 z1 z0))) (*.f64 z0 (PI.f64))) #s(literal 1/8 binary64)) (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64))))) z1) < -9.9999874849559983e-319

   1. Initial program 99.6%

      \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

      4. lower-PI.f64 71.0%

         \[\leadsto \frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\pi}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

      5. lower-*.f64 70.9%

         \[\leadsto \frac{0.25}{\left(z0 \cdot z1\right) \cdot \pi} \]

   6. Applied rewrites 70.9%

      \[\leadsto \frac{0.25}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \pi} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(\pi \cdot \color{blue}{z1}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot \pi\right) \cdot \color{blue}{z1}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot \pi\right) \cdot z1} \]

      7. lower-*.f64 70.9%

         \[\leadsto \frac{0.25}{\left(z0 \cdot \pi\right) \cdot \color{blue}{z1}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot \pi\right) \cdot z1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot z0\right) \cdot z1} \]

      10. lower-*.f64 70.9%

          \[\leadsto \frac{0.25}{\left(\pi \cdot z0\right) \cdot z1} \]

   8. Applied rewrites 70.9%

      \[\leadsto \frac{0.25}{\left(\pi \cdot z0\right) \cdot \color{blue}{z1}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\pi \cdot z0\right) \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot z0\right) \cdot \color{blue}{z1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \color{blue}{\left(\pi \cdot z0\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \left(\pi \cdot \color{blue}{z0}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \left(z0 \cdot \color{blue}{\pi}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \left(z0 \cdot \color{blue}{\pi}\right)} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{\color{blue}{z0 \cdot \pi}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{\color{blue}{z0 \cdot \pi}} \]

      9. lower-/.f64 71.0%

         \[\leadsto \frac{\frac{0.25}{z1}}{\color{blue}{z0} \cdot \pi} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{z0 \cdot \color{blue}{\pi}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{\pi \cdot \color{blue}{z0}} \]

      12. lift-*.f64 71.0%

          \[\leadsto \frac{\frac{0.25}{z1}}{\pi \cdot \color{blue}{z0}} \]

   10. Applied rewrites 71.0%

       \[\leadsto \frac{\frac{0.25}{z1}}{\color{blue}{\pi \cdot z0}} \]

   if -9.9999874849559983e-319 < (/.f64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 #s(literal -3333333333333333/10000000000000000 binary64) (/.f64 z1 z0))) (*.f64 z0 (PI.f64))) #s(literal 1/8 binary64)) (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64))))) z1) < 0.0

   1. Initial program 99.6%

      \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)}}{z1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      10. lower-PI.f64 87.8%

          \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}{z1} \]

   4. Applied rewrites 87.8%

      \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]

   6. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\frac{\left(e^{-0.3333333333333333 \cdot \frac{z1}{z0}} \cdot 0.125\right) \cdot z1 - \left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot \frac{-0.125}{\left(\left(\pi \cdot \frac{z1}{z0}\right) \cdot 0.5 + \pi\right) \cdot z1 + \pi \cdot z0}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1}} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot z1}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1} \]
   8. Step-by-step derivation

      1. lower-*.f64 62.8%

         \[\leadsto \frac{0.25 \cdot \color{blue}{z1}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1} \]

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\color{blue}{0.25 \cdot z1}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1} \]

   if 0.0 < (/.f64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 #s(literal -3333333333333333/10000000000000000 binary64) (/.f64 z1 z0))) (*.f64 z0 (PI.f64))) #s(literal 1/8 binary64)) (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64))))) z1)

   1. Initial program 99.6%

      \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi}} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{\color{blue}{z0 \cdot \pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi}} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}}{z1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \color{blue}{\left(z0 \cdot \pi\right)}}}{z1} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \frac{\frac{1}{8}}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot z0\right) \cdot \pi}}}{z1} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0}}{\pi} + \color{blue}{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

      12. div-add-rev N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}} \cdot \frac{1}{8}}{z0} + \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\frac{0.125 \cdot e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{z0} + \frac{0.125}{e^{\frac{z1}{z0}} \cdot z0}}{\pi}}}{z1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{4}}{z0}}}{\pi}}{z1} \]
   5. Step-by-step derivation

      1. lower-/.f64 71.0%

         \[\leadsto \frac{\frac{\frac{0.25}{\color{blue}{z0}}}{\pi}}{z1} \]

   6. Applied rewrites 71.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{0.25}{z0}}}{\pi}}{z1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\frac{0.25}{z1}}{\pi \cdot z0}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.25 \cdot z1}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{z0 \cdot \pi}}{z1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (/
         (+
          (*
           (/ (exp (* -0.3333333333333333 (/ z1 z0))) (* z0 PI))
           0.125)
          (/ 0.125 (* (exp (/ z1 z0)) (* z0 PI))))
         z1)))
  (if (<= t_0 -1e-318)
    (/ (/ 0.25 z1) (* PI z0))
    (if (<= t_0 0.0)
      (/ (* 0.25 z1) (* (* (* PI z1) z0) z1))
      (/ (/ 0.25 (* z0 PI)) z1)))))

C

double code(double z1, double z0) {
	double t_0 = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * ((double) M_PI))) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * ((double) M_PI))))) / z1;
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (0.25 / z1) / (((double) M_PI) * z0);
	} else if (t_0 <= 0.0) {
		tmp = (0.25 * z1) / (((((double) M_PI) * z1) * z0) * z1);
	} else {
		tmp = (0.25 / (z0 * ((double) M_PI))) / z1;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (((Math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * Math.PI)) * 0.125) + (0.125 / (Math.exp((z1 / z0)) * (z0 * Math.PI)))) / z1;
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (0.25 / z1) / (Math.PI * z0);
	} else if (t_0 <= 0.0) {
		tmp = (0.25 * z1) / (((Math.PI * z1) * z0) * z1);
	} else {
		tmp = (0.25 / (z0 * Math.PI)) / z1;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = (((math.exp((-0.3333333333333333 * (z1 / z0))) / (z0 * math.pi)) * 0.125) + (0.125 / (math.exp((z1 / z0)) * (z0 * math.pi)))) / z1
	tmp = 0
	if t_0 <= -1e-318:
		tmp = (0.25 / z1) / (math.pi * z0)
	elif t_0 <= 0.0:
		tmp = (0.25 * z1) / (((math.pi * z1) * z0) * z1)
	else:
		tmp = (0.25 / (z0 * math.pi)) / z1
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(Float64(Float64(exp(Float64(-0.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * pi)) * 0.125) + Float64(0.125 / Float64(exp(Float64(z1 / z0)) * Float64(z0 * pi)))) / z1)
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(0.25 / z1) / Float64(pi * z0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(0.25 * z1) / Float64(Float64(Float64(pi * z1) * z0) * z1));
	else
		tmp = Float64(Float64(0.25 / Float64(z0 * pi)) / z1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (((exp((-0.3333333333333333 * (z1 / z0))) / (z0 * pi)) * 0.125) + (0.125 / (exp((z1 / z0)) * (z0 * pi)))) / z1;
	tmp = 0.0;
	if (t_0 <= -1e-318)
		tmp = (0.25 / z1) / (pi * z0);
	elseif (t_0 <= 0.0)
		tmp = (0.25 * z1) / (((pi * z1) * z0) * z1);
	else
		tmp = (0.25 / (z0 * pi)) / z1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[(N[(N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 / N[(N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision] * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(0.25 / z1), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(0.25 * z1), $MachinePrecision] / N[(N[(N[(Pi * z1), $MachinePrecision] * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 #s(literal -3333333333333333/10000000000000000 binary64) (/.f64 z1 z0))) (*.f64 z0 (PI.f64))) #s(literal 1/8 binary64)) (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64))))) z1) < -9.9999874849559983e-319

   1. Initial program 99.6%

      \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

      4. lower-PI.f64 71.0%

         \[\leadsto \frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\pi}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

      5. lower-*.f64 70.9%

         \[\leadsto \frac{0.25}{\left(z0 \cdot z1\right) \cdot \pi} \]

   6. Applied rewrites 70.9%

      \[\leadsto \frac{0.25}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \pi} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(\pi \cdot \color{blue}{z1}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot \pi\right) \cdot \color{blue}{z1}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot \pi\right) \cdot z1} \]

      7. lower-*.f64 70.9%

         \[\leadsto \frac{0.25}{\left(z0 \cdot \pi\right) \cdot \color{blue}{z1}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot \pi\right) \cdot z1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot z0\right) \cdot z1} \]

      10. lower-*.f64 70.9%

          \[\leadsto \frac{0.25}{\left(\pi \cdot z0\right) \cdot z1} \]

   8. Applied rewrites 70.9%

      \[\leadsto \frac{0.25}{\left(\pi \cdot z0\right) \cdot \color{blue}{z1}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\pi \cdot z0\right) \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot z0\right) \cdot \color{blue}{z1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \color{blue}{\left(\pi \cdot z0\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \left(\pi \cdot \color{blue}{z0}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \left(z0 \cdot \color{blue}{\pi}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{z1 \cdot \left(z0 \cdot \color{blue}{\pi}\right)} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{\color{blue}{z0 \cdot \pi}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{\color{blue}{z0 \cdot \pi}} \]

      9. lower-/.f64 71.0%

         \[\leadsto \frac{\frac{0.25}{z1}}{\color{blue}{z0} \cdot \pi} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{z0 \cdot \color{blue}{\pi}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{z1}}{\pi \cdot \color{blue}{z0}} \]

      12. lift-*.f64 71.0%

          \[\leadsto \frac{\frac{0.25}{z1}}{\pi \cdot \color{blue}{z0}} \]

   10. Applied rewrites 71.0%

       \[\leadsto \frac{\frac{0.25}{z1}}{\color{blue}{\pi \cdot z0}} \]

   if -9.9999874849559983e-319 < (/.f64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 #s(literal -3333333333333333/10000000000000000 binary64) (/.f64 z1 z0))) (*.f64 z0 (PI.f64))) #s(literal 1/8 binary64)) (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64))))) z1) < 0.0

   1. Initial program 99.6%

      \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}}{z1} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)}}{z1} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)}}{z1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{e^{\frac{-3333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot \frac{1}{8} + \frac{\frac{1}{8}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}}{z1} \]

      10. lower-PI.f64 87.8%

          \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}{z1} \]

   4. Applied rewrites 87.8%

      \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}}}{z1} \]

   6. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\frac{\left(e^{-0.3333333333333333 \cdot \frac{z1}{z0}} \cdot 0.125\right) \cdot z1 - \left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot \frac{-0.125}{\left(\left(\pi \cdot \frac{z1}{z0}\right) \cdot 0.5 + \pi\right) \cdot z1 + \pi \cdot z0}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1}} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot z1}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1} \]
   8. Step-by-step derivation

      1. lower-*.f64 62.8%

         \[\leadsto \frac{0.25 \cdot \color{blue}{z1}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1} \]

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\color{blue}{0.25 \cdot z1}}{\left(\left(\pi \cdot z1\right) \cdot z0\right) \cdot z1} \]

   if 0.0 < (/.f64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 #s(literal -3333333333333333/10000000000000000 binary64) (/.f64 z1 z0))) (*.f64 z0 (PI.f64))) #s(literal 1/8 binary64)) (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64))))) z1)

   1. Initial program 99.6%

      \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{z0 \cdot \pi}}}{z1} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{z0 \cdot \mathsf{PI}\left(\right)}}}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{z1} \]

      3. lower-PI.f64 71.0%

         \[\leadsto \frac{\frac{0.25}{z0 \cdot \pi}}{z1} \]

   4. Applied rewrites 71.0%

      \[\leadsto \frac{\color{blue}{\frac{0.25}{z0 \cdot \pi}}}{z1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.1% accurate, 10.1× speedup

Math

\[\frac{\frac{0.25}{\pi \cdot z1}}{z0} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ (/ 0.25 (* PI z1)) z0))

C

double code(double z1, double z0) {
	return (0.25 / (((double) M_PI) * z1)) / z0;
}

Java

public static double code(double z1, double z0) {
	return (0.25 / (Math.PI * z1)) / z0;
}

Python

def code(z1, z0):
	return (0.25 / (math.pi * z1)) / z0

Julia

function code(z1, z0)
	return Float64(Float64(0.25 / Float64(pi * z1)) / z0)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.25 / (pi * z1)) / z0;
end

Wolfram

code[z1_, z0_] := N[(N[(0.25 / N[(Pi * z1), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f64 71.0%

      \[\leadsto \frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)} \]

4. Applied rewrites 71.0%

   \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{z0 \cdot \left(z1 \cdot \pi\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(z1 \cdot \pi\right) \cdot \color{blue}{z0}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{z1 \cdot \pi}}{\color{blue}{z0}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{z1 \cdot \pi}}{\color{blue}{z0}} \]

   6. lower-/.f64 71.1%

      \[\leadsto \frac{\frac{0.25}{z1 \cdot \pi}}{z0} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{z1 \cdot \pi}}{z0} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot z1}}{z0} \]

   9. lower-*.f64 71.1%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot z1}}{z0} \]

6. Applied rewrites 71.1%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot z1}}{\color{blue}{z0}} \]
7. Add Preprocessing

Alternative 15: 71.0% accurate, 12.9× speedup

Math

\[\frac{0.25}{\left(z0 \cdot z1\right) \cdot \pi} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 0.25 (* (* z0 z1) PI)))

C

double code(double z1, double z0) {
	return 0.25 / ((z0 * z1) * ((double) M_PI));
}

Java

public static double code(double z1, double z0) {
	return 0.25 / ((z0 * z1) * Math.PI);
}

Python

def code(z1, z0):
	return 0.25 / ((z0 * z1) * math.pi)

Julia

function code(z1, z0)
	return Float64(0.25 / Float64(Float64(z0 * z1) * pi))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 0.25 / ((z0 * z1) * pi);
end

Wolfram

code[z1_, z0_] := N[(0.25 / N[(N[(z0 * z1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f64 71.0%

      \[\leadsto \frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)} \]

4. Applied rewrites 71.0%

   \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\pi}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]

   5. lower-*.f64 70.9%

      \[\leadsto \frac{0.25}{\left(z0 \cdot z1\right) \cdot \pi} \]

6. Applied rewrites 70.9%

   \[\leadsto \frac{0.25}{\left(z0 \cdot z1\right) \cdot \color{blue}{\pi}} \]
7. Add Preprocessing

Alternative 16: 70.9% accurate, 12.9× speedup

Math

\[\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 0.25 (* z0 (* z1 PI))))

C

double code(double z1, double z0) {
	return 0.25 / (z0 * (z1 * ((double) M_PI)));
}

Java

public static double code(double z1, double z0) {
	return 0.25 / (z0 * (z1 * Math.PI));
}

Python

def code(z1, z0):
	return 0.25 / (z0 * (z1 * math.pi))

Julia

function code(z1, z0)
	return Float64(0.25 / Float64(z0 * Float64(z1 * pi)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 0.25 / (z0 * (z1 * pi));
end

Wolfram

code[z1_, z0_] := N[(0.25 / N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\frac{e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{z0 \cdot \pi} \cdot 0.125 + \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}}{z1} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \color{blue}{\left(z1 \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4}}{z0 \cdot \left(z1 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f64 71.0%

      \[\leadsto \frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)} \]

4. Applied rewrites 71.0%

   \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(/ (+ (* (/ (exp (* -3333333333333333/10000000000000000 (/ z1 z0))) (* z0 PI)) 1/8) (/ 1/8 (* (exp (/ z1 z0)) (* z0 PI)))) z1)"
  :precision binary64
  (/ (+ (* (/ (exp (* -0.3333333333333333 (/ z1 z0))) (* z0 PI)) 0.125) (/ 0.125 (* (exp (/ z1 z0)) (* z0 PI)))) z1))