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

Percentage Accurate: 99.7% → 99.6%

Time: 3.4s

Alternatives: 17

Speedup: N/A×

• 69.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

3. Applied rewrites 99.6%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{z1}{z0} \cdot -0.3333333333333333}\\ \mathbf{if}\;\frac{\frac{t\_0 \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -2 \cdot 10^{+292}:\\ \;\;\;\;\frac{0.125}{z0} \cdot \frac{1 + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}}{\pi \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{-z1}{z0}} + t\_0}{z0 \cdot \left(z1 \cdot \pi\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (exp (* (/ z1 z0) -0.3333333333333333))))
  (if (<=
       (/
        (/ (+ (* t_0 0.125) (* 0.125 (exp (- (/ z1 z0))))) (* z0 PI))
        z1)
       -2e+292)
    (*
     (/ 0.125 z0)
     (/ (+ 1.0 (exp (* -0.3333333333333333 (/ z1 z0)))) (* PI z1)))
    (* 0.125 (/ (+ (exp (/ (- z1) z0)) t_0) (* z0 (* z1 PI)))))))

C

double code(double z1, double z0) {
	double t_0 = exp(((z1 / z0) * -0.3333333333333333));
	double tmp;
	if (((((t_0 * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -2e+292) {
		tmp = (0.125 / z0) * ((1.0 + exp((-0.3333333333333333 * (z1 / z0)))) / (((double) M_PI) * z1));
	} else {
		tmp = 0.125 * ((exp((-z1 / z0)) + t_0) / (z0 * (z1 * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.exp(((z1 / z0) * -0.3333333333333333));
	double tmp;
	if (((((t_0 * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -2e+292) {
		tmp = (0.125 / z0) * ((1.0 + Math.exp((-0.3333333333333333 * (z1 / z0)))) / (Math.PI * z1));
	} else {
		tmp = 0.125 * ((Math.exp((-z1 / z0)) + t_0) / (z0 * (z1 * Math.PI)));
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = math.exp(((z1 / z0) * -0.3333333333333333))
	tmp = 0
	if ((((t_0 * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -2e+292:
		tmp = (0.125 / z0) * ((1.0 + math.exp((-0.3333333333333333 * (z1 / z0)))) / (math.pi * z1))
	else:
		tmp = 0.125 * ((math.exp((-z1 / z0)) + t_0) / (z0 * (z1 * math.pi)))
	return tmp

Julia

function code(z1, z0)
	t_0 = exp(Float64(Float64(z1 / z0) * -0.3333333333333333))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t_0 * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= -2e+292)
		tmp = Float64(Float64(0.125 / z0) * Float64(Float64(1.0 + exp(Float64(-0.3333333333333333 * Float64(z1 / z0)))) / Float64(pi * z1)));
	else
		tmp = Float64(0.125 * Float64(Float64(exp(Float64(Float64(-z1) / z0)) + t_0) / Float64(z0 * Float64(z1 * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = exp(((z1 / z0) * -0.3333333333333333));
	tmp = 0.0;
	if (((((t_0 * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -2e+292)
		tmp = (0.125 / z0) * ((1.0 + exp((-0.3333333333333333 * (z1 / z0)))) / (pi * z1));
	else
		tmp = 0.125 * ((exp((-z1 / z0)) + t_0) / (z0 * (z1 * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(t$95$0 * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], -2e+292], N[(N[(0.125 / z0), $MachinePrecision] * N[(N[(1.0 + N[Exp[N[(-0.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(N[(N[Exp[N[((-z1) / z0), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -2e292

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around 0

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

   6. Applied rewrites 78.2%

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

   if -2e292 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

      8. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot \left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right)}{z0 \cdot {\pi}^{0.6666666666666666}}}{\sqrt[3]{\pi} \cdot z1}} \]

   4. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 93.7%

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 93.7%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.7%

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

   8. Applied rewrites 93.7%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{-z1}{z0}} + e^{\frac{z1}{z0} \cdot -0.3333333333333333}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 0.125 \cdot e^{-\frac{z1}{z0}}\\ t_1 := \frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + t\_0}{z0 \cdot \pi}}{z1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-320}:\\ \;\;\;\;\frac{\frac{0.125 + t\_0}{z0 \cdot \pi}}{z1}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\left(\pi \cdot z0\right) \cdot z1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.25}{\pi \cdot z1} - \left(\frac{0.16666666666666666}{\pi \cdot z0} - \frac{z1 \cdot 0.06944444444444445}{\left(\pi \cdot z0\right) \cdot z0}\right)\right) \cdot \frac{1}{z0}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* 0.125 (exp (- (/ z1 z0)))))
       (t_1
        (/
         (/
          (+ (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125) t_0)
          (* z0 PI))
         z1)))
  (if (<= t_1 -2e-320)
    (/ (/ (+ 0.125 t_0) (* z0 PI)) z1)
    (if (<= t_1 0.0)
      (/ 0.25 (log (exp (* (* PI z0) z1))))
      (*
       (-
        (/ 0.25 (* PI z1))
        (-
         (/ 0.16666666666666666 (* PI z0))
         (/ (* z1 0.06944444444444445) (* (* PI z0) z0))))
       (/ 1.0 z0))))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 * exp(-(z1 / z0));
	double t_1 = (((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + t_0) / (z0 * ((double) M_PI))) / z1;
	double tmp;
	if (t_1 <= -2e-320) {
		tmp = ((0.125 + t_0) / (z0 * ((double) M_PI))) / z1;
	} else if (t_1 <= 0.0) {
		tmp = 0.25 / log(exp(((((double) M_PI) * z0) * z1)));
	} else {
		tmp = ((0.25 / (((double) M_PI) * z1)) - ((0.16666666666666666 / (((double) M_PI) * z0)) - ((z1 * 0.06944444444444445) / ((((double) M_PI) * z0) * z0)))) * (1.0 / z0);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 * Math.exp(-(z1 / z0));
	double t_1 = (((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + t_0) / (z0 * Math.PI)) / z1;
	double tmp;
	if (t_1 <= -2e-320) {
		tmp = ((0.125 + t_0) / (z0 * Math.PI)) / z1;
	} else if (t_1 <= 0.0) {
		tmp = 0.25 / Math.log(Math.exp(((Math.PI * z0) * z1)));
	} else {
		tmp = ((0.25 / (Math.PI * z1)) - ((0.16666666666666666 / (Math.PI * z0)) - ((z1 * 0.06944444444444445) / ((Math.PI * z0) * z0)))) * (1.0 / z0);
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 * math.exp(-(z1 / z0))
	t_1 = (((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + t_0) / (z0 * math.pi)) / z1
	tmp = 0
	if t_1 <= -2e-320:
		tmp = ((0.125 + t_0) / (z0 * math.pi)) / z1
	elif t_1 <= 0.0:
		tmp = 0.25 / math.log(math.exp(((math.pi * z0) * z1)))
	else:
		tmp = ((0.25 / (math.pi * z1)) - ((0.16666666666666666 / (math.pi * z0)) - ((z1 * 0.06944444444444445) / ((math.pi * z0) * z0)))) * (1.0 / z0)
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 * exp(Float64(-Float64(z1 / z0))))
	t_1 = Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + t_0) / Float64(z0 * pi)) / z1)
	tmp = 0.0
	if (t_1 <= -2e-320)
		tmp = Float64(Float64(Float64(0.125 + t_0) / Float64(z0 * pi)) / z1);
	elseif (t_1 <= 0.0)
		tmp = Float64(0.25 / log(exp(Float64(Float64(pi * z0) * z1))));
	else
		tmp = Float64(Float64(Float64(0.25 / Float64(pi * z1)) - Float64(Float64(0.16666666666666666 / Float64(pi * z0)) - Float64(Float64(z1 * 0.06944444444444445) / Float64(Float64(pi * z0) * z0)))) * Float64(1.0 / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 * exp(-(z1 / z0));
	t_1 = (((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + t_0) / (z0 * pi)) / z1;
	tmp = 0.0;
	if (t_1 <= -2e-320)
		tmp = ((0.125 + t_0) / (z0 * pi)) / z1;
	elseif (t_1 <= 0.0)
		tmp = 0.25 / log(exp(((pi * z0) * z1)));
	else
		tmp = ((0.25 / (pi * z1)) - ((0.16666666666666666 / (pi * z0)) - ((z1 * 0.06944444444444445) / ((pi * z0) * z0)))) * (1.0 / z0);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-320], N[(N[(N[(0.125 + t$95$0), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(0.25 / N[Log[N[Exp[N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 / N[(Pi * z1), $MachinePrecision]), $MachinePrecision] - N[(N[(0.16666666666666666 / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] - N[(N[(z1 * 0.06944444444444445), $MachinePrecision] / N[(N[(Pi * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -1.999977734365366e-320

   1. Initial program 99.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8}} + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \]
   3. Step-by-step derivation

   4. Applied rewrites 78.3%

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

   if -1.999977734365366e-320 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < 0.0

   1. Initial program 99.7%

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

   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 57.1%

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   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. lift-PI.f64 N/A

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

      5. add-log-exp N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(z0 \cdot z1\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

      6. log-pow-rev N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(z0 \cdot z1\right)}\right)} \]

      7. lower-log.f64 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(z0 \cdot z1\right)}\right)} \]

      8. lift-PI.f64 N/A

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

      9. pow-exp N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-exp.f64 23.9%

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

   6. Applied rewrites 23.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 67.8%

      \[\leadsto \left(\frac{0.25}{\pi \cdot z1} - \left(\frac{0.16666666666666666}{\pi \cdot z0} - \frac{z1 \cdot 0.06944444444444445}{\left(\pi \cdot z0\right) \cdot z0}\right)\right) \cdot \color{blue}{\frac{1}{z0}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in z1 around 0

   \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8}} + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \]
3. Step-by-step derivation

4. Applied rewrites 78.3%

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

Alternative 5: 75.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

   8. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot \left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right)}{z0 \cdot {\pi}^{0.6666666666666666}}}{\sqrt[3]{\pi} \cdot z1}} \]

4. Taylor expanded in z1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 93.7%

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

6. Applied rewrites 93.7%

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

7. Taylor expanded in z1 around 0

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

9. Applied rewrites 75.1%

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

Alternative 6: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -4 \cdot 10^{-286}:\\ \;\;\;\;-1 \cdot \frac{\frac{\left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - z0 \cdot 0.25}{\left(\pi \cdot z0\right) \cdot z1}}{z0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{z0}}{\pi \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<=
     (/
      (/
       (+
        (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125)
        (* 0.125 (exp (- (/ z1 z0)))))
       (* z0 PI))
      z1)
     -4e-286)
  (*
   -1.0
   (/
    (/
     (-
      (*
       (-
        (/ 0.16666666666666666 PI)
        (* 0.06944444444444445 (/ z1 (* PI z0))))
       (* PI z1))
      (* z0 0.25))
     (* (* PI z0) z1))
    z0))
  (/ (/ 0.25 z0) (* PI z1))))

C

double code(double z1, double z0) {
	double tmp;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -4e-286) {
		tmp = -1.0 * ((((((0.16666666666666666 / ((double) M_PI)) - (0.06944444444444445 * (z1 / (((double) M_PI) * z0)))) * (((double) M_PI) * z1)) - (z0 * 0.25)) / ((((double) M_PI) * z0) * z1)) / z0);
	} else {
		tmp = (0.25 / z0) / (((double) M_PI) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (((((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -4e-286) {
		tmp = -1.0 * ((((((0.16666666666666666 / Math.PI) - (0.06944444444444445 * (z1 / (Math.PI * z0)))) * (Math.PI * z1)) - (z0 * 0.25)) / ((Math.PI * z0) * z1)) / z0);
	} else {
		tmp = (0.25 / z0) / (Math.PI * z1);
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if ((((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -4e-286:
		tmp = -1.0 * ((((((0.16666666666666666 / math.pi) - (0.06944444444444445 * (z1 / (math.pi * z0)))) * (math.pi * z1)) - (z0 * 0.25)) / ((math.pi * z0) * z1)) / z0)
	else:
		tmp = (0.25 / z0) / (math.pi * z1)
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= -4e-286)
		tmp = Float64(-1.0 * Float64(Float64(Float64(Float64(Float64(Float64(0.16666666666666666 / pi) - Float64(0.06944444444444445 * Float64(z1 / Float64(pi * z0)))) * Float64(pi * z1)) - Float64(z0 * 0.25)) / Float64(Float64(pi * z0) * z1)) / z0));
	else
		tmp = Float64(Float64(0.25 / z0) / Float64(pi * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -4e-286)
		tmp = -1.0 * ((((((0.16666666666666666 / pi) - (0.06944444444444445 * (z1 / (pi * z0)))) * (pi * z1)) - (z0 * 0.25)) / ((pi * z0) * z1)) / z0);
	else
		tmp = (0.25 / z0) / (pi * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], -4e-286], N[(-1.0 * N[(N[(N[(N[(N[(N[(0.16666666666666666 / Pi), $MachinePrecision] - N[(0.06944444444444445 * N[(z1 / N[(Pi * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision] - N[(z0 * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / z0), $MachinePrecision] / N[(Pi * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -4.0000000000000002e-286

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 68.3%

      \[\leadsto -1 \cdot \frac{\frac{\left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - z0 \cdot 0.25}{\left(\pi \cdot z0\right) \cdot z1}}{z0} \]

   if -4.0000000000000002e-286 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

   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 57.1%

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 57.2%

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

   6. Applied rewrites 57.2%

      \[\leadsto \frac{\frac{0.25}{z0}}{\color{blue}{\pi \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{0.25 \cdot z0 - \left(\frac{0.16666666666666666}{\pi} - \frac{z1}{\pi \cdot z0} \cdot 0.06944444444444445\right) \cdot \left(\pi \cdot z1\right)}{\pi \cdot z0}}{z0 \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{z0}}{\pi \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<=
     (/
      (/
       (+
        (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125)
        (* 0.125 (exp (- (/ z1 z0)))))
       (* z0 PI))
      z1)
     -2e-279)
  (/
   (/
    (-
     (* 0.25 z0)
     (*
      (-
       (/ 0.16666666666666666 PI)
       (* (/ z1 (* PI z0)) 0.06944444444444445))
      (* PI z1)))
    (* PI z0))
   (* z0 z1))
  (/ (/ 0.25 z0) (* PI z1))))

C

double code(double z1, double z0) {
	double tmp;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -2e-279) {
		tmp = (((0.25 * z0) - (((0.16666666666666666 / ((double) M_PI)) - ((z1 / (((double) M_PI) * z0)) * 0.06944444444444445)) * (((double) M_PI) * z1))) / (((double) M_PI) * z0)) / (z0 * z1);
	} else {
		tmp = (0.25 / z0) / (((double) M_PI) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (((((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -2e-279) {
		tmp = (((0.25 * z0) - (((0.16666666666666666 / Math.PI) - ((z1 / (Math.PI * z0)) * 0.06944444444444445)) * (Math.PI * z1))) / (Math.PI * z0)) / (z0 * z1);
	} else {
		tmp = (0.25 / z0) / (Math.PI * z1);
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if ((((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -2e-279:
		tmp = (((0.25 * z0) - (((0.16666666666666666 / math.pi) - ((z1 / (math.pi * z0)) * 0.06944444444444445)) * (math.pi * z1))) / (math.pi * z0)) / (z0 * z1)
	else:
		tmp = (0.25 / z0) / (math.pi * z1)
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= -2e-279)
		tmp = Float64(Float64(Float64(Float64(0.25 * z0) - Float64(Float64(Float64(0.16666666666666666 / pi) - Float64(Float64(z1 / Float64(pi * z0)) * 0.06944444444444445)) * Float64(pi * z1))) / Float64(pi * z0)) / Float64(z0 * z1));
	else
		tmp = Float64(Float64(0.25 / z0) / Float64(pi * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -2e-279)
		tmp = (((0.25 * z0) - (((0.16666666666666666 / pi) - ((z1 / (pi * z0)) * 0.06944444444444445)) * (pi * z1))) / (pi * z0)) / (z0 * z1);
	else
		tmp = (0.25 / z0) / (pi * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], -2e-279], N[(N[(N[(N[(0.25 * z0), $MachinePrecision] - N[(N[(N[(0.16666666666666666 / Pi), $MachinePrecision] - N[(N[(z1 / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] * 0.06944444444444445), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] / N[(z0 * z1), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / z0), $MachinePrecision] / N[(Pi * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -2.0000000000000001e-279

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 56.9%

      \[\leadsto \frac{\left(0.25 \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot \color{blue}{z0}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\left(\pi \cdot z0\right) \cdot \color{blue}{\left(z1 \cdot z0\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\pi \cdot z0}}{\color{blue}{z1 \cdot z0}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\pi \cdot z0}}{z0 \cdot \color{blue}{z1}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\pi \cdot z0}}{\color{blue}{z0 \cdot z1}} \]

   8. Applied rewrites 68.5%

      \[\leadsto \frac{\frac{0.25 \cdot z0 - \left(\frac{0.16666666666666666}{\pi} - \frac{z1}{\pi \cdot z0} \cdot 0.06944444444444445\right) \cdot \left(\pi \cdot z1\right)}{\pi \cdot z0}}{\color{blue}{z0 \cdot z1}} \]

   if -2.0000000000000001e-279 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

   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 57.1%

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 57.2%

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

   6. Applied rewrites 57.2%

      \[\leadsto \frac{\frac{0.25}{z0}}{\color{blue}{\pi \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -\infty:\\ \;\;\;\;\frac{1}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot z1} \cdot \left(0.25 \cdot z0 - \left(\frac{0.16666666666666666}{\pi} - \frac{z1}{\pi \cdot z0} \cdot 0.06944444444444445\right) \cdot \left(\pi \cdot z1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<=
     (/
      (/
       (+
        (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125)
        (* 0.125 (exp (- (/ z1 z0)))))
       (* z0 PI))
      z1)
     (- INFINITY))
  (*
   (/ 1.0 (* (* (* PI z0) z0) z1))
   (-
    (* 0.25 z0)
    (*
     (-
      (/ 0.16666666666666666 PI)
      (* (/ z1 (* PI z0)) 0.06944444444444445))
     (* PI z1))))
  (*
   0.125
   (/ (+ 2.0 (* -1.3333333333333333 (/ z1 z0))) (* z0 (* z1 PI))))))

C

double code(double z1, double z0) {
	double tmp;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -((double) INFINITY)) {
		tmp = (1.0 / (((((double) M_PI) * z0) * z0) * z1)) * ((0.25 * z0) - (((0.16666666666666666 / ((double) M_PI)) - ((z1 / (((double) M_PI) * z0)) * 0.06944444444444445)) * (((double) M_PI) * z1)));
	} else {
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (((((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (((Math.PI * z0) * z0) * z1)) * ((0.25 * z0) - (((0.16666666666666666 / Math.PI) - ((z1 / (Math.PI * z0)) * 0.06944444444444445)) * (Math.PI * z1)));
	} else {
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * Math.PI)));
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if ((((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -math.inf:
		tmp = (1.0 / (((math.pi * z0) * z0) * z1)) * ((0.25 * z0) - (((0.16666666666666666 / math.pi) - ((z1 / (math.pi * z0)) * 0.06944444444444445)) * (math.pi * z1)))
	else:
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * math.pi)))
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(pi * z0) * z0) * z1)) * Float64(Float64(0.25 * z0) - Float64(Float64(Float64(0.16666666666666666 / pi) - Float64(Float64(z1 / Float64(pi * z0)) * 0.06944444444444445)) * Float64(pi * z1))));
	else
		tmp = Float64(0.125 * Float64(Float64(2.0 + Float64(-1.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * Float64(z1 * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -Inf)
		tmp = (1.0 / (((pi * z0) * z0) * z1)) * ((0.25 * z0) - (((0.16666666666666666 / pi) - ((z1 / (pi * z0)) * 0.06944444444444445)) * (pi * z1)));
	else
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], (-Infinity)], N[(N[(1.0 / N[(N[(N[(Pi * z0), $MachinePrecision] * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * z0), $MachinePrecision] - N[(N[(N[(0.16666666666666666 / Pi), $MachinePrecision] - N[(N[(z1 / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] * 0.06944444444444445), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(N[(2.0 + N[(-1.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -inf.0

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 56.9%

      \[\leadsto \frac{\left(0.25 \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1\right) \cdot \color{blue}{\frac{1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1\right)} \]

      5. lower-/.f64 56.6%

         \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \cdot \left(\color{blue}{\left(0.25 \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right)\right)} \cdot 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \cdot \left(\left(\frac{1}{4} \cdot z0 - \color{blue}{\left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)}\right) \cdot 1\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\left(\pi \cdot z0\right) \cdot z1\right)} \cdot \left(\left(\frac{1}{4} \cdot z0 - \color{blue}{\left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)}\right) \cdot 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\left(\pi \cdot z0\right) \cdot z1\right)} \cdot \left(\left(\frac{1}{4} \cdot z0 - \left(\pi \cdot z1\right) \cdot \color{blue}{\left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)}\right) \cdot 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \frac{1}{\left(z0 \cdot \left(\pi \cdot z0\right)\right) \cdot z1} \cdot \left(\left(\frac{1}{4} \cdot z0 - \color{blue}{\left(\pi \cdot z1\right) \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)}\right) \cdot 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot z1} \cdot \left(\left(\frac{1}{4} \cdot z0 - \color{blue}{\left(\pi \cdot z1\right)} \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot z1} \cdot \left(\left(\frac{1}{4} \cdot z0 - \color{blue}{\left(\pi \cdot z1\right)} \cdot \left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi} - \frac{111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1\right) \]

      12. lower-*.f64 50.8%

          \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot z1} \cdot \left(\left(0.25 \cdot z0 - \color{blue}{\left(\pi \cdot z1\right) \cdot \left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right)}\right) \cdot 1\right) \]

   8. Applied rewrites 50.8%

      \[\leadsto \frac{1}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot z1} \cdot \color{blue}{\left(0.25 \cdot z0 - \left(\frac{0.16666666666666666}{\pi} - \frac{z1}{\pi \cdot z0} \cdot 0.06944444444444445\right) \cdot \left(\pi \cdot z1\right)\right)} \]

   if -inf.0 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

      8. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot \left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right)}{z0 \cdot {\pi}^{0.6666666666666666}}}{\sqrt[3]{\pi} \cdot z1}} \]

   4. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 93.7%

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

   6. Applied rewrites 93.7%

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

   7. Taylor expanded in z1 around 0

      \[\leadsto 0.125 \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

      3. lower-/.f64 65.9%

         \[\leadsto 0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

   9. Applied rewrites 65.9%

      \[\leadsto 0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{0.25 - \left(\frac{0.16666666666666666}{\pi \cdot z0} - \frac{0.06944444444444445}{\left(\pi \cdot z0\right) \cdot z0} \cdot z1\right) \cdot \left(\pi \cdot z1\right)}{\left(\pi \cdot z0\right) \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{z0}}{\pi \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<=
     (/
      (/
       (+
        (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125)
        (* 0.125 (exp (- (/ z1 z0)))))
       (* z0 PI))
      z1)
     -2e-279)
  (/
   (-
    0.25
    (*
     (-
      (/ 0.16666666666666666 (* PI z0))
      (* (/ 0.06944444444444445 (* (* PI z0) z0)) z1))
     (* PI z1)))
   (* (* PI z0) z1))
  (/ (/ 0.25 z0) (* PI z1))))

C

double code(double z1, double z0) {
	double tmp;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -2e-279) {
		tmp = (0.25 - (((0.16666666666666666 / (((double) M_PI) * z0)) - ((0.06944444444444445 / ((((double) M_PI) * z0) * z0)) * z1)) * (((double) M_PI) * z1))) / ((((double) M_PI) * z0) * z1);
	} else {
		tmp = (0.25 / z0) / (((double) M_PI) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (((((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -2e-279) {
		tmp = (0.25 - (((0.16666666666666666 / (Math.PI * z0)) - ((0.06944444444444445 / ((Math.PI * z0) * z0)) * z1)) * (Math.PI * z1))) / ((Math.PI * z0) * z1);
	} else {
		tmp = (0.25 / z0) / (Math.PI * z1);
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if ((((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -2e-279:
		tmp = (0.25 - (((0.16666666666666666 / (math.pi * z0)) - ((0.06944444444444445 / ((math.pi * z0) * z0)) * z1)) * (math.pi * z1))) / ((math.pi * z0) * z1)
	else:
		tmp = (0.25 / z0) / (math.pi * z1)
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= -2e-279)
		tmp = Float64(Float64(0.25 - Float64(Float64(Float64(0.16666666666666666 / Float64(pi * z0)) - Float64(Float64(0.06944444444444445 / Float64(Float64(pi * z0) * z0)) * z1)) * Float64(pi * z1))) / Float64(Float64(pi * z0) * z1));
	else
		tmp = Float64(Float64(0.25 / z0) / Float64(pi * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -2e-279)
		tmp = (0.25 - (((0.16666666666666666 / (pi * z0)) - ((0.06944444444444445 / ((pi * z0) * z0)) * z1)) * (pi * z1))) / ((pi * z0) * z1);
	else
		tmp = (0.25 / z0) / (pi * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], -2e-279], N[(N[(0.25 - N[(N[(N[(0.16666666666666666 / N[(Pi * z0), $MachinePrecision]), $MachinePrecision] - N[(N[(0.06944444444444445 / N[(N[(Pi * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / z0), $MachinePrecision] / N[(Pi * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -2.0000000000000001e-279

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 69.9%

      \[\leadsto -1 \cdot \frac{\left(\left(\frac{0.16666666666666666}{\pi \cdot z0} - \frac{z1 \cdot 0.06944444444444445}{\left(\pi \cdot z0\right) \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - 0.25\right) \cdot 1}{\color{blue}{\left(\pi \cdot z0\right) \cdot z1}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\left(\left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi \cdot z0} - \frac{z1 \cdot \frac{111111111111111108888888888888889}{1600000000000000000000000000000000}}{\left(\pi \cdot z0\right) \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - \frac{1}{4}\right) \cdot 1}{\left(\pi \cdot z0\right) \cdot z1}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(\left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi \cdot z0} - \frac{z1 \cdot \frac{111111111111111108888888888888889}{1600000000000000000000000000000000}}{\left(\pi \cdot z0\right) \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - \frac{1}{4}\right) \cdot 1}{\left(\pi \cdot z0\right) \cdot z1}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(\left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi \cdot z0} - \frac{z1 \cdot \frac{111111111111111108888888888888889}{1600000000000000000000000000000000}}{\left(\pi \cdot z0\right) \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - \frac{1}{4}\right) \cdot 1}{\left(\pi \cdot z0\right) \cdot z1}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi \cdot z0} - \frac{z1 \cdot \frac{111111111111111108888888888888889}{1600000000000000000000000000000000}}{\left(\pi \cdot z0\right) \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - \frac{1}{4}\right) \cdot 1\right)}{\color{blue}{\left(\pi \cdot z0\right) \cdot z1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\left(\frac{\frac{13333333333333333}{80000000000000000}}{\pi \cdot z0} - \frac{z1 \cdot \frac{111111111111111108888888888888889}{1600000000000000000000000000000000}}{\left(\pi \cdot z0\right) \cdot z0}\right) \cdot \left(\pi \cdot z1\right) - \frac{1}{4}\right) \cdot 1\right)}{\color{blue}{\left(\pi \cdot z0\right) \cdot z1}} \]

   8. Applied rewrites 69.9%

      \[\leadsto \frac{0.25 - \left(\frac{0.16666666666666666}{\pi \cdot z0} - \frac{0.06944444444444445}{\left(\pi \cdot z0\right) \cdot z0} \cdot z1\right) \cdot \left(\pi \cdot z1\right)}{\color{blue}{\left(\pi \cdot z0\right) \cdot z1}} \]

   if -2.0000000000000001e-279 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

   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 57.1%

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 57.2%

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

   6. Applied rewrites 57.2%

      \[\leadsto \frac{\frac{0.25}{z0}}{\color{blue}{\pi \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -\infty:\\ \;\;\;\;\frac{\left(0.25 \cdot z0 - z1 \cdot \left(0.16666666666666666 + -0.06944444444444445 \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<=
     (/
      (/
       (+
        (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125)
        (* 0.125 (exp (- (/ z1 z0)))))
       (* z0 PI))
      z1)
     (- INFINITY))
  (/
   (*
    (-
     (* 0.25 z0)
     (*
      z1
      (+ 0.16666666666666666 (* -0.06944444444444445 (/ z1 z0)))))
    1.0)
   (* (* (* PI z0) z1) z0))
  (*
   0.125
   (/ (+ 2.0 (* -1.3333333333333333 (/ z1 z0))) (* z0 (* z1 PI))))))

C

double code(double z1, double z0) {
	double tmp;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -((double) INFINITY)) {
		tmp = (((0.25 * z0) - (z1 * (0.16666666666666666 + (-0.06944444444444445 * (z1 / z0))))) * 1.0) / (((((double) M_PI) * z0) * z1) * z0);
	} else {
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (((((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -Double.POSITIVE_INFINITY) {
		tmp = (((0.25 * z0) - (z1 * (0.16666666666666666 + (-0.06944444444444445 * (z1 / z0))))) * 1.0) / (((Math.PI * z0) * z1) * z0);
	} else {
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * Math.PI)));
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if ((((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -math.inf:
		tmp = (((0.25 * z0) - (z1 * (0.16666666666666666 + (-0.06944444444444445 * (z1 / z0))))) * 1.0) / (((math.pi * z0) * z1) * z0)
	else:
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * math.pi)))
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(0.25 * z0) - Float64(z1 * Float64(0.16666666666666666 + Float64(-0.06944444444444445 * Float64(z1 / z0))))) * 1.0) / Float64(Float64(Float64(pi * z0) * z1) * z0));
	else
		tmp = Float64(0.125 * Float64(Float64(2.0 + Float64(-1.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * Float64(z1 * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -Inf)
		tmp = (((0.25 * z0) - (z1 * (0.16666666666666666 + (-0.06944444444444445 * (z1 / z0))))) * 1.0) / (((pi * z0) * z1) * z0);
	else
		tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(0.25 * z0), $MachinePrecision] - N[(z1 * N[(0.16666666666666666 + N[(-0.06944444444444445 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(N[(2.0 + N[(-1.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -inf.0

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 56.9%

      \[\leadsto \frac{\left(0.25 \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \frac{\left(0.25 \cdot z0 - z1 \cdot \left(\frac{13333333333333333}{80000000000000000} + \frac{-111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot \color{blue}{z0}\right) \cdot z1\right) \cdot z0} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - z1 \cdot \left(\frac{13333333333333333}{80000000000000000} + \frac{-111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - z1 \cdot \left(\frac{13333333333333333}{80000000000000000} + \frac{-111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot z0 - z1 \cdot \left(\frac{13333333333333333}{80000000000000000} + \frac{-111111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \]

      4. lower-/.f64 56.9%

         \[\leadsto \frac{\left(0.25 \cdot z0 - z1 \cdot \left(0.16666666666666666 + -0.06944444444444445 \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0} \]

   9. Applied rewrites 56.9%

      \[\leadsto \frac{\left(0.25 \cdot z0 - z1 \cdot \left(0.16666666666666666 + -0.06944444444444445 \cdot \frac{z1}{z0}\right)\right) \cdot 1}{\left(\left(\pi \cdot \color{blue}{z0}\right) \cdot z1\right) \cdot z0} \]

   if -inf.0 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

      8. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot \left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right)}{z0 \cdot {\pi}^{0.6666666666666666}}}{\sqrt[3]{\pi} \cdot z1}} \]

   4. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 93.7%

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

   6. Applied rewrites 93.7%

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

   7. Taylor expanded in z1 around 0

      \[\leadsto 0.125 \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

      3. lower-/.f64 65.9%

         \[\leadsto 0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

   9. Applied rewrites 65.9%

      \[\leadsto 0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.9% accurate, 5.9× speedup

Math

\[0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 0.125
 (/ (+ 2.0 (* -1.3333333333333333 (/ z1 z0))) (* z0 (* z1 PI)))))

C

double code(double z1, double z0) {
	return 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * ((double) M_PI))));
}

Java

public static double code(double z1, double z0) {
	return 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * Math.PI)));
}

Python

def code(z1, z0):
	return 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * math.pi)))

Julia

function code(z1, z0)
	return Float64(0.125 * Float64(Float64(2.0 + Float64(-1.3333333333333333 * Float64(z1 / z0))) / Float64(z0 * Float64(z1 * pi))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 0.125 * ((2.0 + (-1.3333333333333333 * (z1 / z0))) / (z0 * (z1 * pi)));
end

Wolfram

code[z1_, z0_] := N[(0.125 * N[(N[(2.0 + N[(-1.3333333333333333 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

      \[\leadsto \frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}}{z1} \]

   8. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{z1}{z0} \cdot \frac{-3333333333333333}{10000000000000000}} \cdot \frac{1}{8} + \frac{1}{8} \cdot e^{-\frac{z1}{z0}}}{z0}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1}} \]

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot \left(e^{\frac{-z1}{z0}} + e^{-0.3333333333333333 \cdot \frac{z1}{z0}}\right)}{z0 \cdot {\pi}^{0.6666666666666666}}}{\sqrt[3]{\pi} \cdot z1}} \]

4. Taylor expanded in z1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 93.7%

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

6. Applied rewrites 93.7%

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

7. Taylor expanded in z1 around 0

   \[\leadsto 0.125 \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{1}{8} \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{8} \cdot \frac{2 + \frac{-13333333333333333}{10000000000000000} \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

   3. lower-/.f64 65.9%

      \[\leadsto 0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{z0 \cdot \left(z1 \cdot \pi\right)} \]

9. Applied rewrites 65.9%

   \[\leadsto 0.125 \cdot \frac{2 + -1.3333333333333333 \cdot \frac{z1}{z0}}{\color{blue}{z0} \cdot \left(z1 \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 12: 65.8% accurate, 6.6× speedup

Math

\[\frac{0.25 + -0.16666666666666666 \cdot \frac{z1}{z0}}{\left(\pi \cdot z0\right) \cdot z1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ (+ 0.25 (* -0.16666666666666666 (/ z1 z0))) (* (* PI z0) z1)))

C

double code(double z1, double z0) {
	return (0.25 + (-0.16666666666666666 * (z1 / z0))) / ((((double) M_PI) * z0) * z1);
}

Java

public static double code(double z1, double z0) {
	return (0.25 + (-0.16666666666666666 * (z1 / z0))) / ((Math.PI * z0) * z1);
}

Python

def code(z1, z0):
	return (0.25 + (-0.16666666666666666 * (z1 / z0))) / ((math.pi * z0) * z1)

Julia

function code(z1, z0)
	return Float64(Float64(0.25 + Float64(-0.16666666666666666 * Float64(z1 / z0))) / Float64(Float64(pi * z0) * z1))
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.25 + (-0.16666666666666666 * (z1 / z0))) / ((pi * z0) * z1);
end

Wolfram

code[z1_, z0_] := N[(N[(0.25 + N[(-0.16666666666666666 * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 93.7%

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

4. Taylor expanded in z1 around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{4} + \frac{-13333333333333333}{80000000000000000} \cdot \frac{z1}{z0}}}{\left(\pi \cdot z0\right) \cdot z1} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{-13333333333333333}{80000000000000000} \cdot \frac{z1}{z0}}}{\left(\pi \cdot z0\right) \cdot z1} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} + \frac{-13333333333333333}{80000000000000000} \cdot \color{blue}{\frac{z1}{z0}}}{\left(\pi \cdot z0\right) \cdot z1} \]

   3. lower-/.f64 65.8%

      \[\leadsto \frac{0.25 + -0.16666666666666666 \cdot \frac{z1}{\color{blue}{z0}}}{\left(\pi \cdot z0\right) \cdot z1} \]

6. Applied rewrites 65.8%

   \[\leadsto \frac{\color{blue}{0.25 + -0.16666666666666666 \cdot \frac{z1}{z0}}}{\left(\pi \cdot z0\right) \cdot z1} \]
7. Add Preprocessing

Alternative 13: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{e^{\frac{z1}{z0} \cdot -0.3333333333333333} \cdot 0.125 + 0.125 \cdot e^{-\frac{z1}{z0}}}{z0 \cdot \pi}}{z1} \leq -\infty:\\ \;\;\;\;\frac{\left(0.25 \cdot z0\right) \cdot 1}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{z0}}{\pi \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<=
     (/
      (/
       (+
        (* (exp (* (/ z1 z0) -0.3333333333333333)) 0.125)
        (* 0.125 (exp (- (/ z1 z0)))))
       (* z0 PI))
      z1)
     (- INFINITY))
  (/ (* (* 0.25 z0) 1.0) (* (* (* PI z0) z1) z0))
  (/ (/ 0.25 z0) (* PI z1))))

C

double code(double z1, double z0) {
	double tmp;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * ((double) M_PI))) / z1) <= -((double) INFINITY)) {
		tmp = ((0.25 * z0) * 1.0) / (((((double) M_PI) * z0) * z1) * z0);
	} else {
		tmp = (0.25 / z0) / (((double) M_PI) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (((((Math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * Math.exp(-(z1 / z0)))) / (z0 * Math.PI)) / z1) <= -Double.POSITIVE_INFINITY) {
		tmp = ((0.25 * z0) * 1.0) / (((Math.PI * z0) * z1) * z0);
	} else {
		tmp = (0.25 / z0) / (Math.PI * z1);
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if ((((math.exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * math.exp(-(z1 / z0)))) / (z0 * math.pi)) / z1) <= -math.inf:
		tmp = ((0.25 * z0) * 1.0) / (((math.pi * z0) * z1) * z0)
	else:
		tmp = (0.25 / z0) / (math.pi * z1)
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(exp(Float64(Float64(z1 / z0) * -0.3333333333333333)) * 0.125) + Float64(0.125 * exp(Float64(-Float64(z1 / z0))))) / Float64(z0 * pi)) / z1) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(0.25 * z0) * 1.0) / Float64(Float64(Float64(pi * z0) * z1) * z0));
	else
		tmp = Float64(Float64(0.25 / z0) / Float64(pi * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (((((exp(((z1 / z0) * -0.3333333333333333)) * 0.125) + (0.125 * exp(-(z1 / z0)))) / (z0 * pi)) / z1) <= -Inf)
		tmp = ((0.25 * z0) * 1.0) / (((pi * z0) * z1) * z0);
	else
		tmp = (0.25 / z0) / (pi * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[(N[(N[(N[Exp[N[(N[(z1 / z0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[(0.125 * N[Exp[(-N[(z1 / z0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], (-Infinity)], N[(N[(N[(0.25 * z0), $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / z0), $MachinePrecision] / N[(Pi * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1) < -inf.0

   1. Initial program 99.7%

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

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \pi} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \pi}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\pi}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\left(\frac{11111111111111108888888888888889}{1600000000000000000000000000000000} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{z1}{z0 \cdot \mathsf{PI}\left(\right)}\right) - \frac{13333333333333333}{80000000000000000} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{z0} - \frac{1}{4} \cdot \frac{1}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}} \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444443 \cdot \frac{z1}{z0 \cdot \pi} + 0.0625 \cdot \frac{z1}{z0 \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{z0} - 0.25 \cdot \frac{1}{z1 \cdot \pi}}{z0}} \]

   6. Applied rewrites 56.9%

      \[\leadsto \frac{\left(0.25 \cdot z0 - \left(\pi \cdot z1\right) \cdot \left(\frac{0.16666666666666666}{\pi} - 0.06944444444444445 \cdot \frac{z1}{\pi \cdot z0}\right)\right) \cdot 1}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right) \cdot z0}} \]

   7. Taylor expanded in z1 around 0

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

      1. lower-*.f64 50.6%

         \[\leadsto \frac{\left(0.25 \cdot z0\right) \cdot 1}{\left(\left(\pi \cdot \color{blue}{z0}\right) \cdot z1\right) \cdot z0} \]

   9. Applied rewrites 50.6%

      \[\leadsto \frac{\left(0.25 \cdot z0\right) \cdot 1}{\left(\color{blue}{\left(\pi \cdot z0\right)} \cdot z1\right) \cdot z0} \]

   if -inf.0 < (/.f64 (/.f64 (+.f64 (*.f64 (exp.f64 (*.f64 (/.f64 z1 z0) #s(literal -3333333333333333/10000000000000000 binary64))) #s(literal 1/8 binary64)) (*.f64 #s(literal 1/8 binary64) (exp.f64 (neg.f64 (/.f64 z1 z0))))) (*.f64 z0 (PI.f64))) z1)

   1. Initial program 99.7%

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

   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 57.1%

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 57.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 57.2%

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

   6. Applied rewrites 57.2%

      \[\leadsto \frac{\frac{0.25}{z0}}{\color{blue}{\pi \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.1% accurate, 7.9× speedup

Math

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

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(Float64(Float64(0.25 / z0) / z1) / pi)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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 57.1%

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

4. Applied rewrites 57.1%

   \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
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. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 57.1%

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

6. Applied rewrites 57.1%

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

Alternative 15: 57.1% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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 57.1%

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

4. Applied rewrites 57.1%

   \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
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. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 57.1%

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

6. Applied rewrites 57.1%

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

Alternative 16: 57.1% accurate, 10.0× speedup

Math

\[\frac{1}{\left(\pi \cdot z0\right) \cdot z1} \cdot 0.25 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* (/ 1.0 (* (* PI z0) z1)) 0.25))

C

double code(double z1, double z0) {
	return (1.0 / ((((double) M_PI) * z0) * z1)) * 0.25;
}

Java

public static double code(double z1, double z0) {
	return (1.0 / ((Math.PI * z0) * z1)) * 0.25;
}

Python

def code(z1, z0):
	return (1.0 / ((math.pi * z0) * z1)) * 0.25

Julia

function code(z1, z0)
	return Float64(Float64(1.0 / Float64(Float64(pi * z0) * z1)) * 0.25)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (1.0 / ((pi * z0) * z1)) * 0.25;
end

Wolfram

code[z1_, z0_] := N[(N[(1.0 / N[(N[(Pi * z0), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

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 57.1%

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

4. Applied rewrites 57.1%

   \[\leadsto \color{blue}{\frac{0.25}{z0 \cdot \left(z1 \cdot \pi\right)}} \]
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. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 57.0%

       \[\leadsto \frac{1}{\left(\pi \cdot z0\right) \cdot z1} \cdot 0.25 \]

6. Applied rewrites 57.0%

   \[\leadsto \frac{1}{\left(\pi \cdot z0\right) \cdot z1} \cdot \color{blue}{0.25} \]
7. Add Preprocessing

Alternative 17: 57.0% accurate, 12.3× 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 99.7%

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

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 57.1%

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

4. Applied rewrites 57.1%

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

Reproduce

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