(/ 1/8 (* (exp (/ z1 z0)) (* z0 PI)))

Percentage Accurate: 99.6% → 99.6%

Time: 2.6s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. z1 = 2.51297174034425e+107
  2. z0 = -4.1004137753342967e-274

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\frac{\frac{0.125}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ (* (/ 0.125 (* PI (exp (/ z1 z0)))) 3.0) (* z0 3.0)))

C

double code(double z1, double z0) {
	return ((0.125 / (((double) M_PI) * exp((z1 / z0)))) * 3.0) / (z0 * 3.0);
}

Java

public static double code(double z1, double z0) {
	return ((0.125 / (Math.PI * Math.exp((z1 / z0)))) * 3.0) / (z0 * 3.0);
}

Python

def code(z1, z0):
	return ((0.125 / (math.pi * math.exp((z1 / z0)))) * 3.0) / (z0 * 3.0)

Julia

function code(z1, z0)
	return Float64(Float64(Float64(0.125 / Float64(pi * exp(Float64(z1 / z0)))) * 3.0) / Float64(z0 * 3.0))
end

MATLAB

function tmp = code(z1, z0)
	tmp = ((0.125 / (pi * exp((z1 / z0)))) * 3.0) / (z0 * 3.0);
end

Wolfram

code[z1_, z0_] := N[(N[(N[(0.125 / N[(Pi * N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] / N[(z0 * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. mult-flip N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi}}{z0} \cdot \color{blue}{\frac{3}{3}} \]

   14. frac-times N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

   15. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{0.125}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3}} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot e^{\frac{z1}{z0}}} \cdot 0.375 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* (/ 0.3333333333333333 (* (* z0 PI) (exp (/ z1 z0)))) 0.375))

C

double code(double z1, double z0) {
	return (0.3333333333333333 / ((z0 * ((double) M_PI)) * exp((z1 / z0)))) * 0.375;
}

Java

public static double code(double z1, double z0) {
	return (0.3333333333333333 / ((z0 * Math.PI) * Math.exp((z1 / z0)))) * 0.375;
}

Python

def code(z1, z0):
	return (0.3333333333333333 / ((z0 * math.pi) * math.exp((z1 / z0)))) * 0.375

Julia

function code(z1, z0)
	return Float64(Float64(0.3333333333333333 / Float64(Float64(z0 * pi) * exp(Float64(z1 / z0)))) * 0.375)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.3333333333333333 / ((z0 * pi) * exp((z1 / z0)))) * 0.375;
end

Wolfram

code[z1_, z0_] := N[(N[(0.3333333333333333 / N[(N[(z0 * Pi), $MachinePrecision] * N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. mult-flip N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi}}{z0} \cdot \color{blue}{\frac{3}{3}} \]

   14. frac-times N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

   15. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

3. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}}{z0 \cdot 3} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot \frac{3}{z0 \cdot 3}} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \frac{3}{\color{blue}{z0 \cdot 3}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \color{blue}{\frac{\frac{3}{z0}}{3}} \]

   9. frac-times N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi} \cdot \frac{3}{z0}}{e^{\frac{z1}{z0}} \cdot 3}} \]

   10. times-frac N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot 3}{\pi \cdot z0}}}{e^{\frac{z1}{z0}} \cdot 3} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{3}{8}}}{\pi \cdot z0}}{e^{\frac{z1}{z0}} \cdot 3} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

   14. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(z0 \cdot \pi\right) \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

   18. associate-*l* N/A

       \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot e^{\frac{z1}{z0}}\right) \cdot 3}} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot e^{\frac{z1}{z0}}} \cdot 0.375} \]
6. Add Preprocessing

Alternative 3: 86.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{z1 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.16 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.375}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(1.5 \cdot t\_0 + 3 \cdot \pi\right)}\\ \mathbf{elif}\;z0 \leq 2.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.125}{z0 \cdot \left(z0 \cdot \frac{\left(z1 + z0\right) \cdot \pi}{z0 \cdot z0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot t\_0\right)} \cdot 0.375\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ (* z1 PI) z0)))
  (if (<= z0 -1.16e-116)
    (/ 0.375 (+ (* 3.0 (* z0 PI)) (* z1 (+ (* 1.5 t_0) (* 3.0 PI)))))
    (if (<= z0 2.1e-156)
      (/ 0.125 (* z0 (* z0 (/ (* (+ z1 z0) PI) (* z0 z0)))))
      (*
       (/ 0.3333333333333333 (+ (* z0 PI) (* z1 (+ PI (* 0.5 t_0)))))
       0.375)))))

C

double code(double z1, double z0) {
	double t_0 = (z1 * ((double) M_PI)) / z0;
	double tmp;
	if (z0 <= -1.16e-116) {
		tmp = 0.375 / ((3.0 * (z0 * ((double) M_PI))) + (z1 * ((1.5 * t_0) + (3.0 * ((double) M_PI)))));
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * ((double) M_PI)) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / ((z0 * ((double) M_PI)) + (z1 * (((double) M_PI) + (0.5 * t_0))))) * 0.375;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (z1 * Math.PI) / z0;
	double tmp;
	if (z0 <= -1.16e-116) {
		tmp = 0.375 / ((3.0 * (z0 * Math.PI)) + (z1 * ((1.5 * t_0) + (3.0 * Math.PI))));
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * Math.PI) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / ((z0 * Math.PI) + (z1 * (Math.PI + (0.5 * t_0))))) * 0.375;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = (z1 * math.pi) / z0
	tmp = 0
	if z0 <= -1.16e-116:
		tmp = 0.375 / ((3.0 * (z0 * math.pi)) + (z1 * ((1.5 * t_0) + (3.0 * math.pi))))
	elif z0 <= 2.1e-156:
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * math.pi) / (z0 * z0))))
	else:
		tmp = (0.3333333333333333 / ((z0 * math.pi) + (z1 * (math.pi + (0.5 * t_0))))) * 0.375
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(z1 * pi) / z0)
	tmp = 0.0
	if (z0 <= -1.16e-116)
		tmp = Float64(0.375 / Float64(Float64(3.0 * Float64(z0 * pi)) + Float64(z1 * Float64(Float64(1.5 * t_0) + Float64(3.0 * pi)))));
	elseif (z0 <= 2.1e-156)
		tmp = Float64(0.125 / Float64(z0 * Float64(z0 * Float64(Float64(Float64(z1 + z0) * pi) / Float64(z0 * z0)))));
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(z0 * pi) + Float64(z1 * Float64(pi + Float64(0.5 * t_0))))) * 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (z1 * pi) / z0;
	tmp = 0.0;
	if (z0 <= -1.16e-116)
		tmp = 0.375 / ((3.0 * (z0 * pi)) + (z1 * ((1.5 * t_0) + (3.0 * pi))));
	elseif (z0 <= 2.1e-156)
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * pi) / (z0 * z0))));
	else
		tmp = (0.3333333333333333 / ((z0 * pi) + (z1 * (pi + (0.5 * t_0))))) * 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(z1 * Pi), $MachinePrecision] / z0), $MachinePrecision]}, If[LessEqual[z0, -1.16e-116], N[(0.375 / N[(N[(3.0 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] + N[(z1 * N[(N[(1.5 * t$95$0), $MachinePrecision] + N[(3.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.1e-156], N[(0.125 / N[(z0 * N[(z0 * N[(N[(N[(z1 + z0), $MachinePrecision] * Pi), $MachinePrecision] / N[(z0 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(z0 * Pi), $MachinePrecision] + N[(z1 * N[(Pi + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.16e-116

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{8}}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)} \cdot \color{blue}{\frac{3}{3}} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot 3}{\left(e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)\right) \cdot 3}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot 3}{\left(e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)\right) \cdot 3}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{3}{8}}}{\left(e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)\right) \cdot 3} \]

      12. lower-*.f64 99.4%

          \[\leadsto \frac{0.375}{\color{blue}{\left(e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)\right) \cdot 3}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{0.375}{\left(\left(\pi \cdot z0\right) \cdot e^{\frac{z1}{z0}}\right) \cdot 3}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.375}{\color{blue}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \pi}{z0} + 3 \cdot \pi\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z1} \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)} \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \color{blue}{\left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + \color{blue}{3 \cdot \mathsf{PI}\left(\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + \color{blue}{3} \cdot \mathsf{PI}\left(\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \pi}{z0} + 3 \cdot \mathsf{PI}\left(\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(\frac{3}{2} \cdot \frac{z1 \cdot \pi}{z0} + 3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

      12. lower-PI.f64 82.7%

          \[\leadsto \frac{0.375}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(1.5 \cdot \frac{z1 \cdot \pi}{z0} + 3 \cdot \pi\right)} \]

   6. Applied rewrites 82.7%

      \[\leadsto \frac{0.375}{\color{blue}{3 \cdot \left(z0 \cdot \pi\right) + z1 \cdot \left(1.5 \cdot \frac{z1 \cdot \pi}{z0} + 3 \cdot \pi\right)}} \]

   if -1.16e-116 < z0 < 2.1000000000000001e-156

   1. Initial program 99.6%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      6. lower-PI.f64 74.3%

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

   4. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{\pi \cdot z0 + z1 \cdot \pi}{\color{blue}{z0}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 48.2%

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

   6. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 52.6%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.6%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   if 2.1000000000000001e-156 < z0

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi}}{z0} \cdot \color{blue}{\frac{3}{3}} \]

      14. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}}{z0 \cdot 3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot \frac{3}{z0 \cdot 3}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \frac{3}{\color{blue}{z0 \cdot 3}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \color{blue}{\frac{\frac{3}{z0}}{3}} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi} \cdot \frac{3}{z0}}{e^{\frac{z1}{z0}} \cdot 3}} \]

      10. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot 3}{\pi \cdot z0}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{3}{8}}}{\pi \cdot z0}}{e^{\frac{z1}{z0}} \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      14. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(z0 \cdot \pi\right) \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot e^{\frac{z1}{z0}}\right) \cdot 3}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot e^{\frac{z1}{z0}}} \cdot 0.375} \]

   6. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi}} \cdot 0.375 \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \frac{3}{8} \]

      2. lower-PI.f64 64.7%

         \[\leadsto \frac{0.3333333333333333}{z0 \cdot \pi} \cdot 0.375 \]

   8. Applied rewrites 64.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi}} \cdot 0.375 \]

   9. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}} \cdot 0.375 \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \cdot \frac{3}{8} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       3. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \cdot \frac{3}{8} \]

       5. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)} \cdot \frac{3}{8} \]

       6. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)} \cdot \frac{3}{8} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \cdot \frac{3}{8} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       10. lower-PI.f64 82.8%

           \[\leadsto \frac{0.3333333333333333}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)} \cdot 0.375 \]

   11. Applied rewrites 82.8%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}} \cdot 0.375 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -1.16 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.375}{\left(\left(0.5 \cdot \left(\pi \cdot \frac{z1}{z0}\right) + \pi\right) \cdot z1 + z0 \cdot \pi\right) \cdot 3}\\ \mathbf{elif}\;z0 \leq 2.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.125}{z0 \cdot \left(z0 \cdot \frac{\left(z1 + z0\right) \cdot \pi}{z0 \cdot z0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)} \cdot 0.375\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= z0 -1.16e-116)
  (/
   0.375
   (* (+ (* (+ (* 0.5 (* PI (/ z1 z0))) PI) z1) (* z0 PI)) 3.0))
  (if (<= z0 2.1e-156)
    (/ 0.125 (* z0 (* z0 (/ (* (+ z1 z0) PI) (* z0 z0)))))
    (*
     (/
      0.3333333333333333
      (+ (* z0 PI) (* z1 (+ PI (* 0.5 (/ (* z1 PI) z0))))))
     0.375))))

C

double code(double z1, double z0) {
	double tmp;
	if (z0 <= -1.16e-116) {
		tmp = 0.375 / (((((0.5 * (((double) M_PI) * (z1 / z0))) + ((double) M_PI)) * z1) + (z0 * ((double) M_PI))) * 3.0);
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * ((double) M_PI)) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / ((z0 * ((double) M_PI)) + (z1 * (((double) M_PI) + (0.5 * ((z1 * ((double) M_PI)) / z0)))))) * 0.375;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (z0 <= -1.16e-116) {
		tmp = 0.375 / (((((0.5 * (Math.PI * (z1 / z0))) + Math.PI) * z1) + (z0 * Math.PI)) * 3.0);
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * Math.PI) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / ((z0 * Math.PI) + (z1 * (Math.PI + (0.5 * ((z1 * Math.PI) / z0)))))) * 0.375;
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if z0 <= -1.16e-116:
		tmp = 0.375 / (((((0.5 * (math.pi * (z1 / z0))) + math.pi) * z1) + (z0 * math.pi)) * 3.0)
	elif z0 <= 2.1e-156:
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * math.pi) / (z0 * z0))))
	else:
		tmp = (0.3333333333333333 / ((z0 * math.pi) + (z1 * (math.pi + (0.5 * ((z1 * math.pi) / z0)))))) * 0.375
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (z0 <= -1.16e-116)
		tmp = Float64(0.375 / Float64(Float64(Float64(Float64(Float64(0.5 * Float64(pi * Float64(z1 / z0))) + pi) * z1) + Float64(z0 * pi)) * 3.0));
	elseif (z0 <= 2.1e-156)
		tmp = Float64(0.125 / Float64(z0 * Float64(z0 * Float64(Float64(Float64(z1 + z0) * pi) / Float64(z0 * z0)))));
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(z0 * pi) + Float64(z1 * Float64(pi + Float64(0.5 * Float64(Float64(z1 * pi) / z0)))))) * 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (z0 <= -1.16e-116)
		tmp = 0.375 / (((((0.5 * (pi * (z1 / z0))) + pi) * z1) + (z0 * pi)) * 3.0);
	elseif (z0 <= 2.1e-156)
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * pi) / (z0 * z0))));
	else
		tmp = (0.3333333333333333 / ((z0 * pi) + (z1 * (pi + (0.5 * ((z1 * pi) / z0)))))) * 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[z0, -1.16e-116], N[(0.375 / N[(N[(N[(N[(N[(0.5 * N[(Pi * N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision] * z1), $MachinePrecision] + N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.1e-156], N[(0.125 / N[(z0 * N[(z0 * N[(N[(N[(z1 + z0), $MachinePrecision] * Pi), $MachinePrecision] / N[(z0 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(z0 * Pi), $MachinePrecision] + N[(z1 * N[(Pi + N[(0.5 * N[(N[(z1 * Pi), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.16e-116

   1. Initial program 99.6%

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

   2. Taylor expanded in z1 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 82.9%

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

   4. Applied rewrites 82.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{1}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)} \cdot \color{blue}{\frac{\frac{3}{8}}{3}} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{3}{8}}{\left(z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{3}{8}}}{\left(z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)\right) \cdot 3} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)\right) \cdot 3}} \]

      8. lower-*.f64 82.7%

         \[\leadsto \frac{0.375}{\color{blue}{\left(z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)\right) \cdot 3}} \]

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{0.375}{\left(\left(0.5 \cdot \left(\pi \cdot \frac{z1}{z0}\right) + \pi\right) \cdot z1 + z0 \cdot \pi\right) \cdot 3}} \]

   if -1.16e-116 < z0 < 2.1000000000000001e-156

   1. Initial program 99.6%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      6. lower-PI.f64 74.3%

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

   4. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{\pi \cdot z0 + z1 \cdot \pi}{\color{blue}{z0}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 48.2%

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

   6. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 52.6%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.6%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   if 2.1000000000000001e-156 < z0

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi}}{z0} \cdot \color{blue}{\frac{3}{3}} \]

      14. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}}{z0 \cdot 3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot \frac{3}{z0 \cdot 3}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \frac{3}{\color{blue}{z0 \cdot 3}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \color{blue}{\frac{\frac{3}{z0}}{3}} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi} \cdot \frac{3}{z0}}{e^{\frac{z1}{z0}} \cdot 3}} \]

      10. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot 3}{\pi \cdot z0}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{3}{8}}}{\pi \cdot z0}}{e^{\frac{z1}{z0}} \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      14. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(z0 \cdot \pi\right) \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot e^{\frac{z1}{z0}}\right) \cdot 3}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot e^{\frac{z1}{z0}}} \cdot 0.375} \]

   6. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi}} \cdot 0.375 \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \frac{3}{8} \]

      2. lower-PI.f64 64.7%

         \[\leadsto \frac{0.3333333333333333}{z0 \cdot \pi} \cdot 0.375 \]

   8. Applied rewrites 64.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi}} \cdot 0.375 \]

   9. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}} \cdot 0.375 \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \cdot \frac{3}{8} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       3. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \cdot \frac{3}{8} \]

       5. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)} \cdot \frac{3}{8} \]

       6. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)} \cdot \frac{3}{8} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \cdot \frac{3}{8} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       10. lower-PI.f64 82.8%

           \[\leadsto \frac{0.3333333333333333}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)} \cdot 0.375 \]

   11. Applied rewrites 82.8%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}} \cdot 0.375 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)\\ \mathbf{if}\;z0 \leq -1.7 \cdot 10^{-125}:\\ \;\;\;\;\frac{0.125}{t\_0}\\ \mathbf{elif}\;z0 \leq 2.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.125}{z0 \cdot \left(z0 \cdot \frac{\left(z1 + z0\right) \cdot \pi}{z0 \cdot z0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{t\_0} \cdot 0.375\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (+ (* z0 PI) (* z1 (+ PI (* 0.5 (/ (* z1 PI) z0)))))))
  (if (<= z0 -1.7e-125)
    (/ 0.125 t_0)
    (if (<= z0 2.1e-156)
      (/ 0.125 (* z0 (* z0 (/ (* (+ z1 z0) PI) (* z0 z0)))))
      (* (/ 0.3333333333333333 t_0) 0.375)))))

C

double code(double z1, double z0) {
	double t_0 = (z0 * ((double) M_PI)) + (z1 * (((double) M_PI) + (0.5 * ((z1 * ((double) M_PI)) / z0))));
	double tmp;
	if (z0 <= -1.7e-125) {
		tmp = 0.125 / t_0;
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * ((double) M_PI)) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / t_0) * 0.375;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (z0 * Math.PI) + (z1 * (Math.PI + (0.5 * ((z1 * Math.PI) / z0))));
	double tmp;
	if (z0 <= -1.7e-125) {
		tmp = 0.125 / t_0;
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * Math.PI) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / t_0) * 0.375;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = (z0 * math.pi) + (z1 * (math.pi + (0.5 * ((z1 * math.pi) / z0))))
	tmp = 0
	if z0 <= -1.7e-125:
		tmp = 0.125 / t_0
	elif z0 <= 2.1e-156:
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * math.pi) / (z0 * z0))))
	else:
		tmp = (0.3333333333333333 / t_0) * 0.375
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(z0 * pi) + Float64(z1 * Float64(pi + Float64(0.5 * Float64(Float64(z1 * pi) / z0)))))
	tmp = 0.0
	if (z0 <= -1.7e-125)
		tmp = Float64(0.125 / t_0);
	elseif (z0 <= 2.1e-156)
		tmp = Float64(0.125 / Float64(z0 * Float64(z0 * Float64(Float64(Float64(z1 + z0) * pi) / Float64(z0 * z0)))));
	else
		tmp = Float64(Float64(0.3333333333333333 / t_0) * 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (z0 * pi) + (z1 * (pi + (0.5 * ((z1 * pi) / z0))));
	tmp = 0.0;
	if (z0 <= -1.7e-125)
		tmp = 0.125 / t_0;
	elseif (z0 <= 2.1e-156)
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * pi) / (z0 * z0))));
	else
		tmp = (0.3333333333333333 / t_0) * 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(z0 * Pi), $MachinePrecision] + N[(z1 * N[(Pi + N[(0.5 * N[(N[(z1 * Pi), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.7e-125], N[(0.125 / t$95$0), $MachinePrecision], If[LessEqual[z0, 2.1e-156], N[(0.125 / N[(z0 * N[(z0 * N[(N[(N[(z1 + z0), $MachinePrecision] * Pi), $MachinePrecision] / N[(z0 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / t$95$0), $MachinePrecision] * 0.375), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.6999999999999999e-125

   1. Initial program 99.6%

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

   2. Taylor expanded in z1 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 82.9%

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

   4. Applied rewrites 82.9%

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

   if -1.6999999999999999e-125 < z0 < 2.1000000000000001e-156

   1. Initial program 99.6%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      6. lower-PI.f64 74.3%

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

   4. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{\pi \cdot z0 + z1 \cdot \pi}{\color{blue}{z0}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 48.2%

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

   6. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 52.6%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.6%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   if 2.1000000000000001e-156 < z0

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi}}{z0} \cdot \color{blue}{\frac{3}{3}} \]

      14. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}}{z0 \cdot 3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot \frac{3}{z0 \cdot 3}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \frac{3}{\color{blue}{z0 \cdot 3}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \color{blue}{\frac{\frac{3}{z0}}{3}} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi} \cdot \frac{3}{z0}}{e^{\frac{z1}{z0}} \cdot 3}} \]

      10. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot 3}{\pi \cdot z0}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{3}{8}}}{\pi \cdot z0}}{e^{\frac{z1}{z0}} \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      14. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(z0 \cdot \pi\right) \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot e^{\frac{z1}{z0}}\right) \cdot 3}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot e^{\frac{z1}{z0}}} \cdot 0.375} \]

   6. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi}} \cdot 0.375 \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \frac{3}{8} \]

      2. lower-PI.f64 64.7%

         \[\leadsto \frac{0.3333333333333333}{z0 \cdot \pi} \cdot 0.375 \]

   8. Applied rewrites 64.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi}} \cdot 0.375 \]

   9. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \pi}{z0}\right)}} \cdot 0.375 \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \cdot \frac{3}{8} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       3. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \cdot \frac{3}{8} \]

       5. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)} \cdot \frac{3}{8} \]

       6. lower-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \color{blue}{\frac{1}{2}} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \color{blue}{\frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}}\right)} \cdot \frac{3}{8} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \cdot \frac{3}{8} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{z0 \cdot \pi + z1 \cdot \left(\pi + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \cdot \frac{3}{8} \]

       10. lower-PI.f64 82.8%

           \[\leadsto \frac{0.3333333333333333}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)} \cdot 0.375 \]

   11. Applied rewrites 82.8%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}} \cdot 0.375 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.125}{z0 \cdot \pi + z1 \cdot \left(\pi + 0.5 \cdot \frac{z1 \cdot \pi}{z0}\right)}\\ \mathbf{if}\;z0 \leq -1.7 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 2.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.125}{z0 \cdot \left(z0 \cdot \frac{\left(z1 + z0\right) \cdot \pi}{z0 \cdot z0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (/
         0.125
         (+ (* z0 PI) (* z1 (+ PI (* 0.5 (/ (* z1 PI) z0))))))))
  (if (<= z0 -1.7e-125)
    t_0
    (if (<= z0 2.1e-156)
      (/ 0.125 (* z0 (* z0 (/ (* (+ z1 z0) PI) (* z0 z0)))))
      t_0))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 / ((z0 * ((double) M_PI)) + (z1 * (((double) M_PI) + (0.5 * ((z1 * ((double) M_PI)) / z0)))));
	double tmp;
	if (z0 <= -1.7e-125) {
		tmp = t_0;
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * ((double) M_PI)) / (z0 * z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 / ((z0 * Math.PI) + (z1 * (Math.PI + (0.5 * ((z1 * Math.PI) / z0)))));
	double tmp;
	if (z0 <= -1.7e-125) {
		tmp = t_0;
	} else if (z0 <= 2.1e-156) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * Math.PI) / (z0 * z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 / ((z0 * math.pi) + (z1 * (math.pi + (0.5 * ((z1 * math.pi) / z0)))))
	tmp = 0
	if z0 <= -1.7e-125:
		tmp = t_0
	elif z0 <= 2.1e-156:
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * math.pi) / (z0 * z0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 / Float64(Float64(z0 * pi) + Float64(z1 * Float64(pi + Float64(0.5 * Float64(Float64(z1 * pi) / z0))))))
	tmp = 0.0
	if (z0 <= -1.7e-125)
		tmp = t_0;
	elseif (z0 <= 2.1e-156)
		tmp = Float64(0.125 / Float64(z0 * Float64(z0 * Float64(Float64(Float64(z1 + z0) * pi) / Float64(z0 * z0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 / ((z0 * pi) + (z1 * (pi + (0.5 * ((z1 * pi) / z0)))));
	tmp = 0.0;
	if (z0 <= -1.7e-125)
		tmp = t_0;
	elseif (z0 <= 2.1e-156)
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * pi) / (z0 * z0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 / N[(N[(z0 * Pi), $MachinePrecision] + N[(z1 * N[(Pi + N[(0.5 * N[(N[(z1 * Pi), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.7e-125], t$95$0, If[LessEqual[z0, 2.1e-156], N[(0.125 / N[(z0 * N[(z0 * N[(N[(N[(z1 + z0), $MachinePrecision] * Pi), $MachinePrecision] / N[(z0 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1.6999999999999999e-125 or 2.1000000000000001e-156 < z0

   1. Initial program 99.6%

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

   2. Taylor expanded in z1 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 82.9%

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

   4. Applied rewrites 82.9%

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

   if -1.6999999999999999e-125 < z0 < 2.1000000000000001e-156

   1. Initial program 99.6%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      6. lower-PI.f64 74.3%

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

   4. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{\pi \cdot z0 + z1 \cdot \pi}{\color{blue}{z0}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 48.2%

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

   6. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 52.6%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.6%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 52.6%

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

   8. Applied rewrites 52.6%

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

Alternative 7: 86.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z1 around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1 \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \mathsf{PI}\left(\right) + \color{blue}{z1} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 82.9%

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

4. Applied rewrites 82.9%

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

   1. lift-+.f64 N/A

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

   2. sum-to-mult N/A

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

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

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

6. Applied rewrites 86.7%

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

Alternative 8: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.125}{e^{\frac{z1}{z0}} \cdot \left(z0 \cdot \pi\right)}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-309}:\\ \;\;\;\;\frac{0.375}{\left(z0 + z1\right) \cdot \pi} \cdot 0.3333333333333333\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.125}{z0 \cdot \left(z0 \cdot \frac{\left(z1 + z0\right) \cdot \pi}{z0 \cdot z0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot \left(1 + \frac{z1}{z0}\right)} \cdot 0.375\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ 0.125 (* (exp (/ z1 z0)) (* z0 PI)))))
  (if (<= t_0 -4e-309)
    (* (/ 0.375 (* (+ z0 z1) PI)) 0.3333333333333333)
    (if (<= t_0 0.0)
      (/ 0.125 (* z0 (* z0 (/ (* (+ z1 z0) PI) (* z0 z0)))))
      (*
       (/ 0.3333333333333333 (* (* z0 PI) (+ 1.0 (/ z1 z0))))
       0.375)))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 / (exp((z1 / z0)) * (z0 * ((double) M_PI)));
	double tmp;
	if (t_0 <= -4e-309) {
		tmp = (0.375 / ((z0 + z1) * ((double) M_PI))) * 0.3333333333333333;
	} else if (t_0 <= 0.0) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * ((double) M_PI)) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / ((z0 * ((double) M_PI)) * (1.0 + (z1 / z0)))) * 0.375;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 / (Math.exp((z1 / z0)) * (z0 * Math.PI));
	double tmp;
	if (t_0 <= -4e-309) {
		tmp = (0.375 / ((z0 + z1) * Math.PI)) * 0.3333333333333333;
	} else if (t_0 <= 0.0) {
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * Math.PI) / (z0 * z0))));
	} else {
		tmp = (0.3333333333333333 / ((z0 * Math.PI) * (1.0 + (z1 / z0)))) * 0.375;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 / (math.exp((z1 / z0)) * (z0 * math.pi))
	tmp = 0
	if t_0 <= -4e-309:
		tmp = (0.375 / ((z0 + z1) * math.pi)) * 0.3333333333333333
	elif t_0 <= 0.0:
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * math.pi) / (z0 * z0))))
	else:
		tmp = (0.3333333333333333 / ((z0 * math.pi) * (1.0 + (z1 / z0)))) * 0.375
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 / Float64(exp(Float64(z1 / z0)) * Float64(z0 * pi)))
	tmp = 0.0
	if (t_0 <= -4e-309)
		tmp = Float64(Float64(0.375 / Float64(Float64(z0 + z1) * pi)) * 0.3333333333333333);
	elseif (t_0 <= 0.0)
		tmp = Float64(0.125 / Float64(z0 * Float64(z0 * Float64(Float64(Float64(z1 + z0) * pi) / Float64(z0 * z0)))));
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(z0 * pi) * Float64(1.0 + Float64(z1 / z0)))) * 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 / (exp((z1 / z0)) * (z0 * pi));
	tmp = 0.0;
	if (t_0 <= -4e-309)
		tmp = (0.375 / ((z0 + z1) * pi)) * 0.3333333333333333;
	elseif (t_0 <= 0.0)
		tmp = 0.125 / (z0 * (z0 * (((z1 + z0) * pi) / (z0 * z0))));
	else
		tmp = (0.3333333333333333 / ((z0 * pi) * (1.0 + (z1 / z0)))) * 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 / N[(N[Exp[N[(z1 / z0), $MachinePrecision]], $MachinePrecision] * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-309], N[(N[(0.375 / N[(N[(z0 + z1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.125 / N[(z0 * N[(z0 * N[(N[(N[(z1 + z0), $MachinePrecision] * Pi), $MachinePrecision] / N[(z0 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(z0 * Pi), $MachinePrecision] * N[(1.0 + N[(z1 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64)))) < -3.9999999999999977e-309

   1. Initial program 99.6%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      6. lower-PI.f64 74.3%

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

   4. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.3%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.3%

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 74.3%

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

   6. Applied rewrites 74.3%

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

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{0.375}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)}}{3}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)} \cdot \frac{1}{3}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)} \cdot \color{blue}{\frac{1}{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)} \cdot \frac{1}{3}} \]

   10. Applied rewrites 66.2%

       \[\leadsto \color{blue}{\frac{0.375}{\left(z0 + z1\right) \cdot \pi} \cdot 0.3333333333333333} \]

   if -3.9999999999999977e-309 < (/.f64 #s(literal 1/8 binary64) (*.f64 (exp.f64 (/.f64 z1 z0)) (*.f64 z0 (PI.f64)))) < 0.0

   1. Initial program 99.6%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

      6. lower-PI.f64 74.3%

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

   4. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{\pi \cdot z0 + z1 \cdot \pi}{\color{blue}{z0}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \frac{z0 \cdot \pi + z1 \cdot \pi}{z0}} \]

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 48.2%

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

   6. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 52.6%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 52.6%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 52.6%

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

   8. Applied rewrites 52.6%

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

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

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi}}{z0} \cdot \color{blue}{\frac{3}{3}} \]

      14. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{e^{\frac{z1}{z0}}}}{\pi} \cdot 3}{z0 \cdot 3}} \]

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}{z0 \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot 3}}{z0 \cdot 3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}} \cdot \frac{3}{z0 \cdot 3}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\pi \cdot e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}}} \cdot \frac{3}{z0 \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \frac{3}{\color{blue}{z0 \cdot 3}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{8}}{\pi}}{e^{\frac{z1}{z0}}} \cdot \color{blue}{\frac{\frac{3}{z0}}{3}} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi} \cdot \frac{3}{z0}}{e^{\frac{z1}{z0}} \cdot 3}} \]

      10. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot 3}{\pi \cdot z0}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{3}{8}}}{\pi \cdot z0}}{e^{\frac{z1}{z0}} \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{3}{8}}{\color{blue}{z0 \cdot \pi}}}{e^{\frac{z1}{z0}} \cdot 3} \]

      14. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(z0 \cdot \pi\right) \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\pi \cdot z0\right)} \cdot \left(e^{\frac{z1}{z0}} \cdot 3\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot e^{\frac{z1}{z0}}\right) \cdot 3}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot e^{\frac{z1}{z0}}} \cdot 0.375} \]

   6. Taylor expanded in z1 around 0

      \[\leadsto \frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot \color{blue}{\left(1 + \frac{z1}{z0}\right)}} \cdot 0.375 \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{\left(z0 \cdot \pi\right) \cdot \left(1 + \color{blue}{\frac{z1}{z0}}\right)} \cdot \frac{3}{8} \]

      2. lower-/.f64 74.2%

         \[\leadsto \frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot \left(1 + \frac{z1}{\color{blue}{z0}}\right)} \cdot 0.375 \]

   8. Applied rewrites 74.2%

      \[\leadsto \frac{0.3333333333333333}{\left(z0 \cdot \pi\right) \cdot \color{blue}{\left(1 + \frac{z1}{z0}\right)}} \cdot 0.375 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z0 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

   6. lower-PI.f64 74.3%

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

4. Applied rewrites 74.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 74.3%

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

6. Applied rewrites 74.3%

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

Alternative 10: 66.3% accurate, 5.3× speedup

Math

\[\frac{0.375}{\left(z0 + z1\right) \cdot \pi} \cdot 0.3333333333333333 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* (/ 0.375 (* (+ z0 z1) PI)) 0.3333333333333333))

C

double code(double z1, double z0) {
	return (0.375 / ((z0 + z1) * ((double) M_PI))) * 0.3333333333333333;
}

Java

public static double code(double z1, double z0) {
	return (0.375 / ((z0 + z1) * Math.PI)) * 0.3333333333333333;
}

Python

def code(z1, z0):
	return (0.375 / ((z0 + z1) * math.pi)) * 0.3333333333333333

Julia

function code(z1, z0)
	return Float64(Float64(0.375 / Float64(Float64(z0 + z1) * pi)) * 0.3333333333333333)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.375 / ((z0 + z1) * pi)) * 0.3333333333333333;
end

Wolfram

code[z1_, z0_] := N[(N[(0.375 / N[(N[(z0 + z1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z0 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

   6. lower-PI.f64 74.3%

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

4. Applied rewrites 74.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 74.3%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 74.3%

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 74.3%

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

6. Applied rewrites 74.3%

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

8. Applied rewrites 66.1%

   \[\leadsto \color{blue}{\frac{0.375}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)}}{3}} \]

   4. mult-flip N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)} \cdot \frac{1}{3}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)} \cdot \color{blue}{\frac{1}{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{8}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)} \cdot \frac{1}{3}} \]

10. Applied rewrites 66.2%

    \[\leadsto \color{blue}{\frac{0.375}{\left(z0 + z1\right) \cdot \pi} \cdot 0.3333333333333333} \]
11. Add Preprocessing

Alternative 11: 66.2% accurate, 5.3× speedup

Math

\[\frac{0.3333333333333333}{\left(z0 + z1\right) \cdot \pi} \cdot 0.375 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* (/ 0.3333333333333333 (* (+ z0 z1) PI)) 0.375))

C

double code(double z1, double z0) {
	return (0.3333333333333333 / ((z0 + z1) * ((double) M_PI))) * 0.375;
}

Java

public static double code(double z1, double z0) {
	return (0.3333333333333333 / ((z0 + z1) * Math.PI)) * 0.375;
}

Python

def code(z1, z0):
	return (0.3333333333333333 / ((z0 + z1) * math.pi)) * 0.375

Julia

function code(z1, z0)
	return Float64(Float64(0.3333333333333333 / Float64(Float64(z0 + z1) * pi)) * 0.375)
end

MATLAB

function tmp = code(z1, z0)
	tmp = (0.3333333333333333 / ((z0 + z1) * pi)) * 0.375;
end

Wolfram

code[z1_, z0_] := N[(N[(0.3333333333333333 / N[(N[(z0 + z1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z0 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

   6. lower-PI.f64 74.3%

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

4. Applied rewrites 74.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 74.3%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 74.3%

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 74.3%

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

6. Applied rewrites 74.3%

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

8. Applied rewrites 66.1%

   \[\leadsto \color{blue}{\frac{0.375}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\frac{3}{8} \cdot \frac{1}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3} \cdot \frac{3}{8}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3} \cdot \frac{3}{8}} \]

10. Applied rewrites 66.2%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(z0 + z1\right) \cdot \pi} \cdot 0.375} \]
11. Add Preprocessing

Alternative 12: 66.2% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z0 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{\color{blue}{z1 \cdot \mathsf{PI}\left(\right)}}{z0}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \left(\pi + \frac{z1 \cdot \mathsf{PI}\left(\right)}{z0}\right)} \]

   6. lower-PI.f64 74.3%

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

4. Applied rewrites 74.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 74.3%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 74.3%

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 74.3%

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

6. Applied rewrites 74.3%

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

8. Applied rewrites 66.1%

   \[\leadsto \color{blue}{\frac{0.375}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{8}}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{3}{8}}{\color{blue}{\left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right) \cdot 3}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{3}{8}}{\color{blue}{3 \cdot \left(1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{3}{8}}{3}}{1 \cdot \left(\left(z1 + z0\right) \cdot \pi\right)}} \]

   5. metadata-eval N/A

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

   6. lower-/.f64 66.3%

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 66.3%

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

   12. lift-*.f64 N/A

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

   13. *-lft-identity 66.3%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 66.3%

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

10. Applied rewrites 66.3%

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

Alternative 13: 64.7% accurate, 7.8× speedup

Math

\[\frac{0.125}{z0 \cdot \pi} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z1 around 0

   \[\leadsto \frac{0.125}{\color{blue}{z0 \cdot \pi}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{8}}{z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

   2. lower-PI.f64 64.7%

      \[\leadsto \frac{0.125}{z0 \cdot \pi} \]

4. Applied rewrites 64.7%

   \[\leadsto \frac{0.125}{\color{blue}{z0 \cdot \pi}} \]
5. Add Preprocessing

Reproduce

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