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

Percentage Accurate: 99.5% → 99.6%

Time: 2.6s

Alternatives: 16

Speedup: 1.0×

• could not determine a ground truth

  1. z1 = -4.343299629486535e+126
  2. z0 = 5.6035429208413516e+138

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lift-exp.f64 N/A

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

   8. rec-exp N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. distribute-neg-frac N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 99.6%

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z1, z0)
	return Float64(0.125 / Float64(Float64(pi * z1) * (exp(1) ^ Float64(z0 / z1))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 0.125 / ((pi * z1) * (2.71828182845904523536 ^ (z0 / z1)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. remove-double-neg N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

   13. mult-flip N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. mult-flip N/A

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

   17. lift-/.f64 N/A

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

   18. exp-prod N/A

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

   19. lower-pow.f64 N/A

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

   20. lower-exp.f64 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-exp.f64 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f64 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 4: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.125}{\pi \cdot \frac{z1 \cdot z1 - z0 \cdot z0}{z1 - z0}}\\ \mathbf{if}\;z1 \leq -8.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{0.125 + -0.125 \cdot \frac{z0}{z1}}{z1 \cdot \pi}\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;\frac{0.125}{z1 \cdot \left(\left(\left(z0 + z1\right) \cdot \frac{\pi}{z1 \cdot z1}\right) \cdot z1\right)}\\ \mathbf{elif}\;z1 \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125}{z1}}{\pi + \frac{z0 \cdot \pi}{z1}}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ 0.125 (* PI (/ (- (* z1 z1) (* z0 z0)) (- z1 z0))))))
  (if (<= z1 -8.2e+160)
    (/ (+ 0.125 (* -0.125 (/ z0 z1))) (* z1 PI))
    (if (<= z1 -5e-39)
      t_0
      (if (<= z1 1.25e-155)
        (/ 0.125 (* z1 (* (* (+ z0 z1) (/ PI (* z1 z1))) z1)))
        (if (<= z1 2.7e+160)
          t_0
          (/ (/ 0.125 z1) (+ PI (/ (* z0 PI) z1)))))))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 / (((double) M_PI) * (((z1 * z1) - (z0 * z0)) / (z1 - z0)));
	double tmp;
	if (z1 <= -8.2e+160) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * ((double) M_PI));
	} else if (z1 <= -5e-39) {
		tmp = t_0;
	} else if (z1 <= 1.25e-155) {
		tmp = 0.125 / (z1 * (((z0 + z1) * (((double) M_PI) / (z1 * z1))) * z1));
	} else if (z1 <= 2.7e+160) {
		tmp = t_0;
	} else {
		tmp = (0.125 / z1) / (((double) M_PI) + ((z0 * ((double) M_PI)) / z1));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 / (Math.PI * (((z1 * z1) - (z0 * z0)) / (z1 - z0)));
	double tmp;
	if (z1 <= -8.2e+160) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * Math.PI);
	} else if (z1 <= -5e-39) {
		tmp = t_0;
	} else if (z1 <= 1.25e-155) {
		tmp = 0.125 / (z1 * (((z0 + z1) * (Math.PI / (z1 * z1))) * z1));
	} else if (z1 <= 2.7e+160) {
		tmp = t_0;
	} else {
		tmp = (0.125 / z1) / (Math.PI + ((z0 * Math.PI) / z1));
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 / (math.pi * (((z1 * z1) - (z0 * z0)) / (z1 - z0)))
	tmp = 0
	if z1 <= -8.2e+160:
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * math.pi)
	elif z1 <= -5e-39:
		tmp = t_0
	elif z1 <= 1.25e-155:
		tmp = 0.125 / (z1 * (((z0 + z1) * (math.pi / (z1 * z1))) * z1))
	elif z1 <= 2.7e+160:
		tmp = t_0
	else:
		tmp = (0.125 / z1) / (math.pi + ((z0 * math.pi) / z1))
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 / Float64(pi * Float64(Float64(Float64(z1 * z1) - Float64(z0 * z0)) / Float64(z1 - z0))))
	tmp = 0.0
	if (z1 <= -8.2e+160)
		tmp = Float64(Float64(0.125 + Float64(-0.125 * Float64(z0 / z1))) / Float64(z1 * pi));
	elseif (z1 <= -5e-39)
		tmp = t_0;
	elseif (z1 <= 1.25e-155)
		tmp = Float64(0.125 / Float64(z1 * Float64(Float64(Float64(z0 + z1) * Float64(pi / Float64(z1 * z1))) * z1)));
	elseif (z1 <= 2.7e+160)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.125 / z1) / Float64(pi + Float64(Float64(z0 * pi) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 / (pi * (((z1 * z1) - (z0 * z0)) / (z1 - z0)));
	tmp = 0.0;
	if (z1 <= -8.2e+160)
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * pi);
	elseif (z1 <= -5e-39)
		tmp = t_0;
	elseif (z1 <= 1.25e-155)
		tmp = 0.125 / (z1 * (((z0 + z1) * (pi / (z1 * z1))) * z1));
	elseif (z1 <= 2.7e+160)
		tmp = t_0;
	else
		tmp = (0.125 / z1) / (pi + ((z0 * pi) / z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 / N[(Pi * N[(N[(N[(z1 * z1), $MachinePrecision] - N[(z0 * z0), $MachinePrecision]), $MachinePrecision] / N[(z1 - z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -8.2e+160], N[(N[(0.125 + N[(-0.125 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z1 * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, -5e-39], t$95$0, If[LessEqual[z1, 1.25e-155], N[(0.125 / N[(z1 * N[(N[(N[(z0 + z1), $MachinePrecision] * N[(Pi / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 2.7e+160], t$95$0, N[(N[(0.125 / z1), $MachinePrecision] / N[(Pi + N[(N[(z0 * Pi), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z1 < -8.2e160

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.3%

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

   6. Applied rewrites 66.3%

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

   if -8.2e160 < z1 < -4.9999999999999998e-39 or 1.25e-155 < z1 < 2.7e160

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 67.3%

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

   6. Applied rewrites 67.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

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

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

      4. lower-unsound-*.f32 N/A

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

      5. lower-*.f32 N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. lower-unsound--.f64 48.7%

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

   8. Applied rewrites 48.7%

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

   if -4.9999999999999998e-39 < z1 < 1.25e-155

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8%

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

   6. Applied rewrites 74.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. add-to-fraction N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. common-denominator N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 52.5%

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

   if 2.7e160 < z1

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 74.9%

         \[\leadsto \frac{\frac{0.125}{z1}}{\pi + \frac{z0 \cdot \pi}{z1}} \]

   6. Applied rewrites 74.9%

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

Alternative 5: 84.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in z0 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 83.3%

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

4. Applied rewrites 83.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-to-mult N/A

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

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

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

6. Applied rewrites 87.0%

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

Alternative 6: 83.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in z0 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 83.3%

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

4. Applied rewrites 83.3%

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

Alternative 7: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot z1\right) \cdot e^{\frac{z0}{z1}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{0.125}{\pi \cdot z0 - \left(-0.5 \cdot \left(\pi \cdot \frac{z0}{z1}\right)\right) \cdot z0}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{0.125 + -0.125 \cdot \frac{z0}{z1}}{z1 \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{z1 \cdot \frac{z1 \cdot \left(\pi \cdot z0\right)}{z1 \cdot z1}}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* PI z1) (exp (/ z0 z1)))))
  (if (<= t_0 (- INFINITY))
    (/ 0.125 (- (* PI z0) (* (* -0.5 (* PI (/ z0 z1))) z0)))
    (if (<= t_0 4e+306)
      (/ (+ 0.125 (* -0.125 (/ z0 z1))) (* z1 PI))
      (/ 0.125 (* z1 (/ (* z1 (* PI z0)) (* z1 z1))))))))

C

double code(double z1, double z0) {
	double t_0 = (((double) M_PI) * z1) * exp((z0 / z1));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.125 / ((((double) M_PI) * z0) - ((-0.5 * (((double) M_PI) * (z0 / z1))) * z0));
	} else if (t_0 <= 4e+306) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * ((double) M_PI));
	} else {
		tmp = 0.125 / (z1 * ((z1 * (((double) M_PI) * z0)) / (z1 * z1)));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (Math.PI * z1) * Math.exp((z0 / z1));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.125 / ((Math.PI * z0) - ((-0.5 * (Math.PI * (z0 / z1))) * z0));
	} else if (t_0 <= 4e+306) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * Math.PI);
	} else {
		tmp = 0.125 / (z1 * ((z1 * (Math.PI * z0)) / (z1 * z1)));
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = (math.pi * z1) * math.exp((z0 / z1))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.125 / ((math.pi * z0) - ((-0.5 * (math.pi * (z0 / z1))) * z0))
	elif t_0 <= 4e+306:
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * math.pi)
	else:
		tmp = 0.125 / (z1 * ((z1 * (math.pi * z0)) / (z1 * z1)))
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(pi * z1) * exp(Float64(z0 / z1)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.125 / Float64(Float64(pi * z0) - Float64(Float64(-0.5 * Float64(pi * Float64(z0 / z1))) * z0)));
	elseif (t_0 <= 4e+306)
		tmp = Float64(Float64(0.125 + Float64(-0.125 * Float64(z0 / z1))) / Float64(z1 * pi));
	else
		tmp = Float64(0.125 / Float64(z1 * Float64(Float64(z1 * Float64(pi * z0)) / Float64(z1 * z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (pi * z1) * exp((z0 / z1));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.125 / ((pi * z0) - ((-0.5 * (pi * (z0 / z1))) * z0));
	elseif (t_0 <= 4e+306)
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * pi);
	else
		tmp = 0.125 / (z1 * ((z1 * (pi * z0)) / (z1 * z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(Pi * z1), $MachinePrecision] * N[Exp[N[(z0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.125 / N[(N[(Pi * z0), $MachinePrecision] - N[(N[(-0.5 * N[(Pi * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(N[(0.125 + N[(-0.125 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z1 * Pi), $MachinePrecision]), $MachinePrecision], N[(0.125 / N[(z1 * N[(N[(z1 * N[(Pi * z0), $MachinePrecision]), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (PI.f64) z1) (exp.f64 (/.f64 z0 z1))) < -inf.0

   1. Initial program 99.5%

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

   2. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 83.3%

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

   4. Applied rewrites 83.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in z1 around 0

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

   9. Applied rewrites 19.7%

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

   if -inf.0 < (*.f64 (*.f64 (PI.f64) z1) (exp.f64 (/.f64 z0 z1))) < 4.0000000000000001e306

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.3%

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

   6. Applied rewrites 66.3%

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

   if 4.0000000000000001e306 < (*.f64 (*.f64 (PI.f64) z1) (exp.f64 (/.f64 z0 z1)))

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in z1 around 0

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

   9. Applied rewrites 16.0%

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

Alternative 8: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.125}{\left(\pi \cdot z1\right) \cdot e^{\frac{z0}{z1}}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0.125 + -0.125 \cdot \frac{z0}{z1}}{z1 \cdot \pi}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.125}{z1 \cdot \frac{z1 \cdot \left(\pi \cdot z0\right)}{z1 \cdot z1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125}{z1}}{\pi + \frac{z0 \cdot \pi}{z1}}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ 0.125 (* (* PI z1) (exp (/ z0 z1))))))
  (if (<= t_0 -4e-308)
    (/ (+ 0.125 (* -0.125 (/ z0 z1))) (* z1 PI))
    (if (<= t_0 0.0)
      (/ 0.125 (* z1 (/ (* z1 (* PI z0)) (* z1 z1))))
      (/ (/ 0.125 z1) (+ PI (/ (* z0 PI) z1)))))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 / ((((double) M_PI) * z1) * exp((z0 / z1)));
	double tmp;
	if (t_0 <= -4e-308) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * ((double) M_PI));
	} else if (t_0 <= 0.0) {
		tmp = 0.125 / (z1 * ((z1 * (((double) M_PI) * z0)) / (z1 * z1)));
	} else {
		tmp = (0.125 / z1) / (((double) M_PI) + ((z0 * ((double) M_PI)) / z1));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 / ((Math.PI * z1) * Math.exp((z0 / z1)));
	double tmp;
	if (t_0 <= -4e-308) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * Math.PI);
	} else if (t_0 <= 0.0) {
		tmp = 0.125 / (z1 * ((z1 * (Math.PI * z0)) / (z1 * z1)));
	} else {
		tmp = (0.125 / z1) / (Math.PI + ((z0 * Math.PI) / z1));
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 / ((math.pi * z1) * math.exp((z0 / z1)))
	tmp = 0
	if t_0 <= -4e-308:
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * math.pi)
	elif t_0 <= 0.0:
		tmp = 0.125 / (z1 * ((z1 * (math.pi * z0)) / (z1 * z1)))
	else:
		tmp = (0.125 / z1) / (math.pi + ((z0 * math.pi) / z1))
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 / Float64(Float64(pi * z1) * exp(Float64(z0 / z1))))
	tmp = 0.0
	if (t_0 <= -4e-308)
		tmp = Float64(Float64(0.125 + Float64(-0.125 * Float64(z0 / z1))) / Float64(z1 * pi));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.125 / Float64(z1 * Float64(Float64(z1 * Float64(pi * z0)) / Float64(z1 * z1))));
	else
		tmp = Float64(Float64(0.125 / z1) / Float64(pi + Float64(Float64(z0 * pi) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 / ((pi * z1) * exp((z0 / z1)));
	tmp = 0.0;
	if (t_0 <= -4e-308)
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * pi);
	elseif (t_0 <= 0.0)
		tmp = 0.125 / (z1 * ((z1 * (pi * z0)) / (z1 * z1)));
	else
		tmp = (0.125 / z1) / (pi + ((z0 * pi) / z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 / N[(N[(Pi * z1), $MachinePrecision] * N[Exp[N[(z0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-308], N[(N[(0.125 + N[(-0.125 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z1 * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.125 / N[(z1 * N[(N[(z1 * N[(Pi * z0), $MachinePrecision]), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 / z1), $MachinePrecision] / N[(Pi + N[(N[(z0 * Pi), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1/8 binary64) (*.f64 (*.f64 (PI.f64) z1) (exp.f64 (/.f64 z0 z1)))) < -4.0000000000000001e-308

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.3%

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

   6. Applied rewrites 66.3%

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

   if -4.0000000000000001e-308 < (/.f64 #s(literal 1/8 binary64) (*.f64 (*.f64 (PI.f64) z1) (exp.f64 (/.f64 z0 z1)))) < 0.0

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. div-add N/A

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

      7. common-denominator N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in z1 around 0

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

   9. Applied rewrites 16.0%

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

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 74.9%

         \[\leadsto \frac{\frac{0.125}{z1}}{\pi + \frac{z0 \cdot \pi}{z1}} \]

   6. Applied rewrites 74.9%

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

Alternative 9: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.125}{z1 \cdot \pi + \frac{\left(z1 \cdot \pi\right) \cdot z0}{z1}}\\ t_1 := \frac{\frac{0.125}{z1}}{\pi + \frac{z0 \cdot \pi}{z1}}\\ \mathbf{if}\;z1 \leq -3.6 \cdot 10^{+204}:\\ \;\;\;\;\frac{0.125 + -0.125 \cdot \frac{z0}{z1}}{z1 \cdot \pi}\\ \mathbf{elif}\;z1 \leq -1 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 1.5 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z1 \leq 4.6 \cdot 10^{+164}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ 0.125 (+ (* z1 PI) (/ (* (* z1 PI) z0) z1))))
       (t_1 (/ (/ 0.125 z1) (+ PI (/ (* z0 PI) z1)))))
  (if (<= z1 -3.6e+204)
    (/ (+ 0.125 (* -0.125 (/ z0 z1))) (* z1 PI))
    (if (<= z1 -1e-67)
      t_0
      (if (<= z1 1.5e+43) t_1 (if (<= z1 4.6e+164) t_0 t_1))))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 / ((z1 * ((double) M_PI)) + (((z1 * ((double) M_PI)) * z0) / z1));
	double t_1 = (0.125 / z1) / (((double) M_PI) + ((z0 * ((double) M_PI)) / z1));
	double tmp;
	if (z1 <= -3.6e+204) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * ((double) M_PI));
	} else if (z1 <= -1e-67) {
		tmp = t_0;
	} else if (z1 <= 1.5e+43) {
		tmp = t_1;
	} else if (z1 <= 4.6e+164) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 / ((z1 * Math.PI) + (((z1 * Math.PI) * z0) / z1));
	double t_1 = (0.125 / z1) / (Math.PI + ((z0 * Math.PI) / z1));
	double tmp;
	if (z1 <= -3.6e+204) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * Math.PI);
	} else if (z1 <= -1e-67) {
		tmp = t_0;
	} else if (z1 <= 1.5e+43) {
		tmp = t_1;
	} else if (z1 <= 4.6e+164) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 / ((z1 * math.pi) + (((z1 * math.pi) * z0) / z1))
	t_1 = (0.125 / z1) / (math.pi + ((z0 * math.pi) / z1))
	tmp = 0
	if z1 <= -3.6e+204:
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * math.pi)
	elif z1 <= -1e-67:
		tmp = t_0
	elif z1 <= 1.5e+43:
		tmp = t_1
	elif z1 <= 4.6e+164:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 / Float64(Float64(z1 * pi) + Float64(Float64(Float64(z1 * pi) * z0) / z1)))
	t_1 = Float64(Float64(0.125 / z1) / Float64(pi + Float64(Float64(z0 * pi) / z1)))
	tmp = 0.0
	if (z1 <= -3.6e+204)
		tmp = Float64(Float64(0.125 + Float64(-0.125 * Float64(z0 / z1))) / Float64(z1 * pi));
	elseif (z1 <= -1e-67)
		tmp = t_0;
	elseif (z1 <= 1.5e+43)
		tmp = t_1;
	elseif (z1 <= 4.6e+164)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 / ((z1 * pi) + (((z1 * pi) * z0) / z1));
	t_1 = (0.125 / z1) / (pi + ((z0 * pi) / z1));
	tmp = 0.0;
	if (z1 <= -3.6e+204)
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * pi);
	elseif (z1 <= -1e-67)
		tmp = t_0;
	elseif (z1 <= 1.5e+43)
		tmp = t_1;
	elseif (z1 <= 4.6e+164)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 / N[(N[(z1 * Pi), $MachinePrecision] + N[(N[(N[(z1 * Pi), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.125 / z1), $MachinePrecision] / N[(Pi + N[(N[(z0 * Pi), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -3.6e+204], N[(N[(0.125 + N[(-0.125 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z1 * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, -1e-67], t$95$0, If[LessEqual[z1, 1.5e+43], t$95$1, If[LessEqual[z1, 4.6e+164], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -3.6000000000000002e204

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.3%

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

   6. Applied rewrites 66.3%

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

   if -3.6000000000000002e204 < z1 < -9.9999999999999994e-68 or 1.5000000000000001e43 < z1 < 4.5999999999999999e164

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8%

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

   6. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-PI.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. add-to-fraction N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 63.5%

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

   if -9.9999999999999994e-68 < z1 < 1.5000000000000001e43 or 4.5999999999999999e164 < z1

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 74.9%

         \[\leadsto \frac{\frac{0.125}{z1}}{\pi + \frac{z0 \cdot \pi}{z1}} \]

   6. Applied rewrites 74.9%

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

Alternative 10: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.125}{z1 \cdot \pi + \frac{\left(z1 \cdot \pi\right) \cdot z0}{z1}}\\ \mathbf{if}\;z1 \leq -3.6 \cdot 10^{+204}:\\ \;\;\;\;\frac{0.125 + -0.125 \cdot \frac{z0}{z1}}{z1 \cdot \pi}\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 1.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{0.125}{z1 \cdot \left(\pi + \frac{\pi}{z1} \cdot z0\right)}\\ \mathbf{elif}\;z1 \leq 4.6 \cdot 10^{+164}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125}{z1}}{\pi}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ 0.125 (+ (* z1 PI) (/ (* (* z1 PI) z0) z1)))))
  (if (<= z1 -3.6e+204)
    (/ (+ 0.125 (* -0.125 (/ z0 z1))) (* z1 PI))
    (if (<= z1 -5e+28)
      t_0
      (if (<= z1 1.5e+43)
        (/ 0.125 (* z1 (+ PI (* (/ PI z1) z0))))
        (if (<= z1 4.6e+164) t_0 (/ (/ 0.125 z1) PI)))))))

C

double code(double z1, double z0) {
	double t_0 = 0.125 / ((z1 * ((double) M_PI)) + (((z1 * ((double) M_PI)) * z0) / z1));
	double tmp;
	if (z1 <= -3.6e+204) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * ((double) M_PI));
	} else if (z1 <= -5e+28) {
		tmp = t_0;
	} else if (z1 <= 1.5e+43) {
		tmp = 0.125 / (z1 * (((double) M_PI) + ((((double) M_PI) / z1) * z0)));
	} else if (z1 <= 4.6e+164) {
		tmp = t_0;
	} else {
		tmp = (0.125 / z1) / ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 0.125 / ((z1 * Math.PI) + (((z1 * Math.PI) * z0) / z1));
	double tmp;
	if (z1 <= -3.6e+204) {
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * Math.PI);
	} else if (z1 <= -5e+28) {
		tmp = t_0;
	} else if (z1 <= 1.5e+43) {
		tmp = 0.125 / (z1 * (Math.PI + ((Math.PI / z1) * z0)));
	} else if (z1 <= 4.6e+164) {
		tmp = t_0;
	} else {
		tmp = (0.125 / z1) / Math.PI;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 0.125 / ((z1 * math.pi) + (((z1 * math.pi) * z0) / z1))
	tmp = 0
	if z1 <= -3.6e+204:
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * math.pi)
	elif z1 <= -5e+28:
		tmp = t_0
	elif z1 <= 1.5e+43:
		tmp = 0.125 / (z1 * (math.pi + ((math.pi / z1) * z0)))
	elif z1 <= 4.6e+164:
		tmp = t_0
	else:
		tmp = (0.125 / z1) / math.pi
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(0.125 / Float64(Float64(z1 * pi) + Float64(Float64(Float64(z1 * pi) * z0) / z1)))
	tmp = 0.0
	if (z1 <= -3.6e+204)
		tmp = Float64(Float64(0.125 + Float64(-0.125 * Float64(z0 / z1))) / Float64(z1 * pi));
	elseif (z1 <= -5e+28)
		tmp = t_0;
	elseif (z1 <= 1.5e+43)
		tmp = Float64(0.125 / Float64(z1 * Float64(pi + Float64(Float64(pi / z1) * z0))));
	elseif (z1 <= 4.6e+164)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.125 / z1) / pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 0.125 / ((z1 * pi) + (((z1 * pi) * z0) / z1));
	tmp = 0.0;
	if (z1 <= -3.6e+204)
		tmp = (0.125 + (-0.125 * (z0 / z1))) / (z1 * pi);
	elseif (z1 <= -5e+28)
		tmp = t_0;
	elseif (z1 <= 1.5e+43)
		tmp = 0.125 / (z1 * (pi + ((pi / z1) * z0)));
	elseif (z1 <= 4.6e+164)
		tmp = t_0;
	else
		tmp = (0.125 / z1) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(0.125 / N[(N[(z1 * Pi), $MachinePrecision] + N[(N[(N[(z1 * Pi), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -3.6e+204], N[(N[(0.125 + N[(-0.125 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z1 * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, -5e+28], t$95$0, If[LessEqual[z1, 1.5e+43], N[(0.125 / N[(z1 * N[(Pi + N[(N[(Pi / z1), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 4.6e+164], t$95$0, N[(N[(0.125 / z1), $MachinePrecision] / Pi), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z1 < -3.6000000000000002e204

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.3%

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

   6. Applied rewrites 66.3%

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

   if -3.6000000000000002e204 < z1 < -4.9999999999999996e28 or 1.5000000000000001e43 < z1 < 4.5999999999999999e164

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8%

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

   6. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-PI.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. add-to-fraction N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 63.5%

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

   if -4.9999999999999996e28 < z1 < 1.5000000000000001e43

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8%

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

   6. Applied rewrites 74.8%

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

   if 4.5999999999999999e164 < z1

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{\frac{0.125}{z1}}{\color{blue}{\pi}} \]
   5. Step-by-step derivation

      1. lower-PI.f64 65.9%

         \[\leadsto \frac{\frac{0.125}{z1}}{\pi} \]

   6. Applied rewrites 65.9%

      \[\leadsto \frac{\frac{0.125}{z1}}{\color{blue}{\pi}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z1 \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.125}{z1 \cdot \left(\pi + \frac{\pi}{z1} \cdot z0\right)}\\ \mathbf{elif}\;z1 \leq 4.6 \cdot 10^{+164}:\\ \;\;\;\;\frac{0.125}{\frac{\left(\pi \cdot \left(z0 + z1\right)\right) \cdot z1}{z1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125}{z1}}{\pi}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= z1 5e+33)
  (/ 0.125 (* z1 (+ PI (* (/ PI z1) z0))))
  (if (<= z1 4.6e+164)
    (/ 0.125 (/ (* (* PI (+ z0 z1)) z1) z1))
    (/ (/ 0.125 z1) PI))))

C

double code(double z1, double z0) {
	double tmp;
	if (z1 <= 5e+33) {
		tmp = 0.125 / (z1 * (((double) M_PI) + ((((double) M_PI) / z1) * z0)));
	} else if (z1 <= 4.6e+164) {
		tmp = 0.125 / (((((double) M_PI) * (z0 + z1)) * z1) / z1);
	} else {
		tmp = (0.125 / z1) / ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (z1 <= 5e+33) {
		tmp = 0.125 / (z1 * (Math.PI + ((Math.PI / z1) * z0)));
	} else if (z1 <= 4.6e+164) {
		tmp = 0.125 / (((Math.PI * (z0 + z1)) * z1) / z1);
	} else {
		tmp = (0.125 / z1) / Math.PI;
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if z1 <= 5e+33:
		tmp = 0.125 / (z1 * (math.pi + ((math.pi / z1) * z0)))
	elif z1 <= 4.6e+164:
		tmp = 0.125 / (((math.pi * (z0 + z1)) * z1) / z1)
	else:
		tmp = (0.125 / z1) / math.pi
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (z1 <= 5e+33)
		tmp = Float64(0.125 / Float64(z1 * Float64(pi + Float64(Float64(pi / z1) * z0))));
	elseif (z1 <= 4.6e+164)
		tmp = Float64(0.125 / Float64(Float64(Float64(pi * Float64(z0 + z1)) * z1) / z1));
	else
		tmp = Float64(Float64(0.125 / z1) / pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (z1 <= 5e+33)
		tmp = 0.125 / (z1 * (pi + ((pi / z1) * z0)));
	elseif (z1 <= 4.6e+164)
		tmp = 0.125 / (((pi * (z0 + z1)) * z1) / z1);
	else
		tmp = (0.125 / z1) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[z1, 5e+33], N[(0.125 / N[(z1 * N[(Pi + N[(N[(Pi / z1), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 4.6e+164], N[(0.125 / N[(N[(N[(Pi * N[(z0 + z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 / z1), $MachinePrecision] / Pi), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z1 < 4.9999999999999997e33

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8%

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

   6. Applied rewrites 74.8%

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

   if 4.9999999999999997e33 < z1 < 4.5999999999999999e164

   1. Initial program 99.5%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.8%

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

   4. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 41.4%

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

   6. Applied rewrites 41.4%

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

   if 4.5999999999999999e164 < z1

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{\frac{0.125}{z1}}{\color{blue}{\pi}} \]
   5. Step-by-step derivation

      1. lower-PI.f64 65.9%

         \[\leadsto \frac{\frac{0.125}{z1}}{\pi} \]

   6. Applied rewrites 65.9%

      \[\leadsto \frac{\frac{0.125}{z1}}{\color{blue}{\pi}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in z1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 74.8%

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

4. Applied rewrites 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 74.8%

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

6. Applied rewrites 74.8%

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

Alternative 13: 67.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in z1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 74.8%

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

4. Applied rewrites 74.8%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 74.8%

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

6. Applied rewrites 67.3%

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

Alternative 14: 67.3% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in z1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 74.8%

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

4. Applied rewrites 74.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. associate-*l* N/A

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

   7. lft-mult-inverse N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 67.3%

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

6. Applied rewrites 67.3%

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

Alternative 15: 65.8% accurate, 7.8× speedup

Math

\[\frac{0.125}{z1 \cdot \pi} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in z1 around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 65.8%

      \[\leadsto \frac{0.125}{z1 \cdot \pi} \]

4. Applied rewrites 65.8%

   \[\leadsto \color{blue}{\frac{0.125}{z1 \cdot \pi}} \]
5. Add Preprocessing

Reproduce

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