(- z1 (* (- -1 (exp (/ -7853981852531433/2500000000000000 z0))) (/ (- 1 z1) (- (exp (/ PI z0)) -1))))

Percentage Accurate: 76.8% → 87.7%

Time: 4.0s

Alternatives: 22

Speedup: 2.2×

• 74.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (-
 z1
 (*
  (- -1.0 (exp (/ -3.1415927410125732 z0)))
  (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0)))))

C

double code(double z1, double z0) {
	return z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (exp((((double) M_PI) / z0)) - -1.0)));
}

Java

public static double code(double z1, double z0) {
	return z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (Math.exp((Math.PI / z0)) - -1.0)));
}

Python

def code(z1, z0):
	return z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (math.exp((math.pi / z0)) - -1.0)))

Julia

function code(z1, z0)
	return Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(Float64(1.0 - z1) / Float64(exp(Float64(pi / z0)) - -1.0))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (exp((pi / z0)) - -1.0)));
end

Wolfram

code[z1_, z0_] := N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.8% accurate, 1.0× speedup

Math

\[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (-
 z1
 (*
  (- -1.0 (exp (/ -3.1415927410125732 z0)))
  (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0)))))

C

double code(double z1, double z0) {
	return z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (exp((((double) M_PI) / z0)) - -1.0)));
}

Java

public static double code(double z1, double z0) {
	return z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (Math.exp((Math.PI / z0)) - -1.0)));
}

Python

def code(z1, z0):
	return z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (math.exp((math.pi / z0)) - -1.0)))

Julia

function code(z1, z0)
	return Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(Float64(1.0 - z1) / Float64(exp(Float64(pi / z0)) - -1.0))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (exp((pi / z0)) - -1.0)));
end

Wolfram

code[z1_, z0_] := N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ t_1 := e^{\frac{\pi}{z0}}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;\left(z1 - \frac{z1 - 1}{t\_1 - -1}\right) - \frac{1}{\frac{-1 - t\_1}{-e^{\frac{-3.1415927410125732}{z0}} \cdot \left(z1 - 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0))))
       (t_1 (exp (/ PI z0))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 2.7e+21)
      (-
       (- z1 (/ (- z1 1.0) (- t_1 -1.0)))
       (/
        1.0
        (/
         (- -1.0 t_1)
         (- (* (exp (/ -3.1415927410125732 z0)) (- z1 1.0))))))
      t_0))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double t_1 = exp((((double) M_PI) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (1.0 / ((-1.0 - t_1) / -(exp((-3.1415927410125732 / z0)) * (z1 - 1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double t_1 = Math.exp((Math.PI / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (1.0 / ((-1.0 - t_1) / -(Math.exp((-3.1415927410125732 / z0)) * (z1 - 1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	t_1 = math.exp((math.pi / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= 2.7e+21:
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (1.0 / ((-1.0 - t_1) / -(math.exp((-3.1415927410125732 / z0)) * (z1 - 1.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	t_1 = exp(Float64(pi / z0))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = Float64(Float64(z1 - Float64(Float64(z1 - 1.0) / Float64(t_1 - -1.0))) - Float64(1.0 / Float64(Float64(-1.0 - t_1) / Float64(-Float64(exp(Float64(-3.1415927410125732 / z0)) * Float64(z1 - 1.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	t_1 = exp((pi / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (1.0 / ((-1.0 - t_1) / -(exp((-3.1415927410125732 / z0)) * (z1 - 1.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, 2.7e+21], N[(N[(z1 - N[(N[(z1 - 1.0), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[(-1.0 - t$95$1), $MachinePrecision] / (-N[(N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision] * N[(z1 - 1.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      2. lift-*.f64 N/A

         \[\leadsto z1 - \color{blue}{\left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      3. *-commutative N/A

         \[\leadsto z1 - \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto z1 - \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \color{blue}{\left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)} \]

      5. sub-flip N/A

         \[\leadsto z1 - \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \color{blue}{\left(-1 + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right)\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right)} \]

      7. mul-1-neg N/A

         \[\leadsto z1 - \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right)\right)} + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto z1 - \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}}\right)\right) + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z1 - \left(\color{blue}{\frac{\mathsf{neg}\left(\left(1 - z1\right)\right)}{e^{\frac{\pi}{z0}} - -1}} + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right) \]

      10. *-commutative N/A

          \[\leadsto z1 - \left(\frac{\mathsf{neg}\left(\left(1 - z1\right)\right)}{e^{\frac{\pi}{z0}} - -1} + \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right)}\right) \]

   3. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \left(-e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{z1 - 1}{-1 - e^{\frac{\pi}{z0}}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \color{blue}{\left(-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{z1 - 1}{-1 - e^{\frac{\pi}{z0}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \left(-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \color{blue}{\frac{z1 - 1}{-1 - e^{\frac{\pi}{z0}}}} \]

      3. associate-*r/ N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \color{blue}{\frac{\left(-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(z1 - 1\right)}{-1 - e^{\frac{\pi}{z0}}}} \]

      4. div-flip N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \color{blue}{\frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{\left(-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(z1 - 1\right)}}} \]

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

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \color{blue}{\frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{\left(-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(z1 - 1\right)}}} \]

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

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \frac{1}{\color{blue}{\frac{-1 - e^{\frac{\pi}{z0}}}{\left(-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(z1 - 1\right)}}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{\color{blue}{\left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right)} \cdot \left(z1 - 1\right)}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{\color{blue}{\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}} \cdot \left(z1 - 1\right)\right)}}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{\color{blue}{-e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}} \cdot \left(z1 - 1\right)}}} \]

      10. lower-*.f64 76.8%

          \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{-\color{blue}{e^{\frac{-3.1415927410125732}{z0}} \cdot \left(z1 - 1\right)}}} \]

   5. Applied rewrites 76.8%

      \[\leadsto \left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \color{blue}{\frac{1}{\frac{-1 - e^{\frac{\pi}{z0}}}{-e^{\frac{-3.1415927410125732}{z0}} \cdot \left(z1 - 1\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ t_1 := e^{\frac{\pi}{z0}}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;\left(z1 - \frac{z1 - 1}{t\_1 - -1}\right) - \left(-e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{z1 - 1}{-1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0))))
       (t_1 (exp (/ PI z0))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 2.7e+21)
      (-
       (- z1 (/ (- z1 1.0) (- t_1 -1.0)))
       (*
        (- (exp (/ -3.1415927410125732 z0)))
        (/ (- z1 1.0) (- -1.0 t_1))))
      t_0))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double t_1 = exp((((double) M_PI) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (-exp((-3.1415927410125732 / z0)) * ((z1 - 1.0) / (-1.0 - t_1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double t_1 = Math.exp((Math.PI / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (-Math.exp((-3.1415927410125732 / z0)) * ((z1 - 1.0) / (-1.0 - t_1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	t_1 = math.exp((math.pi / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= 2.7e+21:
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (-math.exp((-3.1415927410125732 / z0)) * ((z1 - 1.0) / (-1.0 - t_1)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	t_1 = exp(Float64(pi / z0))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = Float64(Float64(z1 - Float64(Float64(z1 - 1.0) / Float64(t_1 - -1.0))) - Float64(Float64(-exp(Float64(-3.1415927410125732 / z0))) * Float64(Float64(z1 - 1.0) / Float64(-1.0 - t_1))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	t_1 = exp((pi / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = (z1 - ((z1 - 1.0) / (t_1 - -1.0))) - (-exp((-3.1415927410125732 / z0)) * ((z1 - 1.0) / (-1.0 - t_1)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, 2.7e+21], N[(N[(z1 - N[(N[(z1 - 1.0), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[((-N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]) * N[(N[(z1 - 1.0), $MachinePrecision] / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      2. lift-*.f64 N/A

         \[\leadsto z1 - \color{blue}{\left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      3. *-commutative N/A

         \[\leadsto z1 - \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto z1 - \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \color{blue}{\left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)} \]

      5. sub-flip N/A

         \[\leadsto z1 - \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \color{blue}{\left(-1 + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right)\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right)} \]

      7. mul-1-neg N/A

         \[\leadsto z1 - \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right)\right)} + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto z1 - \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}}\right)\right) + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z1 - \left(\color{blue}{\frac{\mathsf{neg}\left(\left(1 - z1\right)\right)}{e^{\frac{\pi}{z0}} - -1}} + \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\right) \]

      10. *-commutative N/A

          \[\leadsto z1 - \left(\frac{\mathsf{neg}\left(\left(1 - z1\right)\right)}{e^{\frac{\pi}{z0}} - -1} + \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \cdot \left(\mathsf{neg}\left(e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)\right)}\right) \]

   3. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(z1 - \frac{z1 - 1}{e^{\frac{\pi}{z0}} - -1}\right) - \left(-e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{z1 - 1}{-1 - e^{\frac{\pi}{z0}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{-\pi}{z0}}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 2.7e+21)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (/ (- 1.0 z1) (- (/ 1.0 (exp (/ (- PI) z0))) -1.0))))
      t_0))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / ((1.0 / exp((-((double) M_PI) / z0))) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / ((1.0 / Math.exp((-Math.PI / z0))) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= 2.7e+21:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / ((1.0 / math.exp((-math.pi / z0))) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(Float64(1.0 - z1) / Float64(Float64(1.0 / exp(Float64(Float64(-pi) / z0))) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / ((1.0 / exp((-pi / z0))) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, 2.7e+21], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[(1.0 / N[Exp[N[((-Pi) / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 2.7e+21)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0))))
      t_0))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (exp((((double) M_PI) / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (Math.exp((Math.PI / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= 2.7e+21:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (math.exp((math.pi / z0)) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(Float64(1.0 - z1) / Float64(exp(Float64(pi / z0)) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= 2.7e+21)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((1.0 - z1) / (exp((pi / z0)) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, 2.7e+21], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ t_1 := e^{\frac{-3.1415927410125732}{z0}}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -3.9 \cdot 10^{-296}:\\ \;\;\;\;z1 - \left(-1 - t\_1\right) \cdot \left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 15200000000000:\\ \;\;\;\;z1 - \frac{z1 \cdot \left(1 + t\_1\right)}{1 + e^{\frac{\pi}{z0}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0))))
       (t_1 (exp (/ -3.1415927410125732 z0))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 -3.9e-296)
      (-
       z1
       (*
        (- -1.0 t_1)
        (+ (* -0.25 (/ (* PI (- 1.0 z1)) z0)) (* 0.5 (- 1.0 z1)))))
      (if (<= z0 15200000000000.0)
        (- z1 (/ (* z1 (+ 1.0 t_1)) (+ 1.0 (exp (/ PI z0)))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double t_1 = exp((-3.1415927410125732 / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -3.9e-296) {
		tmp = z1 - ((-1.0 - t_1) * ((-0.25 * ((((double) M_PI) * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	} else if (z0 <= 15200000000000.0) {
		tmp = z1 - ((z1 * (1.0 + t_1)) / (1.0 + exp((((double) M_PI) / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double t_1 = Math.exp((-3.1415927410125732 / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -3.9e-296) {
		tmp = z1 - ((-1.0 - t_1) * ((-0.25 * ((Math.PI * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	} else if (z0 <= 15200000000000.0) {
		tmp = z1 - ((z1 * (1.0 + t_1)) / (1.0 + Math.exp((Math.PI / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	t_1 = math.exp((-3.1415927410125732 / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= -3.9e-296:
		tmp = z1 - ((-1.0 - t_1) * ((-0.25 * ((math.pi * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))))
	elif z0 <= 15200000000000.0:
		tmp = z1 - ((z1 * (1.0 + t_1)) / (1.0 + math.exp((math.pi / z0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	t_1 = exp(Float64(-3.1415927410125732 / z0))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -3.9e-296)
		tmp = Float64(z1 - Float64(Float64(-1.0 - t_1) * Float64(Float64(-0.25 * Float64(Float64(pi * Float64(1.0 - z1)) / z0)) + Float64(0.5 * Float64(1.0 - z1)))));
	elseif (z0 <= 15200000000000.0)
		tmp = Float64(z1 - Float64(Float64(z1 * Float64(1.0 + t_1)) / Float64(1.0 + exp(Float64(pi / z0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	t_1 = exp((-3.1415927410125732 / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -3.9e-296)
		tmp = z1 - ((-1.0 - t_1) * ((-0.25 * ((pi * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	elseif (z0 <= 15200000000000.0)
		tmp = z1 - ((z1 * (1.0 + t_1)) / (1.0 + exp((pi / z0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, -3.9e-296], N[(z1 - N[(N[(-1.0 - t$95$1), $MachinePrecision] * N[(N[(-0.25 * N[(N[(Pi * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 15200000000000.0], N[(z1 - N[(N[(z1 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.3500000000000001e53 or 1.52e13 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < -3.9000000000000001e-296

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \color{blue}{\frac{1}{2} \cdot \left(1 - z1\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \color{blue}{\frac{1}{2}} \cdot \left(1 - z1\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      8. lower--.f64 54.0%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.0%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - z1\right)\right)} \]

   if -3.9000000000000001e-296 < z0 < 1.52e13

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. lower--.f64 27.8%

         \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

   4. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      3. lift--.f64 N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      4. sub-negate-rev N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      5. lift--.f64 27.8%

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   6. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto z1 - -1 \]
   8. Step-by-step derivation

   9. Applied rewrites 37.4%

      \[\leadsto z1 - -1 \]

   10. Taylor expanded in z1 around inf

       \[\leadsto z1 - \color{blue}{\frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\pi}{z0}}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{\color{blue}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{\color{blue}{1} + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       3. lower-+.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       4. lower-exp.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       5. lower-/.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       6. lower-+.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{z0}}}} \]

       7. lower-exp.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       8. lower-/.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       9. lower-PI.f64 44.5%

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{-3.1415927410125732}{z0}}\right)}{1 + e^{\frac{\pi}{z0}}} \]

   12. Applied rewrites 44.5%

       \[\leadsto z1 - \color{blue}{\frac{z1 \cdot \left(1 + e^{\frac{-3.1415927410125732}{z0}}\right)}{1 + e^{\frac{\pi}{z0}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -3.9 \cdot 10^{-296}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \left(3.1415927410125732 \cdot \frac{1}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 -3.9e-296)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (+ (* -0.25 (/ (* PI (- 1.0 z1)) z0)) (* 0.5 (- 1.0 z1)))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (*
          (- (* 3.1415927410125732 (/ 1.0 z0)) 2.0)
          (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -3.9e-296) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((-0.25 * ((((double) M_PI) * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (exp((((double) M_PI) / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -3.9e-296) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * ((-0.25 * ((Math.PI * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (Math.exp((Math.PI / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= -3.9e-296:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * ((-0.25 * ((math.pi * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))))
	elif z0 <= 2.7e+21:
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (math.exp((math.pi / z0)) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -3.9e-296)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(Float64(-0.25 * Float64(Float64(pi * Float64(1.0 - z1)) / z0)) + Float64(0.5 * Float64(1.0 - z1)))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(Float64(Float64(3.1415927410125732 * Float64(1.0 / z0)) - 2.0) * Float64(Float64(1.0 - z1) / Float64(exp(Float64(pi / z0)) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -3.9e-296)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * ((-0.25 * ((pi * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (exp((pi / z0)) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, -3.9e-296], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.25 * N[(N[(Pi * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(N[(N[(3.1415927410125732 * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < -3.9000000000000001e-296

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \color{blue}{\frac{1}{2} \cdot \left(1 - z1\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \color{blue}{\frac{1}{2}} \cdot \left(1 - z1\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      8. lower--.f64 54.0%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.0%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - z1\right)\right)} \]

   if -3.9000000000000001e-296 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{\left(\frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \left(\frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z0} - \color{blue}{2}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(\frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      3. lower-/.f64 58.0%

         \[\leadsto z1 - \left(3.1415927410125732 \cdot \frac{1}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   4. Applied rewrites 58.0%

      \[\leadsto z1 - \color{blue}{\left(3.1415927410125732 \cdot \frac{1}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -3.9 \cdot 10^{-296}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \left(3.1415927410125732 \cdot \frac{1}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.0065)
    t_0
    (if (<= z0 -3.9e-296)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (*
          (- (* 3.1415927410125732 (/ 1.0 z0)) 2.0)
          (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -3.9e-296) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (exp((((double) M_PI) / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -3.9e-296) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (Math.exp((Math.PI / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.0065:
		tmp = t_0
	elif z0 <= -3.9e-296:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (math.exp((math.pi / z0)) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -3.9e-296)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(Float64(Float64(3.1415927410125732 * Float64(1.0 / z0)) - 2.0) * Float64(Float64(1.0 - z1) / Float64(exp(Float64(pi / z0)) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -3.9e-296)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (((3.1415927410125732 * (1.0 / z0)) - 2.0) * ((1.0 - z1) / (exp((pi / z0)) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.0065], t$95$0, If[LessEqual[z0, -3.9e-296], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(N[(N[(3.1415927410125732 * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0064999999999999997 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0064999999999999997 < z0 < -3.9000000000000001e-296

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 54.8%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -3.9000000000000001e-296 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{\left(\frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \left(\frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z0} - \color{blue}{2}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(\frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      3. lower-/.f64 58.0%

         \[\leadsto z1 - \left(3.1415927410125732 \cdot \frac{1}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   4. Applied rewrites 58.0%

      \[\leadsto z1 - \color{blue}{\left(3.1415927410125732 \cdot \frac{1}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - -2 \cdot \frac{1 - z1}{\frac{1}{e^{\frac{-\pi}{z0}}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.0065)
    t_0
    (if (<= z0 -2e-310)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (* -2.0 (/ (- 1.0 z1) (- (/ 1.0 (exp (/ (- PI) z0))) -1.0))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -2e-310) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / ((1.0 / exp((-((double) M_PI) / z0))) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -2e-310) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / ((1.0 / Math.exp((-Math.PI / z0))) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.0065:
		tmp = t_0
	elif z0 <= -2e-310:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (-2.0 * ((1.0 - z1) / ((1.0 / math.exp((-math.pi / z0))) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -2e-310)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(-2.0 * Float64(Float64(1.0 - z1) / Float64(Float64(1.0 / exp(Float64(Float64(-pi) / z0))) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -2e-310)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (-2.0 * ((1.0 - z1) / ((1.0 / exp((-pi / z0))) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.0065], t$95$0, If[LessEqual[z0, -2e-310], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(-2.0 * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[(1.0 / N[Exp[N[((-Pi) / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0064999999999999997 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0064999999999999997 < z0 < -1.9999999999999939e-310

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 54.8%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -1.9999999999999939e-310 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-/.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

      8. lift-exp.f64 N/A

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\frac{-\pi}{z0}}}} - -1} \]

      9. lift-/.f64 51.3%

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   6. Applied rewrites 51.3%

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \frac{1}{\frac{e^{\frac{\pi}{z0}} - -1}{-2 \cdot \left(1 - z1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.0065)
    t_0
    (if (<= z0 -2e-310)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (/ 1.0 (/ (- (exp (/ PI z0)) -1.0) (* -2.0 (- 1.0 z1)))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -2e-310) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((exp((((double) M_PI) / z0)) - -1.0) / (-2.0 * (1.0 - z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -2e-310) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((Math.exp((Math.PI / z0)) - -1.0) / (-2.0 * (1.0 - z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.0065:
		tmp = t_0
	elif z0 <= -2e-310:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (1.0 / ((math.exp((math.pi / z0)) - -1.0) / (-2.0 * (1.0 - z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -2e-310)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(1.0 / Float64(Float64(exp(Float64(pi / z0)) - -1.0) / Float64(-2.0 * Float64(1.0 - z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -2e-310)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (1.0 / ((exp((pi / z0)) - -1.0) / (-2.0 * (1.0 - z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.0065], t$95$0, If[LessEqual[z0, -2e-310], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(1.0 / N[(N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / N[(-2.0 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0064999999999999997 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0064999999999999997 < z0 < -1.9999999999999939e-310

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 54.8%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -1.9999999999999939e-310 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - \color{blue}{-2 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - -2 \cdot \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 51.1%

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

   6. Applied rewrites 51.1%

      \[\leadsto z1 - \color{blue}{\frac{1}{\frac{e^{\frac{\pi}{z0}} - -1}{-2 \cdot \left(1 - z1\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - -2 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.0065)
    t_0
    (if (<= z0 -2e-310)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (- z1 (* -2.0 (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -2e-310) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (exp((((double) M_PI) / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -2e-310) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (Math.exp((Math.PI / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.0065:
		tmp = t_0
	elif z0 <= -2e-310:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (-2.0 * ((1.0 - z1) / (math.exp((math.pi / z0)) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -2e-310)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(-2.0 * Float64(Float64(1.0 - z1) / Float64(exp(Float64(pi / z0)) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -2e-310)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (-2.0 * ((1.0 - z1) / (exp((pi / z0)) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.0065], t$95$0, If[LessEqual[z0, -2e-310], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(-2.0 * N[(N[(1.0 - z1), $MachinePrecision] / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0064999999999999997 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0064999999999999997 < z0 < -1.9999999999999939e-310

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 54.8%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -1.9999999999999939e-310 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -3.7 \cdot 10^{-308}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 5.1 \cdot 10^{+77}:\\ \;\;\;\;z1 - -2 \cdot \frac{1}{e^{\frac{\pi}{z0}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.0065)
    t_0
    (if (<= z0 -3.7e-308)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 5.1e+77)
        (- z1 (* -2.0 (/ 1.0 (- (exp (/ PI z0)) -1.0))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -3.7e-308) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 5.1e+77) {
		tmp = z1 - (-2.0 * (1.0 / (exp((((double) M_PI) / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_0;
	} else if (z0 <= -3.7e-308) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 5.1e+77) {
		tmp = z1 - (-2.0 * (1.0 / (Math.exp((Math.PI / z0)) - -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.0065:
		tmp = t_0
	elif z0 <= -3.7e-308:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)))
	elif z0 <= 5.1e+77:
		tmp = z1 - (-2.0 * (1.0 / (math.exp((math.pi / z0)) - -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -3.7e-308)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 5.1e+77)
		tmp = Float64(z1 - Float64(-2.0 * Float64(1.0 / Float64(exp(Float64(pi / z0)) - -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.0065)
		tmp = t_0;
	elseif (z0 <= -3.7e-308)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 5.1e+77)
		tmp = z1 - (-2.0 * (1.0 / (exp((pi / z0)) - -1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.0065], t$95$0, If[LessEqual[z0, -3.7e-308], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 5.1e+77], N[(z1 - N[(-2.0 * N[(1.0 / N[(N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0064999999999999997 or 5.0999999999999997e77 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0064999999999999997 < z0 < -3.7000000000000001e-308

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 54.8%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -3.7000000000000001e-308 < z0 < 5.0999999999999997e77

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z1 around 0

      \[\leadsto z1 - -2 \cdot \frac{\color{blue}{1}}{e^{\frac{\pi}{z0}} - -1} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.9%

      \[\leadsto z1 - -2 \cdot \frac{\color{blue}{1}}{e^{\frac{\pi}{z0}} - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{z0}}\\ t_1 := t\_0 - -1\\ t_2 := e^{\frac{-3.1415927410125732}{z0}}\\ t_3 := -1 - t\_2\\ t_4 := z1 - t\_3 \cdot \frac{1 - z1}{t\_1}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-308}:\\ \;\;\;\;z1 - \frac{z1 \cdot \left(1 + t\_2\right)}{1 + t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{else}:\\ \;\;\;\;z1 - t\_3 \cdot \frac{1}{t\_1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (exp (/ PI z0)))
       (t_1 (- t_0 -1.0))
       (t_2 (exp (/ -3.1415927410125732 z0)))
       (t_3 (- -1.0 t_2))
       (t_4 (- z1 (* t_3 (/ (- 1.0 z1) t_1)))))
  (if (<= t_4 -4e-308)
    (- z1 (/ (* z1 (+ 1.0 t_2)) (+ 1.0 t_0)))
    (if (<= t_4 0.0)
      (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))
      (- z1 (* t_3 (/ 1.0 t_1)))))))

C

double code(double z1, double z0) {
	double t_0 = exp((((double) M_PI) / z0));
	double t_1 = t_0 - -1.0;
	double t_2 = exp((-3.1415927410125732 / z0));
	double t_3 = -1.0 - t_2;
	double t_4 = z1 - (t_3 * ((1.0 - z1) / t_1));
	double tmp;
	if (t_4 <= -4e-308) {
		tmp = z1 - ((z1 * (1.0 + t_2)) / (1.0 + t_0));
	} else if (t_4 <= 0.0) {
		tmp = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	} else {
		tmp = z1 - (t_3 * (1.0 / t_1));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.exp((Math.PI / z0));
	double t_1 = t_0 - -1.0;
	double t_2 = Math.exp((-3.1415927410125732 / z0));
	double t_3 = -1.0 - t_2;
	double t_4 = z1 - (t_3 * ((1.0 - z1) / t_1));
	double tmp;
	if (t_4 <= -4e-308) {
		tmp = z1 - ((z1 * (1.0 + t_2)) / (1.0 + t_0));
	} else if (t_4 <= 0.0) {
		tmp = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	} else {
		tmp = z1 - (t_3 * (1.0 / t_1));
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = math.exp((math.pi / z0))
	t_1 = t_0 - -1.0
	t_2 = math.exp((-3.1415927410125732 / z0))
	t_3 = -1.0 - t_2
	t_4 = z1 - (t_3 * ((1.0 - z1) / t_1))
	tmp = 0
	if t_4 <= -4e-308:
		tmp = z1 - ((z1 * (1.0 + t_2)) / (1.0 + t_0))
	elif t_4 <= 0.0:
		tmp = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	else:
		tmp = z1 - (t_3 * (1.0 / t_1))
	return tmp

Julia

function code(z1, z0)
	t_0 = exp(Float64(pi / z0))
	t_1 = Float64(t_0 - -1.0)
	t_2 = exp(Float64(-3.1415927410125732 / z0))
	t_3 = Float64(-1.0 - t_2)
	t_4 = Float64(z1 - Float64(t_3 * Float64(Float64(1.0 - z1) / t_1)))
	tmp = 0.0
	if (t_4 <= -4e-308)
		tmp = Float64(z1 - Float64(Float64(z1 * Float64(1.0 + t_2)) / Float64(1.0 + t_0)));
	elseif (t_4 <= 0.0)
		tmp = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)));
	else
		tmp = Float64(z1 - Float64(t_3 * Float64(1.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = exp((pi / z0));
	t_1 = t_0 - -1.0;
	t_2 = exp((-3.1415927410125732 / z0));
	t_3 = -1.0 - t_2;
	t_4 = z1 - (t_3 * ((1.0 - z1) / t_1));
	tmp = 0.0;
	if (t_4 <= -4e-308)
		tmp = z1 - ((z1 * (1.0 + t_2)) / (1.0 + t_0));
	elseif (t_4 <= 0.0)
		tmp = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	else
		tmp = z1 - (t_3 * (1.0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[Exp[N[(Pi / z0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(z1 - N[(t$95$3 * N[(N[(1.0 - z1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-308], N[(z1 - N[(N[(z1 * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z1 - N[(t$95$3 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z1 (*.f64 (-.f64 #s(literal -1 binary64) (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z0))) (/.f64 (-.f64 #s(literal 1 binary64) z1) (-.f64 (exp.f64 (/.f64 (PI.f64) z0)) #s(literal -1 binary64))))) < -4.0000000000000001e-308

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. lower--.f64 27.8%

         \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

   4. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      3. lift--.f64 N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      4. sub-negate-rev N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      5. lift--.f64 27.8%

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   6. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto z1 - -1 \]
   8. Step-by-step derivation

   9. Applied rewrites 37.4%

      \[\leadsto z1 - -1 \]

   10. Taylor expanded in z1 around inf

       \[\leadsto z1 - \color{blue}{\frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\pi}{z0}}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{\color{blue}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{\color{blue}{1} + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       3. lower-+.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       4. lower-exp.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       5. lower-/.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       6. lower-+.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{z0}}}} \]

       7. lower-exp.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       8. lower-/.f64 N/A

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{z0}}} \]

       9. lower-PI.f64 44.5%

          \[\leadsto z1 - \frac{z1 \cdot \left(1 + e^{\frac{-3.1415927410125732}{z0}}\right)}{1 + e^{\frac{\pi}{z0}}} \]

   12. Applied rewrites 44.5%

       \[\leadsto z1 - \color{blue}{\frac{z1 \cdot \left(1 + e^{\frac{-3.1415927410125732}{z0}}\right)}{1 + e^{\frac{\pi}{z0}}}} \]

   if -4.0000000000000001e-308 < (-.f64 z1 (*.f64 (-.f64 #s(literal -1 binary64) (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z0))) (/.f64 (-.f64 #s(literal 1 binary64) z1) (-.f64 (exp.f64 (/.f64 (PI.f64) z0)) #s(literal -1 binary64))))) < 0.0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if 0.0 < (-.f64 z1 (*.f64 (-.f64 #s(literal -1 binary64) (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z0))) (/.f64 (-.f64 #s(literal 1 binary64) z1) (-.f64 (exp.f64 (/.f64 (PI.f64) z0)) #s(literal -1 binary64)))))

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{\color{blue}{1}}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 68.1%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{\color{blue}{1}}{e^{\frac{\pi}{z0}} - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \left(1 - z1\right)\\ t_1 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.0065:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z0 \leq -6.5 \cdot 10^{-307}:\\ \;\;\;\;z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \frac{1}{-0.5 \cdot \frac{\pi}{t\_0} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* z0 (- 1.0 z1)))
       (t_1
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.0065)
    t_1
    (if (<= z0 -6.5e-307)
      (-
       z1
       (*
        (- -1.0 (exp (/ -3.1415927410125732 z0)))
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (/
          1.0
          (-
           (* -0.5 (/ PI t_0))
           (+ (/ 1.0 (- 1.0 z1)) (/ 1.5707963705062866 t_0)))))
        t_1)))))

C

double code(double z1, double z0) {
	double t_0 = z0 * (1.0 - z1);
	double t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_1;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((-0.5 * (((double) M_PI) / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = z0 * (1.0 - z1);
	double t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.0065) {
		tmp = t_1;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - ((-1.0 - Math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((-0.5 * (Math.PI / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = z0 * (1.0 - z1)
	t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.0065:
		tmp = t_1
	elif z0 <= -6.5e-307:
		tmp = z1 - ((-1.0 - math.exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (1.0 / ((-0.5 * (math.pi / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(z0 * Float64(1.0 - z1))
	t_1 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.0065)
		tmp = t_1;
	elseif (z0 <= -6.5e-307)
		tmp = Float64(z1 - Float64(Float64(-1.0 - exp(Float64(-3.1415927410125732 / z0))) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(1.0 / Float64(Float64(-0.5 * Float64(pi / t_0)) - Float64(Float64(1.0 / Float64(1.0 - z1)) + Float64(1.5707963705062866 / t_0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = z0 * (1.0 - z1);
	t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.0065)
		tmp = t_1;
	elseif (z0 <= -6.5e-307)
		tmp = z1 - ((-1.0 - exp((-3.1415927410125732 / z0))) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (1.0 / ((-0.5 * (pi / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(z0 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.0065], t$95$1, If[LessEqual[z0, -6.5e-307], N[(z1 - N[(N[(-1.0 - N[Exp[N[(-3.1415927410125732 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(1.0 / N[(N[(-0.5 * N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] + N[(1.5707963705062866 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0064999999999999997 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0064999999999999997 < z0 < -6.5000000000000001e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 54.8%

         \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   4. Applied rewrites 54.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -6.5000000000000001e-307 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - \color{blue}{-2 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - -2 \cdot \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 51.1%

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in z0 around -inf

      \[\leadsto z1 - \frac{1}{\color{blue}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \color{blue}{\left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \left(\color{blue}{\frac{1}{1 - z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{\color{blue}{1 - z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      4. lower-PI.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{\color{blue}{1} - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - \color{blue}{z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \color{blue}{\frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\color{blue}{\frac{7853981852531433}{5000000000000000}}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      9. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{\color{blue}{z0 \cdot \left(1 - z1\right)}}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \color{blue}{\left(1 - z1\right)}}\right)} \]

      12. lower--.f64 45.0%

          \[\leadsto z1 - \frac{1}{-0.5 \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{z0 \cdot \left(1 - \color{blue}{z1}\right)}\right)} \]

   9. Applied rewrites 45.0%

      \[\leadsto z1 - \frac{1}{\color{blue}{-0.5 \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{z0 \cdot \left(1 - z1\right)}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \left(1 - z1\right)\\ t_1 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z0 \leq -6.5 \cdot 10^{-307}:\\ \;\;\;\;z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \frac{1}{-0.5 \cdot \frac{\pi}{t\_0} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* z0 (- 1.0 z1)))
       (t_1
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -1.35e+53)
    t_1
    (if (<= z0 -6.5e-307)
      (-
       z1
       (*
        (-
         (*
          -1.0
          (/
           (-
            (*
             -1.0
             (/
              (- (* 5.167713211464109 (/ 1.0 z0)) 4.9348024751914465)
              z0))
            3.1415927410125732)
           z0))
         2.0)
        (+ (* -0.25 (/ (* PI (- 1.0 z1)) z0)) (* 0.5 (- 1.0 z1)))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (/
          1.0
          (-
           (* -0.5 (/ PI t_0))
           (+ (/ 1.0 (- 1.0 z1)) (/ 1.5707963705062866 t_0)))))
        t_1)))))

C

double code(double z1, double z0) {
	double t_0 = z0 * (1.0 - z1);
	double t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_1;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * ((-0.25 * ((((double) M_PI) * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((-0.5 * (((double) M_PI) / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = z0 * (1.0 - z1);
	double t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_1;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * ((-0.25 * ((Math.PI * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((-0.5 * (Math.PI / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = z0 * (1.0 - z1)
	t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_1
	elif z0 <= -6.5e-307:
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * ((-0.25 * ((math.pi * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))))
	elif z0 <= 2.7e+21:
		tmp = z1 - (1.0 / ((-0.5 * (math.pi / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(z0 * Float64(1.0 - z1))
	t_1 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_1;
	elseif (z0 <= -6.5e-307)
		tmp = Float64(z1 - Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(5.167713211464109 * Float64(1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * Float64(Float64(-0.25 * Float64(Float64(pi * Float64(1.0 - z1)) / z0)) + Float64(0.5 * Float64(1.0 - z1)))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(1.0 / Float64(Float64(-0.5 * Float64(pi / t_0)) - Float64(Float64(1.0 / Float64(1.0 - z1)) + Float64(1.5707963705062866 / t_0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = z0 * (1.0 - z1);
	t_1 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_1;
	elseif (z0 <= -6.5e-307)
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * ((-0.25 * ((pi * (1.0 - z1)) / z0)) + (0.5 * (1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (1.0 / ((-0.5 * (pi / t_0)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(z0 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$1, If[LessEqual[z0, -6.5e-307], N[(z1 - N[(N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(5.167713211464109 * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision] - 4.9348024751914465), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] - 3.1415927410125732), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(N[(-0.25 * N[(N[(Pi * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(1.0 / N[(N[(-0.5 * N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] + N[(1.5707963705062866 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < -6.5000000000000001e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - \color{blue}{2}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      4. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      5. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      6. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      7. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      8. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      9. lower-/.f64 55.3%

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   4. Applied rewrites 55.3%

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \color{blue}{\frac{1}{2} \cdot \left(1 - z1\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \color{blue}{\frac{1}{2}} \cdot \left(1 - z1\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \left(1 - z1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{-1}{4} \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + \frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      8. lower--.f64 48.8%

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   7. Applied rewrites 48.8%

      \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \color{blue}{\left(-0.25 \cdot \frac{\pi \cdot \left(1 - z1\right)}{z0} + 0.5 \cdot \left(1 - z1\right)\right)} \]

   if -6.5000000000000001e-307 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - \color{blue}{-2 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - -2 \cdot \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 51.1%

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in z0 around -inf

      \[\leadsto z1 - \frac{1}{\color{blue}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \color{blue}{\left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \left(\color{blue}{\frac{1}{1 - z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{\color{blue}{1 - z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      4. lower-PI.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{\color{blue}{1} - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - \color{blue}{z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \color{blue}{\frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\color{blue}{\frac{7853981852531433}{5000000000000000}}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      9. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{\color{blue}{z0 \cdot \left(1 - z1\right)}}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \color{blue}{\left(1 - z1\right)}}\right)} \]

      12. lower--.f64 45.0%

          \[\leadsto z1 - \frac{1}{-0.5 \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{z0 \cdot \left(1 - \color{blue}{z1}\right)}\right)} \]

   9. Applied rewrites 45.0%

      \[\leadsto z1 - \frac{1}{\color{blue}{-0.5 \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{z0 \cdot \left(1 - z1\right)}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 74.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ t_1 := z0 \cdot \left(1 - z1\right)\\ \mathbf{if}\;z0 \leq -0.00105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -6.5 \cdot 10^{-307}:\\ \;\;\;\;z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - \frac{1}{-0.5 \cdot \frac{\pi}{t\_1} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0))))
       (t_1 (* z0 (- 1.0 z1))))
  (if (<= z0 -0.00105)
    t_0
    (if (<= z0 -6.5e-307)
      (-
       z1
       (*
        (-
         (*
          -1.0
          (/
           (-
            (*
             -1.0
             (/
              (- (* 5.167713211464109 (/ 1.0 z0)) 4.9348024751914465)
              z0))
            3.1415927410125732)
           z0))
         2.0)
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (-
         z1
         (/
          1.0
          (-
           (* -0.5 (/ PI t_1))
           (+ (/ 1.0 (- 1.0 z1)) (/ 1.5707963705062866 t_1)))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double t_1 = z0 * (1.0 - z1);
	double tmp;
	if (z0 <= -0.00105) {
		tmp = t_0;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((-0.5 * (((double) M_PI) / t_1)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double t_1 = z0 * (1.0 - z1);
	double tmp;
	if (z0 <= -0.00105) {
		tmp = t_0;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (1.0 / ((-0.5 * (Math.PI / t_1)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	t_1 = z0 * (1.0 - z1)
	tmp = 0
	if z0 <= -0.00105:
		tmp = t_0
	elif z0 <= -6.5e-307:
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (1.0 / ((-0.5 * (math.pi / t_1)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	t_1 = Float64(z0 * Float64(1.0 - z1))
	tmp = 0.0
	if (z0 <= -0.00105)
		tmp = t_0;
	elseif (z0 <= -6.5e-307)
		tmp = Float64(z1 - Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(5.167713211464109 * Float64(1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(1.0 / Float64(Float64(-0.5 * Float64(pi / t_1)) - Float64(Float64(1.0 / Float64(1.0 - z1)) + Float64(1.5707963705062866 / t_1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	t_1 = z0 * (1.0 - z1);
	tmp = 0.0;
	if (z0 <= -0.00105)
		tmp = t_0;
	elseif (z0 <= -6.5e-307)
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (1.0 / ((-0.5 * (pi / t_1)) - ((1.0 / (1.0 - z1)) + (1.5707963705062866 / t_1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z0 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.00105], t$95$0, If[LessEqual[z0, -6.5e-307], N[(z1 - N[(N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(5.167713211464109 * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision] - 4.9348024751914465), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] - 3.1415927410125732), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(1.0 / N[(N[(-0.5 * N[(Pi / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] + N[(1.5707963705062866 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0010499999999999999 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0010499999999999999 < z0 < -6.5000000000000001e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - \color{blue}{2}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      4. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      5. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      6. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      7. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      8. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      9. lower-/.f64 55.3%

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   4. Applied rewrites 55.3%

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 47.0%

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   7. Applied rewrites 47.0%

      \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -6.5000000000000001e-307 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - \color{blue}{-2 \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - -2 \cdot \color{blue}{\frac{1 - z1}{e^{\frac{\pi}{z0}} - -1}} \]

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 51.1%

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in z0 around -inf

      \[\leadsto z1 - \frac{1}{\color{blue}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \color{blue}{\left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \left(\color{blue}{\frac{1}{1 - z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{\color{blue}{1 - z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      4. lower-PI.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{\color{blue}{1} - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - \color{blue}{z1}} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \color{blue}{\frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\color{blue}{\frac{7853981852531433}{5000000000000000}}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      9. lower--.f64 N/A

         \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \left(1 - z1\right)}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{\color{blue}{z0 \cdot \left(1 - z1\right)}}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto z1 - \frac{1}{\frac{-1}{2} \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{\frac{7853981852531433}{5000000000000000}}{z0 \cdot \color{blue}{\left(1 - z1\right)}}\right)} \]

      12. lower--.f64 45.0%

          \[\leadsto z1 - \frac{1}{-0.5 \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{z0 \cdot \left(1 - \color{blue}{z1}\right)}\right)} \]

   9. Applied rewrites 45.0%

      \[\leadsto z1 - \frac{1}{\color{blue}{-0.5 \cdot \frac{\pi}{z0 \cdot \left(1 - z1\right)} - \left(\frac{1}{1 - z1} + \frac{1.5707963705062866}{z0 \cdot \left(1 - z1\right)}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 74.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -0.00105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -6.5 \cdot 10^{-307}:\\ \;\;\;\;z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \left(0.5 \cdot \left(1 - z1\right)\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -0.00105)
    t_0
    (if (<= z0 -6.5e-307)
      (-
       z1
       (*
        (-
         (*
          -1.0
          (/
           (-
            (*
             -1.0
             (/
              (- (* 5.167713211464109 (/ 1.0 z0)) 4.9348024751914465)
              z0))
            3.1415927410125732)
           z0))
         2.0)
        (* 0.5 (- 1.0 z1))))
      (if (<= z0 2.7e+21)
        (- z1 (* -2.0 (/ (- 1.0 z1) (+ 2.0 (/ PI z0)))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -0.00105) {
		tmp = t_0;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (((double) M_PI) / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -0.00105) {
		tmp = t_0;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (Math.PI / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -0.00105:
		tmp = t_0
	elif z0 <= -6.5e-307:
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)))
	elif z0 <= 2.7e+21:
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (math.pi / z0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -0.00105)
		tmp = t_0;
	elseif (z0 <= -6.5e-307)
		tmp = Float64(z1 - Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(5.167713211464109 * Float64(1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * Float64(0.5 * Float64(1.0 - z1))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(-2.0 * Float64(Float64(1.0 - z1) / Float64(2.0 + Float64(pi / z0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -0.00105)
		tmp = t_0;
	elseif (z0 <= -6.5e-307)
		tmp = z1 - (((-1.0 * (((-1.0 * (((5.167713211464109 * (1.0 / z0)) - 4.9348024751914465) / z0)) - 3.1415927410125732) / z0)) - 2.0) * (0.5 * (1.0 - z1)));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (pi / z0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -0.00105], t$95$0, If[LessEqual[z0, -6.5e-307], N[(z1 - N[(N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(5.167713211464109 * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision] - 4.9348024751914465), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] - 3.1415927410125732), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(0.5 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(-2.0 * N[(N[(1.0 - z1), $MachinePrecision] / N[(2.0 + N[(Pi / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -0.0010499999999999999 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -0.0010499999999999999 < z0 < -6.5000000000000001e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - \color{blue}{2}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      3. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      4. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      5. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      6. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      7. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      8. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

      9. lower-/.f64 55.3%

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   4. Applied rewrites 55.3%

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right)} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(1 - z1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{\frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z0} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000}}{z0} - \frac{7853981852531433}{2500000000000000}}{z0} - 2\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(1 - z1\right)}\right) \]

      2. lower--.f64 47.0%

         \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \left(0.5 \cdot \left(1 - \color{blue}{z1}\right)\right) \]

   7. Applied rewrites 47.0%

      \[\leadsto z1 - \left(-1 \cdot \frac{-1 \cdot \frac{5.167713211464109 \cdot \frac{1}{z0} - 4.9348024751914465}{z0} - 3.1415927410125732}{z0} - 2\right) \cdot \color{blue}{\left(0.5 \cdot \left(1 - z1\right)\right)} \]

   if -6.5000000000000001e-307 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 44.5%

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}} \]

   7. Applied rewrites 44.5%

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 71.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -1.36 \cdot 10^{-154}:\\ \;\;\;\;\left(1 - \frac{\left(1 - z1\right) \cdot \left(\frac{0.5 \cdot \pi - -1.5707963705062866}{z0} + -1\right)}{z1}\right) \cdot z1\\ \mathbf{elif}\;z0 \leq -6.5 \cdot 10^{-307}:\\ \;\;\;\;z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(-1.5707963705062866 \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\left(1 - z1\right) \cdot \pi\right) \cdot 0.5\right)}{z0 \cdot z0}\right)\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 -1.36e-154)
      (*
       (-
        1.0
        (/
         (*
          (- 1.0 z1)
          (+ (/ (- (* 0.5 PI) -1.5707963705062866) z0) -1.0))
         z1))
       z1)
      (if (<= z0 -6.5e-307)
        (-
         z1
         (+
          (* -1.0 (- 1.0 z1))
          (*
           -1.0
           (/
            (-
             (* (* -1.5707963705062866 (- 1.0 z1)) z0)
             (* z0 (* (* (- 1.0 z1) PI) 0.5)))
            (* z0 z0)))))
        (if (<= z0 2.7e+21)
          (- z1 (* -2.0 (/ (- 1.0 z1) (+ 2.0 (/ PI z0)))))
          t_0))))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -1.36e-154) {
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * ((double) M_PI)) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - ((-1.0 * (1.0 - z1)) + (-1.0 * ((((-1.5707963705062866 * (1.0 - z1)) * z0) - (z0 * (((1.0 - z1) * ((double) M_PI)) * 0.5))) / (z0 * z0))));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (((double) M_PI) / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -1.36e-154) {
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * Math.PI) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1;
	} else if (z0 <= -6.5e-307) {
		tmp = z1 - ((-1.0 * (1.0 - z1)) + (-1.0 * ((((-1.5707963705062866 * (1.0 - z1)) * z0) - (z0 * (((1.0 - z1) * Math.PI) * 0.5))) / (z0 * z0))));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (Math.PI / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= -1.36e-154:
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * math.pi) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1
	elif z0 <= -6.5e-307:
		tmp = z1 - ((-1.0 * (1.0 - z1)) + (-1.0 * ((((-1.5707963705062866 * (1.0 - z1)) * z0) - (z0 * (((1.0 - z1) * math.pi) * 0.5))) / (z0 * z0))))
	elif z0 <= 2.7e+21:
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (math.pi / z0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -1.36e-154)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 - z1) * Float64(Float64(Float64(Float64(0.5 * pi) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1);
	elseif (z0 <= -6.5e-307)
		tmp = Float64(z1 - Float64(Float64(-1.0 * Float64(1.0 - z1)) + Float64(-1.0 * Float64(Float64(Float64(Float64(-1.5707963705062866 * Float64(1.0 - z1)) * z0) - Float64(z0 * Float64(Float64(Float64(1.0 - z1) * pi) * 0.5))) / Float64(z0 * z0)))));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(-2.0 * Float64(Float64(1.0 - z1) / Float64(2.0 + Float64(pi / z0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -1.36e-154)
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * pi) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1;
	elseif (z0 <= -6.5e-307)
		tmp = z1 - ((-1.0 * (1.0 - z1)) + (-1.0 * ((((-1.5707963705062866 * (1.0 - z1)) * z0) - (z0 * (((1.0 - z1) * pi) * 0.5))) / (z0 * z0))));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (pi / z0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, -1.36e-154], N[(N[(1.0 - N[(N[(N[(1.0 - z1), $MachinePrecision] * N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] - -1.5707963705062866), $MachinePrecision] / z0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision], If[LessEqual[z0, -6.5e-307], N[(z1 - N[(N[(-1.0 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(N[(N[(-1.5707963705062866 * N[(1.0 - z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(z0 * N[(N[(N[(1.0 - z1), $MachinePrecision] * Pi), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z0 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(-2.0 * N[(N[(1.0 - z1), $MachinePrecision] / N[(2.0 + N[(Pi / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < -1.36e-154

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \color{blue}{-1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \color{blue}{-1} \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      3. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \color{blue}{\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{\color{blue}{z0}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      8. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      12. lower--.f64 35.6%

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{-1.5707963705062866 \cdot \left(1 - z1\right) - 0.5 \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

   4. Applied rewrites 35.6%

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{-1.5707963705062866 \cdot \left(1 - z1\right) - 0.5 \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \color{blue}{\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)\right)\right) \]

      4. mult-flip N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right) \cdot \frac{1}{z0}\right)\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{z0}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{z0}}\right) \]

   6. Applied rewrites 35.6%

      \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\left(1 - z1\right) \cdot \left(0.5 \cdot \pi - -1.5707963705062866\right)\right) \cdot \color{blue}{\frac{1}{z0}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\left(1 - \frac{\left(1 - z1\right) \cdot \left(\frac{0.5 \cdot \pi - -1.5707963705062866}{z0} + -1\right)}{z1}\right) \cdot z1} \]

   if -1.36e-154 < z0 < -6.5000000000000001e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \color{blue}{-1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \color{blue}{-1} \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      3. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \color{blue}{\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{\color{blue}{z0}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      8. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      12. lower--.f64 35.6%

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{-1.5707963705062866 \cdot \left(1 - z1\right) - 0.5 \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

   4. Applied rewrites 35.6%

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{-1.5707963705062866 \cdot \left(1 - z1\right) - 0.5 \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{\color{blue}{z0}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      3. div-sub N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \left(\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)}{z0} - \color{blue}{\frac{\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}}\right)\right) \]

      4. frac-sub N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)}{\color{blue}{z0 \cdot z0}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)}{\color{blue}{z0 \cdot z0}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)}{\color{blue}{z0} \cdot z0}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)}{z0 \cdot z0}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)}{z0 \cdot z0}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)}{z0 \cdot z0}\right) \]

      10. *-commutative N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\pi \cdot \left(1 - z1\right)\right) \cdot \frac{1}{2}\right)}{z0 \cdot z0}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\pi \cdot \left(1 - z1\right)\right) \cdot \frac{1}{2}\right)}{z0 \cdot z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\pi \cdot \left(1 - z1\right)\right) \cdot \frac{1}{2}\right)}{z0 \cdot z0}\right) \]

      13. *-commutative N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\left(1 - z1\right) \cdot \pi\right) \cdot \frac{1}{2}\right)}{z0 \cdot z0}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\left(1 - z1\right) \cdot \pi\right) \cdot \frac{1}{2}\right)}{z0 \cdot z0}\right) \]

      15. lower-*.f64 42.0%

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(-1.5707963705062866 \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\left(1 - z1\right) \cdot \pi\right) \cdot 0.5\right)}{z0 \cdot \color{blue}{z0}}\right) \]

   6. Applied rewrites 42.0%

      \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\left(-1.5707963705062866 \cdot \left(1 - z1\right)\right) \cdot z0 - z0 \cdot \left(\left(\left(1 - z1\right) \cdot \pi\right) \cdot 0.5\right)}{\color{blue}{z0 \cdot z0}}\right) \]

   if -6.5000000000000001e-307 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 44.5%

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}} \]

   7. Applied rewrites 44.5%

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 69.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -6.5 \cdot 10^{-307}:\\ \;\;\;\;\left(1 - \frac{\left(1 - z1\right) \cdot \left(\frac{0.5 \cdot \pi - -1.5707963705062866}{z0} + -1\right)}{z1}\right) \cdot z1\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -1.35e+53)
    t_0
    (if (<= z0 -6.5e-307)
      (*
       (-
        1.0
        (/
         (*
          (- 1.0 z1)
          (+ (/ (- (* 0.5 PI) -1.5707963705062866) z0) -1.0))
         z1))
       z1)
      (if (<= z0 2.7e+21)
        (- z1 (* -2.0 (/ (- 1.0 z1) (+ 2.0 (/ PI z0)))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -6.5e-307) {
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * ((double) M_PI)) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1;
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (((double) M_PI) / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -1.35e+53) {
		tmp = t_0;
	} else if (z0 <= -6.5e-307) {
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * Math.PI) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1;
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (Math.PI / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -1.35e+53:
		tmp = t_0
	elif z0 <= -6.5e-307:
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * math.pi) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1
	elif z0 <= 2.7e+21:
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (math.pi / z0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -6.5e-307)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 - z1) * Float64(Float64(Float64(Float64(0.5 * pi) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1);
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(-2.0 * Float64(Float64(1.0 - z1) / Float64(2.0 + Float64(pi / z0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -1.35e+53)
		tmp = t_0;
	elseif (z0 <= -6.5e-307)
		tmp = (1.0 - (((1.0 - z1) * ((((0.5 * pi) - -1.5707963705062866) / z0) + -1.0)) / z1)) * z1;
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (pi / z0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+53], t$95$0, If[LessEqual[z0, -6.5e-307], N[(N[(1.0 - N[(N[(N[(1.0 - z1), $MachinePrecision] * N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] - -1.5707963705062866), $MachinePrecision] / z0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(-2.0 * N[(N[(1.0 - z1), $MachinePrecision] / N[(2.0 + N[(Pi / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.3500000000000001e53 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -1.3500000000000001e53 < z0 < -6.5000000000000001e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \color{blue}{-1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \color{blue}{-1} \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      3. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \color{blue}{\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{\color{blue}{z0}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      8. lower--.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

      12. lower--.f64 35.6%

          \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{-1.5707963705062866 \cdot \left(1 - z1\right) - 0.5 \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right) \]

   4. Applied rewrites 35.6%

      \[\leadsto z1 - \color{blue}{\left(-1 \cdot \left(1 - z1\right) + -1 \cdot \frac{-1.5707963705062866 \cdot \left(1 - z1\right) - 0.5 \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + -1 \cdot \color{blue}{\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}}\right) \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\frac{\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)}{z0}\right)\right)\right) \]

      4. mult-flip N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right) \cdot \frac{1}{z0}\right)\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{z0}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} \cdot \left(1 - z1\right) - \frac{1}{2} \cdot \left(\pi \cdot \left(1 - z1\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{z0}}\right) \]

   6. Applied rewrites 35.6%

      \[\leadsto z1 - \left(-1 \cdot \left(1 - z1\right) + \left(\left(1 - z1\right) \cdot \left(0.5 \cdot \pi - -1.5707963705062866\right)\right) \cdot \color{blue}{\frac{1}{z0}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\left(1 - \frac{\left(1 - z1\right) \cdot \left(\frac{0.5 \cdot \pi - -1.5707963705062866}{z0} + -1\right)}{z1}\right) \cdot z1} \]

   if -6.5000000000000001e-307 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 44.5%

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}} \]

   7. Applied rewrites 44.5%

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 61.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -13000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 2.3 \cdot 10^{-300}:\\ \;\;\;\;z1 - \frac{z1 \cdot z1 - 1 \cdot 1}{z1 + 1}\\ \mathbf{elif}\;z0 \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -13000.0)
    t_0
    (if (<= z0 2.3e-300)
      (- z1 (/ (- (* z1 z1) (* 1.0 1.0)) (+ z1 1.0)))
      (if (<= z0 2.7e+21)
        (- z1 (* -2.0 (/ (- 1.0 z1) (+ 2.0 (/ PI z0)))))
        t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -13000.0) {
		tmp = t_0;
	} else if (z0 <= 2.3e-300) {
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (((double) M_PI) / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -13000.0) {
		tmp = t_0;
	} else if (z0 <= 2.3e-300) {
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0));
	} else if (z0 <= 2.7e+21) {
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (Math.PI / z0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -13000.0:
		tmp = t_0
	elif z0 <= 2.3e-300:
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0))
	elif z0 <= 2.7e+21:
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (math.pi / z0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -13000.0)
		tmp = t_0;
	elseif (z0 <= 2.3e-300)
		tmp = Float64(z1 - Float64(Float64(Float64(z1 * z1) - Float64(1.0 * 1.0)) / Float64(z1 + 1.0)));
	elseif (z0 <= 2.7e+21)
		tmp = Float64(z1 - Float64(-2.0 * Float64(Float64(1.0 - z1) / Float64(2.0 + Float64(pi / z0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -13000.0)
		tmp = t_0;
	elseif (z0 <= 2.3e-300)
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0));
	elseif (z0 <= 2.7e+21)
		tmp = z1 - (-2.0 * ((1.0 - z1) / (2.0 + (pi / z0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -13000.0], t$95$0, If[LessEqual[z0, 2.3e-300], N[(z1 - N[(N[(N[(z1 * z1), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(z1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 2.7e+21], N[(z1 - N[(-2.0 * N[(N[(1.0 - z1), $MachinePrecision] / N[(2.0 + N[(Pi / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -13000 or 2.7e21 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -13000 < z0 < 2.3e-300

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. lower--.f64 27.8%

         \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

   4. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      3. lift--.f64 N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      4. sub-negate-rev N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      5. lift--.f64 27.8%

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   6. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      2. flip-- N/A

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

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

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

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

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

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

         \[\leadsto z1 - \frac{z1 \cdot z1 - 1 \cdot 1}{z1 + 1} \]

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

         \[\leadsto z1 - \frac{z1 \cdot z1 - 1 \cdot 1}{z1 + 1} \]

      7. lower-unsound-+.f64 33.7%

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

   8. Applied rewrites 33.7%

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

   if 2.3e-300 < z0 < 2.7e21

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto z1 - \color{blue}{-2} \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 44.5%

         \[\leadsto z1 - -2 \cdot \frac{1 - z1}{2 + \frac{\pi}{z0}} \]

   7. Applied rewrites 44.5%

      \[\leadsto z1 - -2 \cdot \frac{1 - z1}{\color{blue}{2 + \frac{\pi}{z0}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 56.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -13000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq -5.6 \cdot 10^{-307}:\\ \;\;\;\;z1 - \frac{z1 \cdot z1 - 1 \cdot 1}{z1 + 1}\\ \mathbf{elif}\;z0 \leq 5.1 \cdot 10^{+77}:\\ \;\;\;\;z1 - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -13000.0)
    t_0
    (if (<= z0 -5.6e-307)
      (- z1 (/ (- (* z1 z1) (* 1.0 1.0)) (+ z1 1.0)))
      (if (<= z0 5.1e+77) (- z1 -1.0) t_0)))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -13000.0) {
		tmp = t_0;
	} else if (z0 <= -5.6e-307) {
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0));
	} else if (z0 <= 5.1e+77) {
		tmp = z1 - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -13000.0) {
		tmp = t_0;
	} else if (z0 <= -5.6e-307) {
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0));
	} else if (z0 <= 5.1e+77) {
		tmp = z1 - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -13000.0:
		tmp = t_0
	elif z0 <= -5.6e-307:
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0))
	elif z0 <= 5.1e+77:
		tmp = z1 - -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -13000.0)
		tmp = t_0;
	elseif (z0 <= -5.6e-307)
		tmp = Float64(z1 - Float64(Float64(Float64(z1 * z1) - Float64(1.0 * 1.0)) / Float64(z1 + 1.0)));
	elseif (z0 <= 5.1e+77)
		tmp = Float64(z1 - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -13000.0)
		tmp = t_0;
	elseif (z0 <= -5.6e-307)
		tmp = z1 - (((z1 * z1) - (1.0 * 1.0)) / (z1 + 1.0));
	elseif (z0 <= 5.1e+77)
		tmp = z1 - -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -13000.0], t$95$0, If[LessEqual[z0, -5.6e-307], N[(z1 - N[(N[(N[(z1 * z1), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(z1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 5.1e+77], N[(z1 - -1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -13000 or 5.0999999999999997e77 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -13000 < z0 < -5.6000000000000003e-307

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. lower--.f64 27.8%

         \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

   4. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      3. lift--.f64 N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      4. sub-negate-rev N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      5. lift--.f64 27.8%

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   6. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      2. flip-- N/A

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

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

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

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

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

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

         \[\leadsto z1 - \frac{z1 \cdot z1 - 1 \cdot 1}{z1 + 1} \]

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

         \[\leadsto z1 - \frac{z1 \cdot z1 - 1 \cdot 1}{z1 + 1} \]

      7. lower-unsound-+.f64 33.7%

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

   8. Applied rewrites 33.7%

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

   if -5.6000000000000003e-307 < z0 < 5.0999999999999997e77

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. lower--.f64 27.8%

         \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

   4. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      3. lift--.f64 N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      4. sub-negate-rev N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      5. lift--.f64 27.8%

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   6. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto z1 - -1 \]
   8. Step-by-step derivation

   9. Applied rewrites 37.4%

      \[\leadsto z1 - -1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}\\ \mathbf{if}\;z0 \leq -3.8 \cdot 10^{-296}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 5.1 \cdot 10^{+77}:\\ \;\;\;\;z1 - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (+ 1.0 (* -1.0 (/ (- 1.5707963705062866 (* -0.5 PI)) z0)))))
  (if (<= z0 -3.8e-296) t_0 (if (<= z0 5.1e+77) (- z1 -1.0) t_0))))

C

double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * ((double) M_PI))) / z0));
	double tmp;
	if (z0 <= -3.8e-296) {
		tmp = t_0;
	} else if (z0 <= 5.1e+77) {
		tmp = z1 - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * Math.PI)) / z0));
	double tmp;
	if (z0 <= -3.8e-296) {
		tmp = t_0;
	} else if (z0 <= 5.1e+77) {
		tmp = z1 - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * math.pi)) / z0))
	tmp = 0
	if z0 <= -3.8e-296:
		tmp = t_0
	elif z0 <= 5.1e+77:
		tmp = z1 - -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(1.5707963705062866 - Float64(-0.5 * pi)) / z0)))
	tmp = 0.0
	if (z0 <= -3.8e-296)
		tmp = t_0;
	elseif (z0 <= 5.1e+77)
		tmp = Float64(z1 - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = 1.0 + (-1.0 * ((1.5707963705062866 - (-0.5 * pi)) / z0));
	tmp = 0.0;
	if (z0 <= -3.8e-296)
		tmp = t_0;
	elseif (z0 <= 5.1e+77)
		tmp = z1 - -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(1.5707963705062866 - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -3.8e-296], t$95$0, If[LessEqual[z0, 5.1e+77], N[(z1 - -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -3.8000000000000002e-296 or 5.0999999999999997e77 < z0

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{e^{\frac{\pi}{z0}}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\pi}{z0}}} - -1} \]

      3. frac-2neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(z0\right)}}} - -1} \]

      4. distribute-frac-neg2 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\pi\right)}{z0}\right)}} - -1} \]

      5. exp-neg N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      6. distribute-neg-frac N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      7. lift-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      9. lower-exp.f64 N/A

         \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{\pi}{z0}\right)}}} - -1} \]

      10. lift-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{\pi}{z0}}\right)}} - -1} \]

      11. distribute-neg-frac N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      12. lower-/.f64 N/A

          \[\leadsto z1 - \left(-1 - e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{z0}}}} - -1} \]

      13. lower-neg.f64 76.8%

          \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z0}}} - -1} \]

   3. Applied rewrites 76.8%

      \[\leadsto z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z0}}}} - -1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{\color{blue}{1} + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \color{blue}{\frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{\color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1 + e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{z0}}}} \]

      10. lower-PI.f64 58.0%

          \[\leadsto \frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}} \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{1 + e^{\frac{-3.1415927410125732}{z0}}}{1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z0}}}}} \]

   7. Taylor expanded in z0 around -inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \pi}{z0}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{z0}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      4. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{7853981852531433}{5000000000000000} - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{z0} \]

      6. lower-PI.f64 39.5%

         \[\leadsto 1 + -1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0} \]

   9. Applied rewrites 39.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1.5707963705062866 - -0.5 \cdot \pi}{z0}} \]

   if -3.8000000000000002e-296 < z0 < 5.0999999999999997e77

   1. Initial program 76.8%

      \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. lower--.f64 27.8%

         \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

   4. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

      2. mul-1-neg N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      3. lift--.f64 N/A

         \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

      4. sub-negate-rev N/A

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

      5. lift--.f64 27.8%

         \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   6. Applied rewrites 27.8%

      \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto z1 - -1 \]
   8. Step-by-step derivation

   9. Applied rewrites 37.4%

      \[\leadsto z1 - -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 37.4% accurate, 62.8× speedup

Math

\[z1 - -1 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (- z1 -1.0))

C

double code(double z1, double z0) {
	return z1 - -1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = z1 - (-1.0d0)
end function

Java

public static double code(double z1, double z0) {
	return z1 - -1.0;
}

Python

def code(z1, z0):
	return z1 - -1.0

Julia

function code(z1, z0)
	return Float64(z1 - -1.0)
end

MATLAB

function tmp = code(z1, z0)
	tmp = z1 - -1.0;
end

Wolfram

code[z1_, z0_] := N[(z1 - -1.0), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[z1 - \left(-1 - e^{\frac{-3.1415927410125732}{z0}}\right) \cdot \frac{1 - z1}{e^{\frac{\pi}{z0}} - -1} \]

2. Taylor expanded in z0 around inf

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

   1. lower-*.f64 N/A

      \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

   2. lower--.f64 27.8%

      \[\leadsto z1 - -1 \cdot \left(1 - \color{blue}{z1}\right) \]

4. Applied rewrites 27.8%

   \[\leadsto z1 - \color{blue}{-1 \cdot \left(1 - z1\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto z1 - -1 \cdot \color{blue}{\left(1 - z1\right)} \]

   2. mul-1-neg N/A

      \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

   3. lift--.f64 N/A

      \[\leadsto z1 - \left(\mathsf{neg}\left(\left(1 - z1\right)\right)\right) \]

   4. sub-negate-rev N/A

      \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

   5. lift--.f64 27.8%

      \[\leadsto z1 - \left(z1 - \color{blue}{1}\right) \]

6. Applied rewrites 27.8%

   \[\leadsto z1 - \color{blue}{\left(z1 - 1\right)} \]

7. Taylor expanded in z1 around 0

   \[\leadsto z1 - -1 \]
8. Step-by-step derivation

9. Applied rewrites 37.4%

   \[\leadsto z1 - -1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(- z1 (* (- -1 (exp (/ -7853981852531433/2500000000000000 z0))) (/ (- 1 z1) (- (exp (/ PI z0)) -1))))"
  :precision binary64
  (- z1 (* (- -1.0 (exp (/ -3.1415927410125732 z0))) (/ (- 1.0 z1) (- (exp (/ PI z0)) -1.0)))))