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

Percentage Accurate: 78.1% → 99.2%

Time: 4.3s

Alternatives: 21

Speedup: 1.5×

Specification

Math

\[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% accurate, 1.0× speedup

Math

\[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ t_1 := e^{\frac{\pi}{z1}} - -1\\ \mathbf{if}\;z1 \leq -300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{t\_1}}\\ \mathbf{elif}\;z1 \leq 30000000:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1)))))
       (t_1 (- (exp (/ PI z1)) -1.0)))
  (if (<= z1 -300000.0)
    t_0
    (if (<= z1 -4e-306)
      (/
       1.0
       (+
        (/
         z0
         (+
          2.0
          (*
           -1.0
           (/
            (+
             3.1415927410125732
             (*
              -1.0
              (/
               (- 4.9348024751914465 (* 5.167713211464109 (/ 1.0 z1)))
               z1)))
            z1))))
        (/ (- 1.0 z0) t_1)))
      (if (<= z1 30000000.0)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ 1.0 t_1)))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double t_1 = exp((((double) M_PI) / z1)) - -1.0;
	double tmp;
	if (z1 <= -300000.0) {
		tmp = t_0;
	} else if (z1 <= -4e-306) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / t_1));
	} else if (z1 <= 30000000.0) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double t_1 = Math.exp((Math.PI / z1)) - -1.0;
	double tmp;
	if (z1 <= -300000.0) {
		tmp = t_0;
	} else if (z1 <= -4e-306) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / t_1));
	} else if (z1 <= 30000000.0) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	t_1 = math.exp((math.pi / z1)) - -1.0
	tmp = 0
	if z1 <= -300000.0:
		tmp = t_0
	elif z1 <= -4e-306:
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / t_1))
	elif z1 <= 30000000.0:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	t_1 = Float64(exp(Float64(pi / z1)) - -1.0)
	tmp = 0.0
	if (z1 <= -300000.0)
		tmp = t_0;
	elseif (z1 <= -4e-306)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 + Float64(-1.0 * Float64(Float64(3.1415927410125732 + Float64(-1.0 * Float64(Float64(4.9348024751914465 - Float64(5.167713211464109 * Float64(1.0 / z1))) / z1))) / z1)))) + Float64(Float64(1.0 - z0) / t_1)));
	elseif (z1 <= 30000000.0)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(1.0 / t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	t_1 = exp((pi / z1)) - -1.0;
	tmp = 0.0;
	if (z1 <= -300000.0)
		tmp = t_0;
	elseif (z1 <= -4e-306)
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / t_1));
	elseif (z1 <= 30000000.0)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[z1, -300000.0], t$95$0, If[LessEqual[z1, -4e-306], N[(1.0 / N[(N[(z0 / N[(2.0 + N[(-1.0 * N[(N[(3.1415927410125732 + N[(-1.0 * N[(N[(4.9348024751914465 - N[(5.167713211464109 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 30000000.0], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -3e5 or 3e7 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -3e5 < z1 < -4.0000000000000001e-306

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{\color{blue}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      9. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 53.2%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if -4.0000000000000001e-306 < z1 < 3e7

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{\color{blue}{1}}{e^{\frac{\pi}{z1}} - -1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.8%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1}\\ \mathbf{if}\;\frac{1}{t\_0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{t\_0 + \frac{1 - z0}{\frac{1}{e^{\frac{-\pi}{z1}}} - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))))
  (if (<=
       (/ 1.0 (+ t_0 (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
       2e+282)
    (/
     1.0
     (+ t_0 (/ (- 1.0 z0) (- (/ 1.0 (exp (/ (- PI) z1))) -1.0))))
    (/
     1.0
     (+
      0.5
      (*
       -1.0
       (/
        (-
         (* -0.25 (* z0 PI))
         (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
        z1)))))))

C

double code(double z0, double z1) {
	double t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	double tmp;
	if ((1.0 / (t_0 + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)))) <= 2e+282) {
		tmp = 1.0 / (t_0 + ((1.0 - z0) / ((1.0 / exp((-((double) M_PI) / z1))) - -1.0)));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0);
	double tmp;
	if ((1.0 / (t_0 + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)))) <= 2e+282) {
		tmp = 1.0 / (t_0 + ((1.0 - z0) / ((1.0 / Math.exp((-Math.PI / z1))) - -1.0)));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)
	tmp = 0
	if (1.0 / (t_0 + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))) <= 2e+282:
		tmp = 1.0 / (t_0 + ((1.0 - z0) / ((1.0 / math.exp((-math.pi / z1))) - -1.0)))
	else:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0))
	tmp = 0.0
	if (Float64(1.0 / Float64(t_0 + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0)))) <= 2e+282)
		tmp = Float64(1.0 / Float64(t_0 + Float64(Float64(1.0 - z0) / Float64(Float64(1.0 / exp(Float64(Float64(-pi) / z1))) - -1.0))));
	else
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	tmp = 0.0;
	if ((1.0 / (t_0 + ((1.0 - z0) / (exp((pi / z1)) - -1.0)))) <= 2e+282)
		tmp = 1.0 / (t_0 + ((1.0 - z0) / ((1.0 / exp((-pi / z1))) - -1.0)));
	else
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / N[(t$95$0 + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+282], N[(1.0 / N[(t$95$0 + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[(1.0 / N[Exp[N[((-Pi) / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 2.0000000000000001e282

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   if 2.0000000000000001e282 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64)))))

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1}\\ \mathbf{if}\;\frac{1}{t\_0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{t\_0 + \frac{1 - z0}{{23.14069263277927}^{\left(\frac{1}{z1}\right)} - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))))
  (if (<=
       (/ 1.0 (+ t_0 (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
       2e+282)
    (/
     1.0
     (+
      t_0
      (/ (- 1.0 z0) (- (pow 23.14069263277927 (/ 1.0 z1)) -1.0))))
    (/
     1.0
     (+
      0.5
      (*
       -1.0
       (/
        (-
         (* -0.25 (* z0 PI))
         (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
        z1)))))))

C

double code(double z0, double z1) {
	double t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	double tmp;
	if ((1.0 / (t_0 + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)))) <= 2e+282) {
		tmp = 1.0 / (t_0 + ((1.0 - z0) / (pow(23.14069263277927, (1.0 / z1)) - -1.0)));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0);
	double tmp;
	if ((1.0 / (t_0 + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)))) <= 2e+282) {
		tmp = 1.0 / (t_0 + ((1.0 - z0) / (Math.pow(23.14069263277927, (1.0 / z1)) - -1.0)));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)
	tmp = 0
	if (1.0 / (t_0 + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))) <= 2e+282:
		tmp = 1.0 / (t_0 + ((1.0 - z0) / (math.pow(23.14069263277927, (1.0 / z1)) - -1.0)))
	else:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0))
	tmp = 0.0
	if (Float64(1.0 / Float64(t_0 + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0)))) <= 2e+282)
		tmp = Float64(1.0 / Float64(t_0 + Float64(Float64(1.0 - z0) / Float64((23.14069263277927 ^ Float64(1.0 / z1)) - -1.0))));
	else
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	tmp = 0.0;
	if ((1.0 / (t_0 + ((1.0 - z0) / (exp((pi / z1)) - -1.0)))) <= 2e+282)
		tmp = 1.0 / (t_0 + ((1.0 - z0) / ((23.14069263277927 ^ (1.0 / z1)) - -1.0)));
	else
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / N[(t$95$0 + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+282], N[(1.0 / N[(t$95$0 + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Power[23.14069263277927, N[(1.0 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 2.0000000000000001e282

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. mult-flip N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\pi \cdot \frac{1}{z1}}} - -1}} \]

      4. exp-prod N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 78.1%

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

   3. Applied rewrites 78.1%

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

   4. Evaluated real constant 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{{\color{blue}{23.14069263277927}}^{\left(\frac{1}{z1}\right)} - -1}} \]

   if 2.0000000000000001e282 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64)))))

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))))
  (if (<= t_0 2e+282)
    t_0
    (/
     1.0
     (+
      0.5
      (*
       -1.0
       (/
        (-
         (* -0.25 (* z0 PI))
         (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
        z1)))))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	double tmp;
	if (t_0 <= 2e+282) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	double tmp;
	if (t_0 <= 2e+282) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	tmp = 0
	if t_0 <= 2e+282:
		tmp = t_0
	else:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))))
	tmp = 0.0
	if (t_0 <= 2e+282)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	tmp = 0.0;
	if (t_0 <= 2e+282)
		tmp = t_0;
	else
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+282], t$95$0, N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 2.0000000000000001e282

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if 2.0000000000000001e282 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64)))))

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{elif}\;z1 \leq 0.56:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -300000.0)
    t_0
    (if (<= z1 -5e-310)
      (/
       1.0
       (+
        (/
         z0
         (+
          2.0
          (*
           -1.0
           (/
            (+
             3.1415927410125732
             (*
              -1.0
              (/
               (- 4.9348024751914465 (* 5.167713211464109 (/ 1.0 z1)))
               z1)))
            z1))))
        (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
      (if (<= z1 0.56)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ (* -1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -300000.0) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -300000.0) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -300000.0:
		tmp = t_0
	elif z1 <= -5e-310:
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	elif z1 <= 0.56:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -300000.0)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 + Float64(-1.0 * Float64(Float64(3.1415927410125732 + Float64(-1.0 * Float64(Float64(4.9348024751914465 - Float64(5.167713211464109 * Float64(1.0 / z1))) / z1))) / z1)))) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))));
	elseif (z1 <= 0.56)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(-1.0 * z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -300000.0)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	elseif (z1 <= 0.56)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -300000.0], t$95$0, If[LessEqual[z1, -5e-310], N[(1.0 / N[(N[(z0 / N[(2.0 + N[(-1.0 * N[(N[(3.1415927410125732 + N[(-1.0 * N[(N[(4.9348024751914465 - N[(5.167713211464109 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 0.56], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -3e5 or 0.56000000000000005 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -3e5 < z1 < -4.9999999999999847e-310

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{\color{blue}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      9. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 53.2%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if -4.9999999999999847e-310 < z1 < 0.56000000000000005

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{\color{blue}{-1 \cdot z0}}{2 + \frac{\pi}{z1}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 27.0%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot \color{blue}{z0}}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 27.0%

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

Alternative 6: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -280000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 - \frac{3.1415927410125732 - \frac{4.9348024751914465}{z1}}{z1}} - \frac{-1 + z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{elif}\;z1 \leq 0.56:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -280000.0)
    t_0
    (if (<= z1 -5e-310)
      (/
       1.0
       (-
        (/
         z0
         (-
          2.0
          (/ (- 3.1415927410125732 (/ 4.9348024751914465 z1)) z1)))
        (/ (+ -1.0 z0) (- (exp (/ PI z1)) -1.0))))
      (if (<= z1 0.56)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ (* -1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -280000.0) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((z0 / (2.0 - ((3.1415927410125732 - (4.9348024751914465 / z1)) / z1))) - ((-1.0 + z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -280000.0) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((z0 / (2.0 - ((3.1415927410125732 - (4.9348024751914465 / z1)) / z1))) - ((-1.0 + z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -280000.0:
		tmp = t_0
	elif z1 <= -5e-310:
		tmp = 1.0 / ((z0 / (2.0 - ((3.1415927410125732 - (4.9348024751914465 / z1)) / z1))) - ((-1.0 + z0) / (math.exp((math.pi / z1)) - -1.0)))
	elif z1 <= 0.56:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -280000.0)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 - Float64(Float64(3.1415927410125732 - Float64(4.9348024751914465 / z1)) / z1))) - Float64(Float64(-1.0 + z0) / Float64(exp(Float64(pi / z1)) - -1.0))));
	elseif (z1 <= 0.56)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(-1.0 * z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -280000.0)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = 1.0 / ((z0 / (2.0 - ((3.1415927410125732 - (4.9348024751914465 / z1)) / z1))) - ((-1.0 + z0) / (exp((pi / z1)) - -1.0)));
	elseif (z1 <= 0.56)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -280000.0], t$95$0, If[LessEqual[z1, -5e-310], N[(1.0 / N[(N[(z0 / N[(2.0 - N[(N[(3.1415927410125732 - N[(4.9348024751914465 / z1), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 0.56], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -2.8e5 or 0.56000000000000005 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -2.8e5 < z1 < -4.9999999999999847e-310

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{\color{blue}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

      6. lower-/.f64 53.5%

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 - 4.9348024751914465 \cdot \frac{1}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 53.5%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{3.1415927410125732 - 4.9348024751914465 \cdot \frac{1}{z1}}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}} \]

      2. add-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} - \left(\mathsf{neg}\left(\frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}\right)\right)}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} - \left(\mathsf{neg}\left(\frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}\right)\right)}} \]

   6. Applied rewrites 53.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{z0}{2 - \frac{3.1415927410125732 - \frac{4.9348024751914465}{z1}}{z1}} - \frac{-1 + z0}{e^{\frac{\pi}{z1}} - -1}}} \]

   if -4.9999999999999847e-310 < z1 < 0.56000000000000005

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{\color{blue}{-1 \cdot z0}}{2 + \frac{\pi}{z1}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 27.0%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot \color{blue}{z0}}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 27.0%

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

Alternative 7: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 - 3.1415927410125732 \cdot \frac{1}{z1}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{elif}\;z1 \leq 0.56:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -1.4)
    t_0
    (if (<= z1 -5e-310)
      (/
       1.0
       (+
        (/ z0 (- 2.0 (* 3.1415927410125732 (/ 1.0 z1))))
        (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
      (if (<= z1 0.56)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ (* -1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.4) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.4) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -1.4:
		tmp = t_0
	elif z1 <= -5e-310:
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	elif z1 <= 0.56:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -1.4)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 - Float64(3.1415927410125732 * Float64(1.0 / z1)))) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))));
	elseif (z1 <= 0.56)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(-1.0 * z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -1.4)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	elseif (z1 <= 0.56)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -1.4], t$95$0, If[LessEqual[z1, -5e-310], N[(1.0 / N[(N[(z0 / N[(2.0 - N[(3.1415927410125732 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 0.56], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -1.3999999999999999 or 0.56000000000000005 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -1.3999999999999999 < z1 < -4.9999999999999847e-310

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 52.9%

         \[\leadsto \frac{1}{\frac{z0}{2 - 3.1415927410125732 \cdot \frac{1}{\color{blue}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 52.9%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 - 3.1415927410125732 \cdot \frac{1}{z1}}} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if -4.9999999999999847e-310 < z1 < 0.56000000000000005

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{\color{blue}{-1 \cdot z0}}{2 + \frac{\pi}{z1}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 27.0%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot \color{blue}{z0}}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 27.0%

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

Alternative 8: 88.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{elif}\;z1 \leq 0.56:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -1.35)
    t_0
    (if (<= z1 -5e-310)
      (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
      (if (<= z1 0.56)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ (* -1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.35) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.35) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	} else if (z1 <= 0.56) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -1.35:
		tmp = t_0
	elif z1 <= -5e-310:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	elif z1 <= 0.56:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -1.35)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))));
	elseif (z1 <= 0.56)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(-1.0 * z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -1.35)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	elseif (z1 <= 0.56)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((-1.0 * z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -1.35], t$95$0, If[LessEqual[z1, -5e-310], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 0.56], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -1.3500000000000001 or 0.56000000000000005 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -1.3500000000000001 < z1 < -4.9999999999999847e-310

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 46.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 46.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if -4.9999999999999847e-310 < z1 < 0.56000000000000005

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{\color{blue}{-1 \cdot z0}}{2 + \frac{\pi}{z1}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 27.0%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1 \cdot \color{blue}{z0}}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 27.0%

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

Alternative 9: 83.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{elif}\;z1 \leq 30000000:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -1.35)
    t_0
    (if (<= z1 -5e-310)
      (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
      (if (<= z1 30000000.0)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (/ 1.0 (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.35) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	} else if (z1 <= 30000000.0) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.35) {
		tmp = t_0;
	} else if (z1 <= -5e-310) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	} else if (z1 <= 30000000.0) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -1.35:
		tmp = t_0
	elif z1 <= -5e-310:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	elif z1 <= 30000000.0:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -1.35)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))));
	elseif (z1 <= 30000000.0)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(1.0 / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -1.35)
		tmp = t_0;
	elseif (z1 <= -5e-310)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	elseif (z1 <= 30000000.0)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + (1.0 / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -1.35], t$95$0, If[LessEqual[z1, -5e-310], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 30000000.0], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -1.3500000000000001 or 3e7 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -1.3500000000000001 < z1 < -4.9999999999999847e-310

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 46.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 46.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if -4.9999999999999847e-310 < z1 < 3e7

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{\color{blue}{1}}{2 + \frac{\pi}{z1}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 45.4%

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

Alternative 10: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq -1.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{elif}\;z1 \leq 390000000:\\ \;\;\;\;\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + 0.5 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -1.35)
    t_0
    (if (<= z1 -1.2e-306)
      (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))
      (if (<= z1 390000000.0)
        (/
         1.0
         (+
          (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
          (* 0.5 1.0)))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.35) {
		tmp = t_0;
	} else if (z1 <= -1.2e-306) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	} else if (z1 <= 390000000.0) {
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + (0.5 * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -1.35) {
		tmp = t_0;
	} else if (z1 <= -1.2e-306) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	} else if (z1 <= 390000000.0) {
		tmp = 1.0 / ((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + (0.5 * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -1.35:
		tmp = t_0
	elif z1 <= -1.2e-306:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	elif z1 <= 390000000.0:
		tmp = 1.0 / ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + (0.5 * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -1.35)
		tmp = t_0;
	elseif (z1 <= -1.2e-306)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))));
	elseif (z1 <= 390000000.0)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(0.5 * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -1.35)
		tmp = t_0;
	elseif (z1 <= -1.2e-306)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	elseif (z1 <= 390000000.0)
		tmp = 1.0 / ((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + (0.5 * 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -1.35], t$95$0, If[LessEqual[z1, -1.2e-306], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 390000000.0], N[(1.0 / N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -1.3500000000000001 or 3.9e8 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -1.3500000000000001 < z1 < -1.2e-306

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 46.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   4. Applied rewrites 46.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   if -1.2e-306 < z1 < 3.9e8

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + 0.5 \cdot 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + 0.5 \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1}\\ t_1 := \frac{1}{t\_0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{t\_0 + 0.5 \cdot 1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0)))
       (t_1 (/ 1.0 (+ t_0 (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))))
  (if (<= t_1 5e-56)
    (/ 1.0 (+ t_0 (* 0.5 1.0)))
    (if (<= t_1 2e+282)
      (+ 1.0 (/ 1.0 (exp (* -1.0 (/ PI z1)))))
      (/
       1.0
       (+
        0.5
        (*
         -1.0
         (/
          (-
           (* -0.25 (* z0 PI))
           (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
          z1))))))))

C

double code(double z0, double z1) {
	double t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	double t_1 = 1.0 / (t_0 + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	double tmp;
	if (t_1 <= 5e-56) {
		tmp = 1.0 / (t_0 + (0.5 * 1.0));
	} else if (t_1 <= 2e+282) {
		tmp = 1.0 + (1.0 / exp((-1.0 * (((double) M_PI) / z1))));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0);
	double t_1 = 1.0 / (t_0 + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	double tmp;
	if (t_1 <= 5e-56) {
		tmp = 1.0 / (t_0 + (0.5 * 1.0));
	} else if (t_1 <= 2e+282) {
		tmp = 1.0 + (1.0 / Math.exp((-1.0 * (Math.PI / z1))));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)
	t_1 = 1.0 / (t_0 + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	tmp = 0
	if t_1 <= 5e-56:
		tmp = 1.0 / (t_0 + (0.5 * 1.0))
	elif t_1 <= 2e+282:
		tmp = 1.0 + (1.0 / math.exp((-1.0 * (math.pi / z1))))
	else:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0))
	t_1 = Float64(1.0 / Float64(t_0 + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))))
	tmp = 0.0
	if (t_1 <= 5e-56)
		tmp = Float64(1.0 / Float64(t_0 + Float64(0.5 * 1.0)));
	elseif (t_1 <= 2e+282)
		tmp = Float64(1.0 + Float64(1.0 / exp(Float64(-1.0 * Float64(pi / z1)))));
	else
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	t_1 = 1.0 / (t_0 + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	tmp = 0.0;
	if (t_1 <= 5e-56)
		tmp = 1.0 / (t_0 + (0.5 * 1.0));
	elseif (t_1 <= 2e+282)
		tmp = 1.0 + (1.0 / exp((-1.0 * (pi / z1))));
	else
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(t$95$0 + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-56], N[(1.0 / N[(t$95$0 + N[(0.5 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+282], N[(1.0 + N[(1.0 / N[Exp[N[(-1.0 * N[(Pi / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 5e-56

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + 0.5 \cdot 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + 0.5 \cdot 1} \]

   if 5e-56 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 2.0000000000000001e282

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 51.3%

         \[\leadsto 1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z1}}} \]

   6. Applied rewrites 51.3%

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

   if 2.0000000000000001e282 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64)))))

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1}\\ t_1 := \frac{1}{t\_0 + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{t\_0 + -0.5 \cdot z0}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0)))
       (t_1 (/ 1.0 (+ t_0 (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0))))))
  (if (<= t_1 4e-16)
    (/ 1.0 (+ t_0 (* -0.5 z0)))
    (if (<= t_1 2e+282)
      (+ 1.0 (/ 1.0 (exp (* -1.0 (/ PI z1)))))
      (/
       1.0
       (+
        0.5
        (*
         -1.0
         (/
          (-
           (* -0.25 (* z0 PI))
           (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
          z1))))))))

C

double code(double z0, double z1) {
	double t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	double t_1 = 1.0 / (t_0 + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0)));
	double tmp;
	if (t_1 <= 4e-16) {
		tmp = 1.0 / (t_0 + (-0.5 * z0));
	} else if (t_1 <= 2e+282) {
		tmp = 1.0 + (1.0 / exp((-1.0 * (((double) M_PI) / z1))));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0);
	double t_1 = 1.0 / (t_0 + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0)));
	double tmp;
	if (t_1 <= 4e-16) {
		tmp = 1.0 / (t_0 + (-0.5 * z0));
	} else if (t_1 <= 2e+282) {
		tmp = 1.0 + (1.0 / Math.exp((-1.0 * (Math.PI / z1))));
	} else {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)
	t_1 = 1.0 / (t_0 + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0)))
	tmp = 0
	if t_1 <= 4e-16:
		tmp = 1.0 / (t_0 + (-0.5 * z0))
	elif t_1 <= 2e+282:
		tmp = 1.0 + (1.0 / math.exp((-1.0 * (math.pi / z1))))
	else:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0))
	t_1 = Float64(1.0 / Float64(t_0 + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))))
	tmp = 0.0
	if (t_1 <= 4e-16)
		tmp = Float64(1.0 / Float64(t_0 + Float64(-0.5 * z0)));
	elseif (t_1 <= 2e+282)
		tmp = Float64(1.0 + Float64(1.0 / exp(Float64(-1.0 * Float64(pi / z1)))));
	else
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = z0 / (exp((-3.1415927410125732 / z1)) - -1.0);
	t_1 = 1.0 / (t_0 + ((1.0 - z0) / (exp((pi / z1)) - -1.0)));
	tmp = 0.0;
	if (t_1 <= 4e-16)
		tmp = 1.0 / (t_0 + (-0.5 * z0));
	elseif (t_1 <= 2e+282)
		tmp = 1.0 + (1.0 / exp((-1.0 * (pi / z1))));
	else
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(t$95$0 + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-16], N[(1.0 / N[(t$95$0 + N[(-0.5 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+282], N[(1.0 + N[(1.0 / N[Exp[N[(-1.0 * N[(Pi / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 3.9999999999999999e-16

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{-1}{2} \cdot \color{blue}{z0}} \]
   6. Step-by-step derivation

      1. lower-*.f64 9.0%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + -0.5 \cdot z0} \]

   7. Applied rewrites 9.0%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + -0.5 \cdot \color{blue}{z0}} \]

   if 3.9999999999999999e-16 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64))))) < 2.0000000000000001e282

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 51.3%

         \[\leadsto 1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z1}}} \]

   6. Applied rewrites 51.3%

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

   if 2.0000000000000001e282 < (/.f64 #s(literal 1 binary64) (+.f64 (/.f64 z0 (-.f64 (exp.f64 (/.f64 #s(literal -7853981852531433/2500000000000000 binary64) z1)) #s(literal -1 binary64))) (/.f64 (-.f64 #s(literal 1 binary64) z0) (-.f64 (exp.f64 (/.f64 (PI.f64) z1)) #s(literal -1 binary64)))))

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1} \leq 0:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z1}}}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<=
     (+
      (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
      (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0)))
     0.0)
  (/
   1.0
   (+
    0.5
    (*
     -1.0
     (/
      (-
       (* -0.25 (* z0 PI))
       (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
      z1))))
  (+ 1.0 (/ 1.0 (exp (* -1.0 (/ PI z1)))))))

C

double code(double z0, double z1) {
	double tmp;
	if (((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (exp((((double) M_PI) / z1)) - -1.0))) <= 0.0) {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	} else {
		tmp = 1.0 + (1.0 / exp((-1.0 * (((double) M_PI) / z1))));
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double tmp;
	if (((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (Math.exp((Math.PI / z1)) - -1.0))) <= 0.0) {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	} else {
		tmp = 1.0 + (1.0 / Math.exp((-1.0 * (Math.PI / z1))));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (math.exp((math.pi / z1)) - -1.0))) <= 0.0:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	else:
		tmp = 1.0 + (1.0 / math.exp((-1.0 * (math.pi / z1))))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(1.0 - z0) / Float64(exp(Float64(pi / z1)) - -1.0))) <= 0.0)
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	else
		tmp = Float64(1.0 + Float64(1.0 / exp(Float64(-1.0 * Float64(pi / z1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if (((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (exp((pi / z1)) - -1.0))) <= 0.0)
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	else
		tmp = 1.0 + (1.0 / exp((-1.0 * (pi / z1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[Exp[N[(-1.0 * N[(Pi / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

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

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 51.3%

         \[\leadsto 1 + \frac{1}{e^{-1 \cdot \frac{\pi}{z1}}} \]

   6. Applied rewrites 51.3%

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

Alternative 14: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{z1}}\\ \mathbf{if}\;\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{t\_0 - -1} \leq 0:\\ \;\;\;\;\frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (exp (/ PI z1))))
  (if (<=
       (+
        (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0))
        (/ (- 1.0 z0) (- t_0 -1.0)))
       0.0)
    (/
     1.0
     (+
      0.5
      (*
       -1.0
       (/
        (-
         (* -0.25 (* z0 PI))
         (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
        z1))))
    (+ 1.0 t_0))))

C

double code(double z0, double z1) {
	double t_0 = exp((((double) M_PI) / z1));
	double tmp;
	if (((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (t_0 - -1.0))) <= 0.0) {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	} else {
		tmp = 1.0 + t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = Math.exp((Math.PI / z1));
	double tmp;
	if (((z0 / (Math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (t_0 - -1.0))) <= 0.0) {
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	} else {
		tmp = 1.0 + t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = math.exp((math.pi / z1))
	tmp = 0
	if ((z0 / (math.exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (t_0 - -1.0))) <= 0.0:
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	else:
		tmp = 1.0 + t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = exp(Float64(pi / z1))
	tmp = 0.0
	if (Float64(Float64(z0 / Float64(exp(Float64(-3.1415927410125732 / z1)) - -1.0)) + Float64(Float64(1.0 - z0) / Float64(t_0 - -1.0))) <= 0.0)
		tmp = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))));
	else
		tmp = Float64(1.0 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = exp((pi / z1));
	tmp = 0.0;
	if (((z0 / (exp((-3.1415927410125732 / z1)) - -1.0)) + ((1.0 - z0) / (t_0 - -1.0))) <= 0.0)
		tmp = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	else
		tmp = 1.0 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[Exp[N[(Pi / z1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(z0 / N[(N[Exp[N[(-3.1415927410125732 / z1), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

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

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-exp.f64 N/A

         \[\leadsto 1 + e^{\frac{\mathsf{PI}\left(\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + e^{\frac{\mathsf{PI}\left(\right)}{z1}} \]

      4. lower-PI.f64 51.3%

         \[\leadsto 1 + e^{\frac{\pi}{z1}} \]

   4. Applied rewrites 51.3%

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

Alternative 15: 57.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -2400000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 3.55 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}} + 0.5 \cdot \left(1 - z0\right)}\\ \mathbf{elif}\;z1 \leq 350:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -2400000.0)
    t_0
    (if (<= z1 3.55e-298)
      (/
       1.0
       (+
        (/
         z0
         (+
          2.0
          (*
           -1.0
           (/
            (+
             3.1415927410125732
             (*
              -1.0
              (/
               (- 4.9348024751914465 (* 5.167713211464109 (/ 1.0 z1)))
               z1)))
            z1))))
        (* 0.5 (- 1.0 z0))))
      (if (<= z1 350.0)
        (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -2400000.0) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 350.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -2400000.0) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 350.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -2400000.0:
		tmp = t_0
	elif z1 <= 3.55e-298:
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + (0.5 * (1.0 - z0)))
	elif z1 <= 350.0:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -2400000.0)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 + Float64(-1.0 * Float64(Float64(3.1415927410125732 + Float64(-1.0 * Float64(Float64(4.9348024751914465 - Float64(5.167713211464109 * Float64(1.0 / z1))) / z1))) / z1)))) + Float64(0.5 * Float64(1.0 - z0))));
	elseif (z1 <= 350.0)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -2400000.0)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 + (-1.0 * ((4.9348024751914465 - (5.167713211464109 * (1.0 / z1))) / z1))) / z1)))) + (0.5 * (1.0 - z0)));
	elseif (z1 <= 350.0)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -2400000.0], t$95$0, If[LessEqual[z1, 3.55e-298], N[(1.0 / N[(N[(z0 / N[(2.0 + N[(-1.0 * N[(N[(3.1415927410125732 + N[(-1.0 * N[(N[(4.9348024751914465 - N[(5.167713211464109 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 350.0], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -2.4e6 or 350 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -2.4e6 < z1 < 3.5499999999999999e-298

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{\color{blue}{z1}}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} + -1 \cdot \frac{\frac{61685030939893080179390253033489}{12500000000000000000000000000000} - \frac{161491037858253405198930717009894818193991386579}{31250000000000000000000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      9. lower-/.f64 32.5%

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}} + 0.5 \cdot \left(1 - z0\right)} \]

   7. Applied rewrites 32.5%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{3.1415927410125732 + -1 \cdot \frac{4.9348024751914465 - 5.167713211464109 \cdot \frac{1}{z1}}{z1}}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]

   if 3.5499999999999999e-298 < z1 < 350

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-*.f64 31.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 31.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -2400000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 3.55 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 - 4.9348024751914465 \cdot \frac{1}{z1}}{z1}} + 0.5 \cdot \left(1 - z0\right)}\\ \mathbf{elif}\;z1 \leq 350:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -2400000.0)
    t_0
    (if (<= z1 3.55e-298)
      (/
       1.0
       (+
        (/
         z0
         (+
          2.0
          (*
           -1.0
           (/
            (- 3.1415927410125732 (* 4.9348024751914465 (/ 1.0 z1)))
            z1))))
        (* 0.5 (- 1.0 z0))))
      (if (<= z1 350.0)
        (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -2400000.0) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 - (4.9348024751914465 * (1.0 / z1))) / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 350.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -2400000.0) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 - (4.9348024751914465 * (1.0 / z1))) / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 350.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -2400000.0:
		tmp = t_0
	elif z1 <= 3.55e-298:
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 - (4.9348024751914465 * (1.0 / z1))) / z1)))) + (0.5 * (1.0 - z0)))
	elif z1 <= 350.0:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -2400000.0)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 + Float64(-1.0 * Float64(Float64(3.1415927410125732 - Float64(4.9348024751914465 * Float64(1.0 / z1))) / z1)))) + Float64(0.5 * Float64(1.0 - z0))));
	elseif (z1 <= 350.0)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -2400000.0)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = 1.0 / ((z0 / (2.0 + (-1.0 * ((3.1415927410125732 - (4.9348024751914465 * (1.0 / z1))) / z1)))) + (0.5 * (1.0 - z0)));
	elseif (z1 <= 350.0)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -2400000.0], t$95$0, If[LessEqual[z1, 3.55e-298], N[(1.0 / N[(N[(z0 / N[(2.0 + N[(-1.0 * N[(N[(3.1415927410125732 - N[(4.9348024751914465 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 350.0], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -2.4e6 or 350 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -2.4e6 < z1 < 3.5499999999999999e-298

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + \color{blue}{-1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \color{blue}{\frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{\color{blue}{z1}}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{\frac{7853981852531433}{2500000000000000} - \frac{61685030939893080179390253033489}{12500000000000000000000000000000} \cdot \frac{1}{z1}}{z1}} + \frac{1}{2} \cdot \left(1 - z0\right)} \]

      6. lower-/.f64 32.5%

         \[\leadsto \frac{1}{\frac{z0}{2 + -1 \cdot \frac{3.1415927410125732 - 4.9348024751914465 \cdot \frac{1}{z1}}{z1}} + 0.5 \cdot \left(1 - z0\right)} \]

   7. Applied rewrites 32.5%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 + -1 \cdot \frac{3.1415927410125732 - 4.9348024751914465 \cdot \frac{1}{z1}}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]

   if 3.5499999999999999e-298 < z1 < 350

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-*.f64 31.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 31.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}\\ \mathbf{if}\;z1 \leq -2400000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 3.55 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 - 3.1415927410125732 \cdot \frac{1}{z1}} + 0.5 \cdot \left(1 - z0\right)}\\ \mathbf{elif}\;z1 \leq 350:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (+
          0.5
          (*
           -1.0
           (/
            (-
             (* -0.25 (* z0 PI))
             (+ (* -0.25 PI) (* 0.7853981852531433 z0)))
            z1))))))
  (if (<= z1 -2400000.0)
    t_0
    (if (<= z1 3.55e-298)
      (/
       1.0
       (+
        (/ z0 (- 2.0 (* 3.1415927410125732 (/ 1.0 z1))))
        (* 0.5 (- 1.0 z0))))
      (if (<= z1 350.0)
        (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * ((double) M_PI))) - ((-0.25 * ((double) M_PI)) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -2400000.0) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 350.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * Math.PI)) - ((-0.25 * Math.PI) + (0.7853981852531433 * z0))) / z1)));
	double tmp;
	if (z1 <= -2400000.0) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 350.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * math.pi)) - ((-0.25 * math.pi) + (0.7853981852531433 * z0))) / z1)))
	tmp = 0
	if z1 <= -2400000.0:
		tmp = t_0
	elif z1 <= 3.55e-298:
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)))
	elif z1 <= 350.0:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(0.5 + Float64(-1.0 * Float64(Float64(Float64(-0.25 * Float64(z0 * pi)) - Float64(Float64(-0.25 * pi) + Float64(0.7853981852531433 * z0))) / z1))))
	tmp = 0.0
	if (z1 <= -2400000.0)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 - Float64(3.1415927410125732 * Float64(1.0 / z1)))) + Float64(0.5 * Float64(1.0 - z0))));
	elseif (z1 <= 350.0)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / (0.5 + (-1.0 * (((-0.25 * (z0 * pi)) - ((-0.25 * pi) + (0.7853981852531433 * z0))) / z1)));
	tmp = 0.0;
	if (z1 <= -2400000.0)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)));
	elseif (z1 <= 350.0)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(0.5 + N[(-1.0 * N[(N[(N[(-0.25 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * Pi), $MachinePrecision] + N[(0.7853981852531433 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -2400000.0], t$95$0, If[LessEqual[z1, 3.55e-298], N[(1.0 / N[(N[(z0 / N[(2.0 - N[(3.1415927410125732 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 350.0], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -2.4e6 or 350 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + \color{blue}{-1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \color{blue}{\frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{\color{blue}{z1}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      10. lower-PI.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{2} + -1 \cdot \frac{\frac{-1}{4} \cdot \left(z0 \cdot \pi\right) - \left(\frac{-1}{4} \cdot \pi + \frac{7853981852531433}{10000000000000000} \cdot z0\right)}{z1}} \]

      11. lower-*.f64 51.8%

          \[\leadsto \frac{1}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 + -1 \cdot \frac{-0.25 \cdot \left(z0 \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.7853981852531433 \cdot z0\right)}{z1}}} \]

   if -2.4e6 < z1 < 3.5499999999999999e-298

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 - \frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 32.5%

         \[\leadsto \frac{1}{\frac{z0}{2 - 3.1415927410125732 \cdot \frac{1}{\color{blue}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]

   7. Applied rewrites 32.5%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 - 3.1415927410125732 \cdot \frac{1}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]

   if 3.5499999999999999e-298 < z1 < 350

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-*.f64 31.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 31.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 57.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - -0.7853981852531433 \cdot \frac{z0}{z1}}\\ \mathbf{if}\;z1 \leq -0.0118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 3.55 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\frac{z0}{2 - 3.1415927410125732 \cdot \frac{1}{z1}} + 0.5 \cdot \left(1 - z0\right)}\\ \mathbf{elif}\;z1 \leq 400000000:\\ \;\;\;\;\frac{1}{0.5 \cdot z0 + \frac{1 - z0}{2 + \frac{\pi}{z1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (-
          (+ 0.5 (* 0.25 (/ (* z0 PI) z1)))
          (* -0.7853981852531433 (/ z0 z1))))))
  (if (<= z1 -0.0118)
    t_0
    (if (<= z1 3.55e-298)
      (/
       1.0
       (+
        (/ z0 (- 2.0 (* 3.1415927410125732 (/ 1.0 z1))))
        (* 0.5 (- 1.0 z0))))
      (if (<= z1 400000000.0)
        (/ 1.0 (+ (* 0.5 z0) (/ (- 1.0 z0) (+ 2.0 (/ PI z1)))))
        t_0)))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / ((0.5 + (0.25 * ((z0 * ((double) M_PI)) / z1))) - (-0.7853981852531433 * (z0 / z1)));
	double tmp;
	if (z1 <= -0.0118) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 400000000.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (((double) M_PI) / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / ((0.5 + (0.25 * ((z0 * Math.PI) / z1))) - (-0.7853981852531433 * (z0 / z1)));
	double tmp;
	if (z1 <= -0.0118) {
		tmp = t_0;
	} else if (z1 <= 3.55e-298) {
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)));
	} else if (z1 <= 400000000.0) {
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (Math.PI / z1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / ((0.5 + (0.25 * ((z0 * math.pi) / z1))) - (-0.7853981852531433 * (z0 / z1)))
	tmp = 0
	if z1 <= -0.0118:
		tmp = t_0
	elif z1 <= 3.55e-298:
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)))
	elif z1 <= 400000000.0:
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (math.pi / z1))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(Float64(0.5 + Float64(0.25 * Float64(Float64(z0 * pi) / z1))) - Float64(-0.7853981852531433 * Float64(z0 / z1))))
	tmp = 0.0
	if (z1 <= -0.0118)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = Float64(1.0 / Float64(Float64(z0 / Float64(2.0 - Float64(3.1415927410125732 * Float64(1.0 / z1)))) + Float64(0.5 * Float64(1.0 - z0))));
	elseif (z1 <= 400000000.0)
		tmp = Float64(1.0 / Float64(Float64(0.5 * z0) + Float64(Float64(1.0 - z0) / Float64(2.0 + Float64(pi / z1)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / ((0.5 + (0.25 * ((z0 * pi) / z1))) - (-0.7853981852531433 * (z0 / z1)));
	tmp = 0.0;
	if (z1 <= -0.0118)
		tmp = t_0;
	elseif (z1 <= 3.55e-298)
		tmp = 1.0 / ((z0 / (2.0 - (3.1415927410125732 * (1.0 / z1)))) + (0.5 * (1.0 - z0)));
	elseif (z1 <= 400000000.0)
		tmp = 1.0 / ((0.5 * z0) + ((1.0 - z0) / (2.0 + (pi / z1))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(N[(0.5 + N[(0.25 * N[(N[(z0 * Pi), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.7853981852531433 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -0.0118], t$95$0, If[LessEqual[z1, 3.55e-298], N[(1.0 / N[(N[(z0 / N[(2.0 - N[(3.1415927410125732 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z1, 400000000.0], N[(1.0 / N[(N[(0.5 * z0), $MachinePrecision] + N[(N[(1.0 - z0), $MachinePrecision] / N[(2.0 + N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z1 < -0.0118 or 4e8 < z1

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right)}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \color{blue}{\left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\color{blue}{\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{\color{blue}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4}} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z1}}\right)} \]

      12. lower-PI.f64 51.8%

          \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)}} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \frac{1}{4} \cdot \color{blue}{\frac{\pi}{z1}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \frac{-7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{z0}{z1}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{\color{blue}{z1}}} \]

      2. lower-/.f64 52.4%

         \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - -0.7853981852531433 \cdot \frac{z0}{z1}} \]

   10. Applied rewrites 52.4%

       \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - -0.7853981852531433 \cdot \color{blue}{\frac{z0}{z1}}} \]

   if -0.0118 < z1 < 3.5499999999999999e-298

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 - \frac{7853981852531433}{2500000000000000} \cdot \frac{1}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 32.5%

         \[\leadsto \frac{1}{\frac{z0}{2 - 3.1415927410125732 \cdot \frac{1}{\color{blue}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]

   7. Applied rewrites 32.5%

      \[\leadsto \frac{1}{\frac{z0}{\color{blue}{2 - 3.1415927410125732 \cdot \frac{1}{z1}}} + 0.5 \cdot \left(1 - z0\right)} \]

   if 3.5499999999999999e-298 < z1 < 4e8

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]
   2. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{\color{blue}{e^{\frac{\pi}{z1}}} - -1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{\frac{-7853981852531433}{2500000000000000}}{z1}} - -1} + \frac{1 - z0}{e^{\color{blue}{\frac{\pi}{z1}}} - -1}} \]

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 78.1%

          \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\frac{1}{e^{\frac{\color{blue}{-\pi}}{z1}}} - -1}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{\frac{1}{e^{\frac{-\pi}{z1}}}} - -1}} \]

   4. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 46.9%

         \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{\color{blue}{2 + \frac{\pi}{z1}}}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-*.f64 31.8%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]

   9. Applied rewrites 31.8%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot z0} + \frac{1 - z0}{2 + \frac{\pi}{z1}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -4 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\frac{z0 \cdot \left(0.7853981852531433 + 0.25 \cdot \pi\right)}{z1}}\\ \mathbf{elif}\;z0 \leq 6.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.7853981852531433 + \pi \cdot 0.25}{z1} \cdot z0}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= z0 -4e+144)
  (/ 1.0 (/ (* z0 (+ 0.7853981852531433 (* 0.25 PI))) z1))
  (if (<= z0 6.5e+107)
    (/ 1.0 0.5)
    (/ 1.0 (* (/ (+ 0.7853981852531433 (* PI 0.25)) z1) z0)))))

C

double code(double z0, double z1) {
	double tmp;
	if (z0 <= -4e+144) {
		tmp = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * ((double) M_PI)))) / z1);
	} else if (z0 <= 6.5e+107) {
		tmp = 1.0 / 0.5;
	} else {
		tmp = 1.0 / (((0.7853981852531433 + (((double) M_PI) * 0.25)) / z1) * z0);
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double tmp;
	if (z0 <= -4e+144) {
		tmp = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * Math.PI))) / z1);
	} else if (z0 <= 6.5e+107) {
		tmp = 1.0 / 0.5;
	} else {
		tmp = 1.0 / (((0.7853981852531433 + (Math.PI * 0.25)) / z1) * z0);
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if z0 <= -4e+144:
		tmp = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * math.pi))) / z1)
	elif z0 <= 6.5e+107:
		tmp = 1.0 / 0.5
	else:
		tmp = 1.0 / (((0.7853981852531433 + (math.pi * 0.25)) / z1) * z0)
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (z0 <= -4e+144)
		tmp = Float64(1.0 / Float64(Float64(z0 * Float64(0.7853981852531433 + Float64(0.25 * pi))) / z1));
	elseif (z0 <= 6.5e+107)
		tmp = Float64(1.0 / 0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(0.7853981852531433 + Float64(pi * 0.25)) / z1) * z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if (z0 <= -4e+144)
		tmp = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * pi))) / z1);
	elseif (z0 <= 6.5e+107)
		tmp = 1.0 / 0.5;
	else
		tmp = 1.0 / (((0.7853981852531433 + (pi * 0.25)) / z1) * z0);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[z0, -4e+144], N[(1.0 / N[(N[(z0 * N[(0.7853981852531433 + N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 6.5e+107], N[(1.0 / 0.5), $MachinePrecision], N[(1.0 / N[(N[(N[(0.7853981852531433 + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -4.0000000000000001e144

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right)}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \color{blue}{\left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\color{blue}{\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{\color{blue}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4}} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z1}}\right)} \]

      12. lower-PI.f64 51.8%

          \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)}} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{\pi}{z1}}} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} - \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}} \]

      4. lower-PI.f64 40.0%

         \[\leadsto \frac{1}{0.5 - 0.25 \cdot \frac{\pi}{z1}} \]

   7. Applied rewrites 40.0%

      \[\leadsto \frac{1}{0.5 - \color{blue}{0.25 \cdot \frac{\pi}{z1}}} \]

   8. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{1}{z1}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{\color{blue}{1}}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{\color{blue}{z1}}\right)} \]

      7. lower-/.f64 14.9%

         \[\leadsto \frac{1}{z0 \cdot \left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right)} \]

   10. Applied rewrites 14.9%

       \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right)}} \]

   11. Taylor expanded in z1 around 0

       \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \pi\right)}{z1}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       5. lower-PI.f64 14.8%

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(0.7853981852531433 + 0.25 \cdot \pi\right)}{z1}} \]

   13. Applied rewrites 14.8%

       \[\leadsto \frac{1}{\frac{z0 \cdot \left(0.7853981852531433 + 0.25 \cdot \pi\right)}{z1}} \]

   if -4.0000000000000001e144 < z0 < 6.5000000000000006e107

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.1%

      \[\leadsto \frac{1}{\color{blue}{0.5}} \]

   if 6.5000000000000006e107 < z0

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right)}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \color{blue}{\left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\color{blue}{\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{\color{blue}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4}} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z1}}\right)} \]

      12. lower-PI.f64 51.8%

          \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)}} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{\pi}{z1}}} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} - \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}} \]

      4. lower-PI.f64 40.0%

         \[\leadsto \frac{1}{0.5 - 0.25 \cdot \frac{\pi}{z1}} \]

   7. Applied rewrites 40.0%

      \[\leadsto \frac{1}{0.5 - \color{blue}{0.25 \cdot \frac{\pi}{z1}}} \]

   8. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{1}{z1}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{\color{blue}{1}}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{\color{blue}{z1}}\right)} \]

      7. lower-/.f64 14.9%

         \[\leadsto \frac{1}{z0 \cdot \left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right)} \]

   10. Applied rewrites 14.9%

       \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \color{blue}{\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right) \cdot z0} \]

       3. lower-*.f64 14.9%

          \[\leadsto \frac{1}{\left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right) \cdot z0} \]

       4. lift-+.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right) \cdot z0} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\left(\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right) \cdot z0} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right) \cdot z0} \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right) \cdot z0} \]

       8. mult-flip-rev N/A

          \[\leadsto \frac{1}{\left(\frac{\frac{7853981852531433}{10000000000000000}}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right) \cdot z0} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{\frac{7853981852531433}{10000000000000000}}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right) \cdot z0} \]

       10. lift-/.f64 N/A

           \[\leadsto \frac{1}{\left(\frac{\frac{7853981852531433}{10000000000000000}}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right) \cdot z0} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{1}{\left(\frac{\frac{7853981852531433}{10000000000000000}}{z1} + \frac{\frac{1}{4} \cdot \pi}{z1}\right) \cdot z0} \]

       12. lift-PI.f64 N/A

           \[\leadsto \frac{1}{\left(\frac{\frac{7853981852531433}{10000000000000000}}{z1} + \frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}{z1}\right) \cdot z0} \]

       13. div-add-rev N/A

           \[\leadsto \frac{1}{\frac{\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{z1} \cdot z0} \]

       14. lower-/.f64 N/A

           \[\leadsto \frac{1}{\frac{\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{z1} \cdot z0} \]

       15. lower-+.f64 N/A

           \[\leadsto \frac{1}{\frac{\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{z1} \cdot z0} \]

       16. lift-PI.f64 N/A

           \[\leadsto \frac{1}{\frac{\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \pi}{z1} \cdot z0} \]

       17. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\frac{7853981852531433}{10000000000000000} + \pi \cdot \frac{1}{4}}{z1} \cdot z0} \]

       18. lower-*.f64 14.9%

           \[\leadsto \frac{1}{\frac{0.7853981852531433 + \pi \cdot 0.25}{z1} \cdot z0} \]

   12. Applied rewrites 14.9%

       \[\leadsto \frac{1}{\frac{0.7853981852531433 + \pi \cdot 0.25}{z1} \cdot z0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 46.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\frac{z0 \cdot \left(0.7853981852531433 + 0.25 \cdot \pi\right)}{z1}}\\ \mathbf{if}\;z0 \leq -4 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 6.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ 1.0 (/ (* z0 (+ 0.7853981852531433 (* 0.25 PI))) z1))))
  (if (<= z0 -4e+144) t_0 (if (<= z0 6.5e+107) (/ 1.0 0.5) t_0))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * ((double) M_PI)))) / z1);
	double tmp;
	if (z0 <= -4e+144) {
		tmp = t_0;
	} else if (z0 <= 6.5e+107) {
		tmp = 1.0 / 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * Math.PI))) / z1);
	double tmp;
	if (z0 <= -4e+144) {
		tmp = t_0;
	} else if (z0 <= 6.5e+107) {
		tmp = 1.0 / 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * math.pi))) / z1)
	tmp = 0
	if z0 <= -4e+144:
		tmp = t_0
	elif z0 <= 6.5e+107:
		tmp = 1.0 / 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 / Float64(Float64(z0 * Float64(0.7853981852531433 + Float64(0.25 * pi))) / z1))
	tmp = 0.0
	if (z0 <= -4e+144)
		tmp = t_0;
	elseif (z0 <= 6.5e+107)
		tmp = Float64(1.0 / 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 / ((z0 * (0.7853981852531433 + (0.25 * pi))) / z1);
	tmp = 0.0;
	if (z0 <= -4e+144)
		tmp = t_0;
	elseif (z0 <= 6.5e+107)
		tmp = 1.0 / 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(N[(z0 * N[(0.7853981852531433 + N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -4e+144], t$95$0, If[LessEqual[z0, 6.5e+107], N[(1.0 / 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -4.0000000000000001e144 or 6.5000000000000006e107 < z0

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\pi}{z1}\right)}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \color{blue}{\left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\color{blue}{\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{z0}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{\color{blue}{z1}} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \color{blue}{\frac{1}{4}} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{z1}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(\frac{-7853981852531433}{10000000000000000} \cdot \frac{z0}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z1}}\right)} \]

      12. lower-PI.f64 51.8%

          \[\leadsto \frac{1}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{1}{\color{blue}{\left(0.5 + 0.25 \cdot \frac{z0 \cdot \pi}{z1}\right) - \left(-0.7853981852531433 \cdot \frac{z0}{z1} + 0.25 \cdot \frac{\pi}{z1}\right)}} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{\pi}{z1}}} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{2} - \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1}} \]

      4. lower-PI.f64 40.0%

         \[\leadsto \frac{1}{0.5 - 0.25 \cdot \frac{\pi}{z1}} \]

   7. Applied rewrites 40.0%

      \[\leadsto \frac{1}{0.5 - \color{blue}{0.25 \cdot \frac{\pi}{z1}}} \]

   8. Taylor expanded in z0 around inf

      \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \color{blue}{\frac{1}{z1}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{\color{blue}{1}}{z1}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{z1}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \left(\frac{1}{4} \cdot \frac{\pi}{z1} + \frac{7853981852531433}{10000000000000000} \cdot \frac{1}{\color{blue}{z1}}\right)} \]

      7. lower-/.f64 14.9%

         \[\leadsto \frac{1}{z0 \cdot \left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right)} \]

   10. Applied rewrites 14.9%

       \[\leadsto \frac{1}{z0 \cdot \color{blue}{\left(0.25 \cdot \frac{\pi}{z1} + 0.7853981852531433 \cdot \frac{1}{z1}\right)}} \]

   11. Taylor expanded in z1 around 0

       \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \pi\right)}{z1}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(\frac{7853981852531433}{10000000000000000} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{z1}} \]

       5. lower-PI.f64 14.8%

          \[\leadsto \frac{1}{\frac{z0 \cdot \left(0.7853981852531433 + 0.25 \cdot \pi\right)}{z1}} \]

   13. Applied rewrites 14.8%

       \[\leadsto \frac{1}{\frac{z0 \cdot \left(0.7853981852531433 + 0.25 \cdot \pi\right)}{z1}} \]

   if -4.0000000000000001e144 < z0 < 6.5000000000000006e107

   1. Initial program 78.1%

      \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

   2. Taylor expanded in z1 around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.1%

      \[\leadsto \frac{1}{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 41.1% accurate, 22.3× speedup

Math

\[\frac{1}{0.5} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (/ 1.0 0.5))

C

double code(double z0, double z1) {
	return 1.0 / 0.5;
}

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(z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = 1.0d0 / 0.5d0
end function

Java

public static double code(double z0, double z1) {
	return 1.0 / 0.5;
}

Python

def code(z0, z1):
	return 1.0 / 0.5

Julia

function code(z0, z1)
	return Float64(1.0 / 0.5)
end

MATLAB

function tmp = code(z0, z1)
	tmp = 1.0 / 0.5;
end

Wolfram

code[z0_, z1_] := N[(1.0 / 0.5), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\frac{1}{\frac{z0}{e^{\frac{-3.1415927410125732}{z1}} - -1} + \frac{1 - z0}{e^{\frac{\pi}{z1}} - -1}} \]

2. Taylor expanded in z1 around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 41.1%

   \[\leadsto \frac{1}{\color{blue}{0.5}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0 z1)
  :name "(/ 1 (+ (/ z0 (- (exp (/ -7853981852531433/2500000000000000 z1)) -1)) (/ (- 1 z0) (- (exp (/ PI z1)) -1))))"
  :precision binary64
  (/ 1.0 (+ (/ z0 (- (exp (/ -3.1415927410125732 z1)) -1.0)) (/ (- 1.0 z0) (- (exp (/ PI z1)) -1.0)))))