(- z0 (* (- z0 1) (exp (/ -2 z1))))

Percentage Accurate: 76.8% → 99.9%

Time: 3.1s

Alternatives: 22

Speedup: 1.4×

• 74.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (z0 z1)
  :precision binary64
  (- z0 (* (- z0 1.0) (exp (/ -2.0 z1)))))

C

double code(double z0, double z1) {
	return z0 - ((z0 - 1.0) * exp((-2.0 / z1)));
}

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 = z0 - ((z0 - 1.0d0) * exp(((-2.0d0) / z1)))
end function

Java

public static double code(double z0, double z1) {
	return z0 - ((z0 - 1.0) * Math.exp((-2.0 / z1)));
}

Python

def code(z0, z1):
	return z0 - ((z0 - 1.0) * math.exp((-2.0 / z1)))

Julia

function code(z0, z1)
	return Float64(z0 - Float64(Float64(z0 - 1.0) * exp(Float64(-2.0 / z1))))
end

MATLAB

function tmp = code(z0, z1)
	tmp = z0 - ((z0 - 1.0) * exp((-2.0 / z1)));
end

Wolfram

code[z0_, z1_] := N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z0 z1)
  :precision binary64
  (- z0 (* (- z0 1.0) (exp (/ -2.0 z1)))))

C

double code(double z0, double z1) {
	return z0 - ((z0 - 1.0) * exp((-2.0 / z1)));
}

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 = z0 - ((z0 - 1.0d0) * exp(((-2.0d0) / z1)))
end function

Java

public static double code(double z0, double z1) {
	return z0 - ((z0 - 1.0) * Math.exp((-2.0 / z1)));
}

Python

def code(z0, z1):
	return z0 - ((z0 - 1.0) * math.exp((-2.0 / z1)))

Julia

function code(z0, z1)
	return Float64(z0 - Float64(Float64(z0 - 1.0) * exp(Float64(-2.0 / z1))))
end

MATLAB

function tmp = code(z0, z1)
	tmp = z0 - ((z0 - 1.0) * exp((-2.0 / z1)));
end

Wolfram

code[z0_, z1_] := N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}\\ \mathbf{if}\;z1 \leq -13500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 15000000:\\ \;\;\;\;z0 - \left(z0 - 1\right) \cdot {7.38905609893065}^{\left(\frac{-1}{z1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (+
         1.0
         (*
          -1.0
          (/
           (-
            (/ (* z0 (- 2.0 (* 1.3333333333333333 (/ 1.0 z1)))) z1)
            (* 2.0 (- z0 1.0)))
           z1)))))
  (if (<= z1 -13500.0)
    t_0
    (if (<= z1 15000000.0)
      (- z0 (* (- z0 1.0) (pow 7.38905609893065 (/ -1.0 z1))))
      t_0))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	double tmp;
	if (z1 <= -13500.0) {
		tmp = t_0;
	} else if (z1 <= 15000000.0) {
		tmp = z0 - ((z0 - 1.0) * pow(7.38905609893065, (-1.0 / z1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-1.0d0) * ((((z0 * (2.0d0 - (1.3333333333333333d0 * (1.0d0 / z1)))) / z1) - (2.0d0 * (z0 - 1.0d0))) / z1))
    if (z1 <= (-13500.0d0)) then
        tmp = t_0
    else if (z1 <= 15000000.0d0) then
        tmp = z0 - ((z0 - 1.0d0) * (7.38905609893065d0 ** ((-1.0d0) / z1)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	double tmp;
	if (z1 <= -13500.0) {
		tmp = t_0;
	} else if (z1 <= 15000000.0) {
		tmp = z0 - ((z0 - 1.0) * Math.pow(7.38905609893065, (-1.0 / z1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1))
	tmp = 0
	if z1 <= -13500.0:
		tmp = t_0
	elif z1 <= 15000000.0:
		tmp = z0 - ((z0 - 1.0) * math.pow(7.38905609893065, (-1.0 / z1)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(Float64(Float64(z0 * Float64(2.0 - Float64(1.3333333333333333 * Float64(1.0 / z1)))) / z1) - Float64(2.0 * Float64(z0 - 1.0))) / z1)))
	tmp = 0.0
	if (z1 <= -13500.0)
		tmp = t_0;
	elseif (z1 <= 15000000.0)
		tmp = Float64(z0 - Float64(Float64(z0 - 1.0) * (7.38905609893065 ^ Float64(-1.0 / z1))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	tmp = 0.0;
	if (z1 <= -13500.0)
		tmp = t_0;
	elseif (z1 <= 15000000.0)
		tmp = z0 - ((z0 - 1.0) * (7.38905609893065 ^ (-1.0 / z1)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(N[(N[(z0 * N[(2.0 - N[(1.3333333333333333 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision] - N[(2.0 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -13500.0], t$95$0, If[LessEqual[z1, 15000000.0], N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[Power[7.38905609893065, N[(-1.0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z1 < -13500 or 1.5e7 < z1

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   5. Taylor expanded in z0 around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      5. lower-/.f64 65.2%

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

   7. Applied rewrites 65.2%

      \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

   if -13500 < z1 < 1.5e7

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Evaluated real constant 76.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot {\color{blue}{7.38905609893065}}^{\left(\frac{-1}{z1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}\\ \mathbf{if}\;z1 \leq -13500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z1 \leq 15000000:\\ \;\;\;\;z0 - \left(z0 - 1\right) \cdot e^{\frac{-2}{z1}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0
        (+
         1.0
         (*
          -1.0
          (/
           (-
            (/ (* z0 (- 2.0 (* 1.3333333333333333 (/ 1.0 z1)))) z1)
            (* 2.0 (- z0 1.0)))
           z1)))))
  (if (<= z1 -13500.0)
    t_0
    (if (<= z1 15000000.0)
      (- z0 (* (- z0 1.0) (exp (/ -2.0 z1))))
      t_0))))

C

double code(double z0, double z1) {
	double t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	double tmp;
	if (z1 <= -13500.0) {
		tmp = t_0;
	} else if (z1 <= 15000000.0) {
		tmp = z0 - ((z0 - 1.0) * exp((-2.0 / z1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-1.0d0) * ((((z0 * (2.0d0 - (1.3333333333333333d0 * (1.0d0 / z1)))) / z1) - (2.0d0 * (z0 - 1.0d0))) / z1))
    if (z1 <= (-13500.0d0)) then
        tmp = t_0
    else if (z1 <= 15000000.0d0) then
        tmp = z0 - ((z0 - 1.0d0) * exp(((-2.0d0) / z1)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	double tmp;
	if (z1 <= -13500.0) {
		tmp = t_0;
	} else if (z1 <= 15000000.0) {
		tmp = z0 - ((z0 - 1.0) * Math.exp((-2.0 / z1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1))
	tmp = 0
	if z1 <= -13500.0:
		tmp = t_0
	elif z1 <= 15000000.0:
		tmp = z0 - ((z0 - 1.0) * math.exp((-2.0 / z1)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.0 + Float64(-1.0 * Float64(Float64(Float64(Float64(z0 * Float64(2.0 - Float64(1.3333333333333333 * Float64(1.0 / z1)))) / z1) - Float64(2.0 * Float64(z0 - 1.0))) / z1)))
	tmp = 0.0
	if (z1 <= -13500.0)
		tmp = t_0;
	elseif (z1 <= 15000000.0)
		tmp = Float64(z0 - Float64(Float64(z0 - 1.0) * exp(Float64(-2.0 / z1))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	tmp = 0.0;
	if (z1 <= -13500.0)
		tmp = t_0;
	elseif (z1 <= 15000000.0)
		tmp = z0 - ((z0 - 1.0) * exp((-2.0 / z1)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 * N[(N[(N[(N[(z0 * N[(2.0 - N[(1.3333333333333333 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision] - N[(2.0 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z1, -13500.0], t$95$0, If[LessEqual[z1, 15000000.0], N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z1 < -13500 or 1.5e7 < z1

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   5. Taylor expanded in z0 around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      5. lower-/.f64 65.2%

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

   7. Applied rewrites 65.2%

      \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

   if -13500 < z1 < 1.5e7

   1. Initial program 76.8%

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

Alternative 3: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \left(1 - e^{\frac{-2}{z1}}\right)\\ t_1 := 2 \cdot \left(z0 - 1\right)\\ \mathbf{if}\;\frac{-2}{z1} \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{-2}{z1} \leq 0.0002:\\ \;\;\;\;1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - t\_1}{z1} - t\_1}{z1}\\ \mathbf{elif}\;\frac{-2}{z1} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z0 - \left(z0 - 1\right) \cdot \left(\left(\left(z1 + z1\right) \cdot \frac{z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - \frac{2}{z1}\right) - -1\right)\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (* z0 (- 1.0 (exp (/ -2.0 z1)))))
       (t_1 (* 2.0 (- z0 1.0))))
  (if (<= (/ -2.0 z1) -0.02)
    t_0
    (if (<= (/ -2.0 z1) 0.0002)
      (+
       1.0
       (*
        -1.0
        (/
         (-
          (*
           -1.0
           (/ (- (* 1.3333333333333333 (/ (- z0 1.0) z1)) t_1) z1))
          t_1)
         z1)))
      (if (<= (/ -2.0 z1) 2e+62)
        t_0
        (-
         z0
         (*
          (- z0 1.0)
          (-
           (- (* (+ z1 z1) (/ z1 (* (* (* z1 z1) z1) z1))) (/ 2.0 z1))
           -1.0))))))))

C

double code(double z0, double z1) {
	double t_0 = z0 * (1.0 - exp((-2.0 / z1)));
	double t_1 = 2.0 * (z0 - 1.0);
	double tmp;
	if ((-2.0 / z1) <= -0.02) {
		tmp = t_0;
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_1) / z1)) - t_1) / z1));
	} else if ((-2.0 / z1) <= 2e+62) {
		tmp = t_0;
	} else {
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z0 * (1.0d0 - exp(((-2.0d0) / z1)))
    t_1 = 2.0d0 * (z0 - 1.0d0)
    if (((-2.0d0) / z1) <= (-0.02d0)) then
        tmp = t_0
    else if (((-2.0d0) / z1) <= 0.0002d0) then
        tmp = 1.0d0 + ((-1.0d0) * ((((-1.0d0) * (((1.3333333333333333d0 * ((z0 - 1.0d0) / z1)) - t_1) / z1)) - t_1) / z1))
    else if (((-2.0d0) / z1) <= 2d+62) then
        tmp = t_0
    else
        tmp = z0 - ((z0 - 1.0d0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0d0 / z1)) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = z0 * (1.0 - Math.exp((-2.0 / z1)));
	double t_1 = 2.0 * (z0 - 1.0);
	double tmp;
	if ((-2.0 / z1) <= -0.02) {
		tmp = t_0;
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_1) / z1)) - t_1) / z1));
	} else if ((-2.0 / z1) <= 2e+62) {
		tmp = t_0;
	} else {
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = z0 * (1.0 - math.exp((-2.0 / z1)))
	t_1 = 2.0 * (z0 - 1.0)
	tmp = 0
	if (-2.0 / z1) <= -0.02:
		tmp = t_0
	elif (-2.0 / z1) <= 0.0002:
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_1) / z1)) - t_1) / z1))
	elif (-2.0 / z1) <= 2e+62:
		tmp = t_0
	else:
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(z0 * Float64(1.0 - exp(Float64(-2.0 / z1))))
	t_1 = Float64(2.0 * Float64(z0 - 1.0))
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -0.02)
		tmp = t_0;
	elseif (Float64(-2.0 / z1) <= 0.0002)
		tmp = Float64(1.0 + Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(1.3333333333333333 * Float64(Float64(z0 - 1.0) / z1)) - t_1) / z1)) - t_1) / z1)));
	elseif (Float64(-2.0 / z1) <= 2e+62)
		tmp = t_0;
	else
		tmp = Float64(z0 - Float64(Float64(z0 - 1.0) * Float64(Float64(Float64(Float64(z1 + z1) * Float64(z1 / Float64(Float64(Float64(z1 * z1) * z1) * z1))) - Float64(2.0 / z1)) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = z0 * (1.0 - exp((-2.0 / z1)));
	t_1 = 2.0 * (z0 - 1.0);
	tmp = 0.0;
	if ((-2.0 / z1) <= -0.02)
		tmp = t_0;
	elseif ((-2.0 / z1) <= 0.0002)
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_1) / z1)) - t_1) / z1));
	elseif ((-2.0 / z1) <= 2e+62)
		tmp = t_0;
	else
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(z0 * N[(1.0 - N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -0.02], t$95$0, If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 0.0002], N[(1.0 + N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(1.3333333333333333 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 2e+62], t$95$0, N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[(N[(N[(N[(z1 + z1), $MachinePrecision] * N[(z1 / N[(N[(N[(z1 * z1), $MachinePrecision] * z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / z1), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -0.02 or 2.0000000000000001e-4 < (/.f64 #s(literal -2 binary64) z1) < 2.0000000000000001e62

   1. Initial program 76.8%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 44.5%

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

   4. Applied rewrites 44.5%

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

   if -0.02 < (/.f64 #s(literal -2 binary64) z1) < 2.0000000000000001e-4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   if 2.0000000000000001e62 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.7%

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

   4. Applied rewrites 43.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 43.7%

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{2 - 2 \cdot \frac{1}{z1}}{z1}\right)\right) - -1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{2 - 2 \cdot \frac{1}{z1}}{z1}\right)\right) - -1\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      11. lift--.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      12. sub-negate-rev N/A

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

      13. lower--.f64 43.7%

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. mult-flip-rev N/A

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

      17. lower-/.f64 43.7%

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

   6. Applied rewrites 43.7%

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

   8. Applied rewrites 17.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\left(z1 + z1\right) \cdot z1 - \left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot 2}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - -1\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow3 N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      12. associate-*l* N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(z1 \cdot z1\right) \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(z1 \cdot z1\right) \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      14. pow2 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{2} \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      15. pow2 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{2} \cdot {z1}^{2}}\right) - -1\right) \]

      16. pow-prod-up N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{\left(2 + 2\right)}}\right) - -1\right) \]

      17. pow-div N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot {z1}^{\left(3 - \left(2 + 2\right)\right)}\right) - -1\right) \]

      18. metadata-eval N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot {z1}^{\left(3 - 4\right)}\right) - -1\right) \]

      19. metadata-eval N/A

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

      20. inv-pow N/A

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

   10. Applied rewrites 47.7%

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

Alternative 4: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 1.3333333333333333 \cdot \frac{z0 - 1}{z1}\\ t_1 := 2 \cdot \left(z0 - 1\right)\\ \mathbf{if}\;\frac{-2}{z1} \leq -20:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 0.0002:\\ \;\;\;\;1 + -1 \cdot \frac{-1 \cdot \frac{t\_0 - t\_1}{z1} - t\_1}{z1}\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+88}:\\ \;\;\;\;e^{\frac{-2}{z1}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{t\_0 \cdot 1}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (* 1.3333333333333333 (/ (- z0 1.0) z1)))
       (t_1 (* 2.0 (- z0 1.0))))
  (if (<= (/ -2.0 z1) -20.0)
    (- z0 (* -1.0 1.0))
    (if (<= (/ -2.0 z1) 0.0002)
      (+ 1.0 (* -1.0 (/ (- (* -1.0 (/ (- t_0 t_1) z1)) t_1) z1)))
      (if (<= (/ -2.0 z1) 1e+88)
        (exp (/ -2.0 z1))
        (+ 1.0 (/ (* t_0 1.0) (* z1 z1))))))))

C

double code(double z0, double z1) {
	double t_0 = 1.3333333333333333 * ((z0 - 1.0) / z1);
	double t_1 = 2.0 * (z0 - 1.0);
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (-1.0 * (((-1.0 * ((t_0 - t_1) / z1)) - t_1) / z1));
	} else if ((-2.0 / z1) <= 1e+88) {
		tmp = exp((-2.0 / z1));
	} else {
		tmp = 1.0 + ((t_0 * 1.0) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.3333333333333333d0 * ((z0 - 1.0d0) / z1)
    t_1 = 2.0d0 * (z0 - 1.0d0)
    if (((-2.0d0) / z1) <= (-20.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 0.0002d0) then
        tmp = 1.0d0 + ((-1.0d0) * ((((-1.0d0) * ((t_0 - t_1) / z1)) - t_1) / z1))
    else if (((-2.0d0) / z1) <= 1d+88) then
        tmp = exp(((-2.0d0) / z1))
    else
        tmp = 1.0d0 + ((t_0 * 1.0d0) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = 1.3333333333333333 * ((z0 - 1.0) / z1);
	double t_1 = 2.0 * (z0 - 1.0);
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (-1.0 * (((-1.0 * ((t_0 - t_1) / z1)) - t_1) / z1));
	} else if ((-2.0 / z1) <= 1e+88) {
		tmp = Math.exp((-2.0 / z1));
	} else {
		tmp = 1.0 + ((t_0 * 1.0) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 1.3333333333333333 * ((z0 - 1.0) / z1)
	t_1 = 2.0 * (z0 - 1.0)
	tmp = 0
	if (-2.0 / z1) <= -20.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 0.0002:
		tmp = 1.0 + (-1.0 * (((-1.0 * ((t_0 - t_1) / z1)) - t_1) / z1))
	elif (-2.0 / z1) <= 1e+88:
		tmp = math.exp((-2.0 / z1))
	else:
		tmp = 1.0 + ((t_0 * 1.0) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(1.3333333333333333 * Float64(Float64(z0 - 1.0) / z1))
	t_1 = Float64(2.0 * Float64(z0 - 1.0))
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -20.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 0.0002)
		tmp = Float64(1.0 + Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(t_0 - t_1) / z1)) - t_1) / z1)));
	elseif (Float64(-2.0 / z1) <= 1e+88)
		tmp = exp(Float64(-2.0 / z1));
	else
		tmp = Float64(1.0 + Float64(Float64(t_0 * 1.0) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 1.3333333333333333 * ((z0 - 1.0) / z1);
	t_1 = 2.0 * (z0 - 1.0);
	tmp = 0.0;
	if ((-2.0 / z1) <= -20.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 0.0002)
		tmp = 1.0 + (-1.0 * (((-1.0 * ((t_0 - t_1) / z1)) - t_1) / z1));
	elseif ((-2.0 / z1) <= 1e+88)
		tmp = exp((-2.0 / z1));
	else
		tmp = 1.0 + ((t_0 * 1.0) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(1.3333333333333333 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -20.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 0.0002], N[(1.0 + N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(t$95$0 - t$95$1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+88], N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(N[(t$95$0 * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -20

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -20 < (/.f64 #s(literal -2 binary64) z1) < 2.0000000000000001e-4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   if 2.0000000000000001e-4 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999996e87

   1. Initial program 76.8%

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

   2. Taylor expanded in z0 around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto e^{\frac{-2}{z1}} \]

      2. lower-/.f64 58.1%

         \[\leadsto e^{\frac{-2}{z1}} \]

   4. Applied rewrites 58.1%

      \[\leadsto \color{blue}{e^{\frac{-2}{z1}}} \]

   if 9.9999999999999996e87 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

       2. lower-/.f64 N/A

          \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

       3. lower--.f64 57.9%

          \[\leadsto 1 + \frac{\left(1.3333333333333333 \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 57.9%

       \[\leadsto 1 + \frac{\left(1.3333333333333333 \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -20:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}\\ \mathbf{else}:\\ \;\;\;\;z0 - \left(z0 - 1\right) \cdot \left(\left(\left(z1 + z1\right) \cdot \frac{z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - \frac{2}{z1}\right) - -1\right)\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -20.0)
  (- z0 (* -1.0 1.0))
  (if (<= (/ -2.0 z1) 2e+58)
    (+
     1.0
     (*
      -1.0
      (/
       (-
        (/ (* z0 (- 2.0 (* 1.3333333333333333 (/ 1.0 z1)))) z1)
        (* 2.0 (- z0 1.0)))
       z1)))
    (-
     z0
     (*
      (- z0 1.0)
      (-
       (- (* (+ z1 z1) (/ z1 (* (* (* z1 z1) z1) z1))) (/ 2.0 z1))
       -1.0))))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 2e+58) {
		tmp = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	} else {
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-20.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 2d+58) then
        tmp = 1.0d0 + ((-1.0d0) * ((((z0 * (2.0d0 - (1.3333333333333333d0 * (1.0d0 / z1)))) / z1) - (2.0d0 * (z0 - 1.0d0))) / z1))
    else
        tmp = z0 - ((z0 - 1.0d0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0d0 / z1)) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 2e+58) {
		tmp = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	} else {
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -20.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 2e+58:
		tmp = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1))
	else:
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -20.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 2e+58)
		tmp = Float64(1.0 + Float64(-1.0 * Float64(Float64(Float64(Float64(z0 * Float64(2.0 - Float64(1.3333333333333333 * Float64(1.0 / z1)))) / z1) - Float64(2.0 * Float64(z0 - 1.0))) / z1)));
	else
		tmp = Float64(z0 - Float64(Float64(z0 - 1.0) * Float64(Float64(Float64(Float64(z1 + z1) * Float64(z1 / Float64(Float64(Float64(z1 * z1) * z1) * z1))) - Float64(2.0 / z1)) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -20.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 2e+58)
		tmp = 1.0 + (-1.0 * ((((z0 * (2.0 - (1.3333333333333333 * (1.0 / z1)))) / z1) - (2.0 * (z0 - 1.0))) / z1));
	else
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -20.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 2e+58], N[(1.0 + N[(-1.0 * N[(N[(N[(N[(z0 * N[(2.0 - N[(1.3333333333333333 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision] - N[(2.0 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[(N[(N[(N[(z1 + z1), $MachinePrecision] * N[(z1 / N[(N[(N[(z1 * z1), $MachinePrecision] * z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / z1), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -20

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -20 < (/.f64 #s(literal -2 binary64) z1) < 1.9999999999999999e58

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   5. Taylor expanded in z0 around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - \frac{4}{3} \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

      5. lower-/.f64 65.2%

         \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

   7. Applied rewrites 65.2%

      \[\leadsto 1 + -1 \cdot \frac{\frac{z0 \cdot \left(2 - 1.3333333333333333 \cdot \frac{1}{z1}\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} \]

   if 1.9999999999999999e58 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.7%

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

   4. Applied rewrites 43.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 43.7%

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{2 - 2 \cdot \frac{1}{z1}}{z1}\right)\right) - -1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{2 - 2 \cdot \frac{1}{z1}}{z1}\right)\right) - -1\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      11. lift--.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      12. sub-negate-rev N/A

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

      13. lower--.f64 43.7%

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. mult-flip-rev N/A

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

      17. lower-/.f64 43.7%

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

   6. Applied rewrites 43.7%

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

   8. Applied rewrites 17.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\left(z1 + z1\right) \cdot z1 - \left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot 2}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - -1\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow3 N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      12. associate-*l* N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(z1 \cdot z1\right) \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(z1 \cdot z1\right) \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      14. pow2 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{2} \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      15. pow2 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{2} \cdot {z1}^{2}}\right) - -1\right) \]

      16. pow-prod-up N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{\left(2 + 2\right)}}\right) - -1\right) \]

      17. pow-div N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot {z1}^{\left(3 - \left(2 + 2\right)\right)}\right) - -1\right) \]

      18. metadata-eval N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot {z1}^{\left(3 - 4\right)}\right) - -1\right) \]

      19. metadata-eval N/A

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

      20. inv-pow N/A

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

   10. Applied rewrites 47.7%

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

Alternative 6: 82.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -40000:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - 1.3333333333333333 \cdot \frac{1}{z1}}{z1}\right)}{z1}\\ \mathbf{else}:\\ \;\;\;\;z0 - \left(z0 - 1\right) \cdot \left(\left(\left(z1 + z1\right) \cdot \frac{z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - \frac{2}{z1}\right) - -1\right)\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -40000.0)
  (- z0 (* -1.0 1.0))
  (if (<= (/ -2.0 z1) 2e+58)
    (+
     1.0
     (/
      (*
       z0
       (+
        2.0
        (* -1.0 (/ (- 2.0 (* 1.3333333333333333 (/ 1.0 z1))) z1))))
      z1))
    (-
     z0
     (*
      (- z0 1.0)
      (-
       (- (* (+ z1 z1) (/ z1 (* (* (* z1 z1) z1) z1))) (/ 2.0 z1))
       -1.0))))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -40000.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 2e+58) {
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1);
	} else {
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-40000.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 2d+58) then
        tmp = 1.0d0 + ((z0 * (2.0d0 + ((-1.0d0) * ((2.0d0 - (1.3333333333333333d0 * (1.0d0 / z1))) / z1)))) / z1)
    else
        tmp = z0 - ((z0 - 1.0d0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0d0 / z1)) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -40000.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 2e+58) {
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1);
	} else {
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -40000.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 2e+58:
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1)
	else:
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -40000.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 2e+58)
		tmp = Float64(1.0 + Float64(Float64(z0 * Float64(2.0 + Float64(-1.0 * Float64(Float64(2.0 - Float64(1.3333333333333333 * Float64(1.0 / z1))) / z1)))) / z1));
	else
		tmp = Float64(z0 - Float64(Float64(z0 - 1.0) * Float64(Float64(Float64(Float64(z1 + z1) * Float64(z1 / Float64(Float64(Float64(z1 * z1) * z1) * z1))) - Float64(2.0 / z1)) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -40000.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 2e+58)
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1);
	else
		tmp = z0 - ((z0 - 1.0) * ((((z1 + z1) * (z1 / (((z1 * z1) * z1) * z1))) - (2.0 / z1)) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -40000.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 2e+58], N[(1.0 + N[(N[(z0 * N[(2.0 + N[(-1.0 * N[(N[(2.0 - N[(1.3333333333333333 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], N[(z0 - N[(N[(z0 - 1.0), $MachinePrecision] * N[(N[(N[(N[(z1 + z1), $MachinePrecision] * N[(z1 / N[(N[(N[(z1 * z1), $MachinePrecision] * z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / z1), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -4e4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -4e4 < (/.f64 #s(literal -2 binary64) z1) < 1.9999999999999999e58

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   5. Taylor expanded in z0 around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      3. lower-+.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      6. lower--.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      7. lower-*.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      8. lower-/.f64 65.2%

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - 1.3333333333333333 \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

   7. Applied rewrites 65.2%

      \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - 1.3333333333333333 \cdot \frac{1}{z1}}{z1}\right)}{\color{blue}{z1}} \]

   if 1.9999999999999999e58 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.7%

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

   4. Applied rewrites 43.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 43.7%

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{2 - 2 \cdot \frac{1}{z1}}{z1}\right)\right) - -1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{2 - 2 \cdot \frac{1}{z1}}{z1}\right)\right) - -1\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      11. lift--.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\mathsf{neg}\left(\left(2 - 2 \cdot \frac{1}{z1}\right)\right)}{z1} - -1\right) \]

      12. sub-negate-rev N/A

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

      13. lower--.f64 43.7%

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. mult-flip-rev N/A

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

      17. lower-/.f64 43.7%

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

   6. Applied rewrites 43.7%

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

   8. Applied rewrites 17.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\frac{\left(z1 + z1\right) \cdot z1 - \left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot 2}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - -1\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow3 N/A

         \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1}\right) - -1\right) \]

      12. associate-*l* N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(z1 \cdot z1\right) \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{\left(z1 \cdot z1\right) \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      14. pow2 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{2} \cdot \left(z1 \cdot z1\right)}\right) - -1\right) \]

      15. pow2 N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{2} \cdot {z1}^{2}}\right) - -1\right) \]

      16. pow-prod-up N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot \frac{{z1}^{3}}{{z1}^{\left(2 + 2\right)}}\right) - -1\right) \]

      17. pow-div N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot {z1}^{\left(3 - \left(2 + 2\right)\right)}\right) - -1\right) \]

      18. metadata-eval N/A

          \[\leadsto z0 - \left(z0 - 1\right) \cdot \left(\left(\frac{\left(z1 + z1\right) \cdot z1}{\left(\left(z1 \cdot z1\right) \cdot z1\right) \cdot z1} - 2 \cdot {z1}^{\left(3 - 4\right)}\right) - -1\right) \]

      19. metadata-eval N/A

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

      20. inv-pow N/A

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

   10. Applied rewrites 47.7%

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

Alternative 7: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(z0 - 1\right)\\ \mathbf{if}\;e^{\frac{-2}{z1}} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - t\_0}{z1} - t\_0}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (* 2.0 (- z0 1.0))))
  (if (<= (exp (/ -2.0 z1)) 4e-15)
    (- z0 (* -1.0 1.0))
    (+
     1.0
     (*
      -1.0
      (/
       (-
        (*
         -1.0
         (/ (- (* 1.3333333333333333 (/ (- z0 1.0) z1)) t_0) z1))
        t_0)
       z1))))))

C

double code(double z0, double z1) {
	double t_0 = 2.0 * (z0 - 1.0);
	double tmp;
	if (exp((-2.0 / z1)) <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_0) / z1)) - t_0) / z1));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (z0 - 1.0d0)
    if (exp(((-2.0d0) / z1)) <= 4d-15) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else
        tmp = 1.0d0 + ((-1.0d0) * ((((-1.0d0) * (((1.3333333333333333d0 * ((z0 - 1.0d0) / z1)) - t_0) / z1)) - t_0) / z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = 2.0 * (z0 - 1.0);
	double tmp;
	if (Math.exp((-2.0 / z1)) <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_0) / z1)) - t_0) / z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = 2.0 * (z0 - 1.0)
	tmp = 0
	if math.exp((-2.0 / z1)) <= 4e-15:
		tmp = z0 - (-1.0 * 1.0)
	else:
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_0) / z1)) - t_0) / z1))
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(2.0 * Float64(z0 - 1.0))
	tmp = 0.0
	if (exp(Float64(-2.0 / z1)) <= 4e-15)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(1.3333333333333333 * Float64(Float64(z0 - 1.0) / z1)) - t_0) / z1)) - t_0) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = 2.0 * (z0 - 1.0);
	tmp = 0.0;
	if (exp((-2.0 / z1)) <= 4e-15)
		tmp = z0 - (-1.0 * 1.0);
	else
		tmp = 1.0 + (-1.0 * (((-1.0 * (((1.3333333333333333 * ((z0 - 1.0) / z1)) - t_0) / z1)) - t_0) / z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(2.0 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision], 4e-15], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(1.3333333333333333 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 4.0000000000000003e-15

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 4.0000000000000003e-15 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -40000:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - 1.3333333333333333 \cdot \frac{1}{z1}}{z1}\right)}{z1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(1 - \frac{2 - \frac{2}{z1}}{z1}\right) \cdot \left(z0 - 1\right)}{z0}\right) \cdot z0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -40000.0)
  (- z0 (* -1.0 1.0))
  (if (<= (/ -2.0 z1) 2e+58)
    (+
     1.0
     (/
      (*
       z0
       (+
        2.0
        (* -1.0 (/ (- 2.0 (* 1.3333333333333333 (/ 1.0 z1))) z1))))
      z1))
    (*
     (- 1.0 (/ (* (- 1.0 (/ (- 2.0 (/ 2.0 z1)) z1)) (- z0 1.0)) z0))
     z0))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -40000.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 2e+58) {
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1);
	} else {
		tmp = (1.0 - (((1.0 - ((2.0 - (2.0 / z1)) / z1)) * (z0 - 1.0)) / z0)) * z0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-40000.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 2d+58) then
        tmp = 1.0d0 + ((z0 * (2.0d0 + ((-1.0d0) * ((2.0d0 - (1.3333333333333333d0 * (1.0d0 / z1))) / z1)))) / z1)
    else
        tmp = (1.0d0 - (((1.0d0 - ((2.0d0 - (2.0d0 / z1)) / z1)) * (z0 - 1.0d0)) / z0)) * z0
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -40000.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 2e+58) {
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1);
	} else {
		tmp = (1.0 - (((1.0 - ((2.0 - (2.0 / z1)) / z1)) * (z0 - 1.0)) / z0)) * z0;
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -40000.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 2e+58:
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1)
	else:
		tmp = (1.0 - (((1.0 - ((2.0 - (2.0 / z1)) / z1)) * (z0 - 1.0)) / z0)) * z0
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -40000.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 2e+58)
		tmp = Float64(1.0 + Float64(Float64(z0 * Float64(2.0 + Float64(-1.0 * Float64(Float64(2.0 - Float64(1.3333333333333333 * Float64(1.0 / z1))) / z1)))) / z1));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 - Float64(Float64(2.0 - Float64(2.0 / z1)) / z1)) * Float64(z0 - 1.0)) / z0)) * z0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -40000.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 2e+58)
		tmp = 1.0 + ((z0 * (2.0 + (-1.0 * ((2.0 - (1.3333333333333333 * (1.0 / z1))) / z1)))) / z1);
	else
		tmp = (1.0 - (((1.0 - ((2.0 - (2.0 / z1)) / z1)) * (z0 - 1.0)) / z0)) * z0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -40000.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 2e+58], N[(1.0 + N[(N[(z0 * N[(2.0 + N[(-1.0 * N[(N[(2.0 - N[(1.3333333333333333 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(1.0 - N[(N[(2.0 - N[(2.0 / z1), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -4e4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -4e4 < (/.f64 #s(literal -2 binary64) z1) < 1.9999999999999999e58

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   5. Taylor expanded in z0 around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      3. lower-+.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      6. lower--.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      7. lower-*.f64 N/A

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

      8. lower-/.f64 65.2%

         \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - 1.3333333333333333 \cdot \frac{1}{z1}}{z1}\right)}{z1} \]

   7. Applied rewrites 65.2%

      \[\leadsto 1 + \frac{z0 \cdot \left(2 + -1 \cdot \frac{2 - 1.3333333333333333 \cdot \frac{1}{z1}}{z1}\right)}{\color{blue}{z1}} \]

   if 1.9999999999999999e58 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.7%

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

   4. Applied rewrites 43.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

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

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

   6. Applied rewrites 46.6%

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

Alternative 9: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{-2}{z1}}\\ t_1 := \frac{z0 - 1}{z1}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-15}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 1.0001:\\ \;\;\;\;1 + \left(\frac{\left(1 - z0\right) \cdot 2}{z1 \cdot z1} - t\_1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(-2 \cdot z1 - \left(-2 - t\_1 \cdot 1.3333333333333333\right)\right) \cdot 1}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (exp (/ -2.0 z1))) (t_1 (/ (- z0 1.0) z1)))
  (if (<= t_0 4e-15)
    (- z0 (* -1.0 1.0))
    (if (<= t_0 1.0001)
      (+ 1.0 (- (/ (* (- 1.0 z0) 2.0) (* z1 z1)) (* t_1 -2.0)))
      (+
       1.0
       (/
        (* (- (* -2.0 z1) (- -2.0 (* t_1 1.3333333333333333))) 1.0)
        (* z1 z1)))))))

C

double code(double z0, double z1) {
	double t_0 = exp((-2.0 / z1));
	double t_1 = (z0 - 1.0) / z1;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 1.0001) {
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0));
	} else {
		tmp = 1.0 + ((((-2.0 * z1) - (-2.0 - (t_1 * 1.3333333333333333))) * 1.0) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-2.0d0) / z1))
    t_1 = (z0 - 1.0d0) / z1
    if (t_0 <= 4d-15) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (t_0 <= 1.0001d0) then
        tmp = 1.0d0 + ((((1.0d0 - z0) * 2.0d0) / (z1 * z1)) - (t_1 * (-2.0d0)))
    else
        tmp = 1.0d0 + (((((-2.0d0) * z1) - ((-2.0d0) - (t_1 * 1.3333333333333333d0))) * 1.0d0) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = Math.exp((-2.0 / z1));
	double t_1 = (z0 - 1.0) / z1;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 1.0001) {
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0));
	} else {
		tmp = 1.0 + ((((-2.0 * z1) - (-2.0 - (t_1 * 1.3333333333333333))) * 1.0) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = math.exp((-2.0 / z1))
	t_1 = (z0 - 1.0) / z1
	tmp = 0
	if t_0 <= 4e-15:
		tmp = z0 - (-1.0 * 1.0)
	elif t_0 <= 1.0001:
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0))
	else:
		tmp = 1.0 + ((((-2.0 * z1) - (-2.0 - (t_1 * 1.3333333333333333))) * 1.0) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	t_0 = exp(Float64(-2.0 / z1))
	t_1 = Float64(Float64(z0 - 1.0) / z1)
	tmp = 0.0
	if (t_0 <= 4e-15)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (t_0 <= 1.0001)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - z0) * 2.0) / Float64(z1 * z1)) - Float64(t_1 * -2.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(-2.0 * z1) - Float64(-2.0 - Float64(t_1 * 1.3333333333333333))) * 1.0) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = exp((-2.0 / z1));
	t_1 = (z0 - 1.0) / z1;
	tmp = 0.0;
	if (t_0 <= 4e-15)
		tmp = z0 - (-1.0 * 1.0);
	elseif (t_0 <= 1.0001)
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0));
	else
		tmp = 1.0 + ((((-2.0 * z1) - (-2.0 - (t_1 * 1.3333333333333333))) * 1.0) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-15], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0001], N[(1.0 + N[(N[(N[(N[(1.0 - z0), $MachinePrecision] * 2.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(-2.0 * z1), $MachinePrecision] - N[(-2.0 - N[(t$95$1 * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 4.0000000000000003e-15

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 4.0000000000000003e-15 < (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 1.0001

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. fp-cancel-sign-sub-inv N/A

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

       5. div-sub N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower--.f64 N/A

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

   11. Applied rewrites 66.8%

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

   if 1.0001 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(-2 \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.2%

      \[\leadsto 1 + \frac{\left(-2 \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z0 around 0

       \[\leadsto 1 + \frac{\left(-2 \cdot z1 - \left(-2 - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

   12. Applied rewrites 58.3%

       \[\leadsto 1 + \frac{\left(-2 \cdot z1 - \left(-2 - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{z1 \cdot z1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{-2}{z1}}\\ t_1 := \frac{z0 - 1}{z1}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-15}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 + \left(\frac{\left(1 - z0\right) \cdot 2}{z1 \cdot z1} - t\_1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(1.3333333333333333 \cdot t\_1\right) \cdot 1}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (exp (/ -2.0 z1))) (t_1 (/ (- z0 1.0) z1)))
  (if (<= t_0 4e-15)
    (- z0 (* -1.0 1.0))
    (if (<= t_0 2.0)
      (+ 1.0 (- (/ (* (- 1.0 z0) 2.0) (* z1 z1)) (* t_1 -2.0)))
      (+ 1.0 (/ (* (* 1.3333333333333333 t_1) 1.0) (* z1 z1)))))))

C

double code(double z0, double z1) {
	double t_0 = exp((-2.0 / z1));
	double t_1 = (z0 - 1.0) / z1;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0));
	} else {
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-2.0d0) / z1))
    t_1 = (z0 - 1.0d0) / z1
    if (t_0 <= 4d-15) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 + ((((1.0d0 - z0) * 2.0d0) / (z1 * z1)) - (t_1 * (-2.0d0)))
    else
        tmp = 1.0d0 + (((1.3333333333333333d0 * t_1) * 1.0d0) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = Math.exp((-2.0 / z1));
	double t_1 = (z0 - 1.0) / z1;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0));
	} else {
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = math.exp((-2.0 / z1))
	t_1 = (z0 - 1.0) / z1
	tmp = 0
	if t_0 <= 4e-15:
		tmp = z0 - (-1.0 * 1.0)
	elif t_0 <= 2.0:
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0))
	else:
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	t_0 = exp(Float64(-2.0 / z1))
	t_1 = Float64(Float64(z0 - 1.0) / z1)
	tmp = 0.0
	if (t_0 <= 4e-15)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - z0) * 2.0) / Float64(z1 * z1)) - Float64(t_1 * -2.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.3333333333333333 * t_1) * 1.0) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = exp((-2.0 / z1));
	t_1 = (z0 - 1.0) / z1;
	tmp = 0.0;
	if (t_0 <= 4e-15)
		tmp = z0 - (-1.0 * 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0 + ((((1.0 - z0) * 2.0) / (z1 * z1)) - (t_1 * -2.0));
	else
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-15], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 + N[(N[(N[(N[(1.0 - z0), $MachinePrecision] * 2.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(1.3333333333333333 * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 4.0000000000000003e-15

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 4.0000000000000003e-15 < (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 2

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. fp-cancel-sign-sub-inv N/A

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

       5. div-sub N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower--.f64 N/A

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

   11. Applied rewrites 66.8%

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

   if 2 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

       2. lower-/.f64 N/A

          \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

       3. lower--.f64 57.9%

          \[\leadsto 1 + \frac{\left(1.3333333333333333 \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 57.9%

       \[\leadsto 1 + \frac{\left(1.3333333333333333 \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{-2}{z1}}\\ t_1 := \frac{z0 - 1}{z1}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-15}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{-2 \cdot \left(t\_1 - \left(z0 - 1\right)\right)}{z1} - -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(1.3333333333333333 \cdot t\_1\right) \cdot 1}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (exp (/ -2.0 z1))) (t_1 (/ (- z0 1.0) z1)))
  (if (<= t_0 4e-15)
    (- z0 (* -1.0 1.0))
    (if (<= t_0 2.0)
      (- (/ (* -2.0 (- t_1 (- z0 1.0))) z1) -1.0)
      (+ 1.0 (/ (* (* 1.3333333333333333 t_1) 1.0) (* z1 z1)))))))

C

double code(double z0, double z1) {
	double t_0 = exp((-2.0 / z1));
	double t_1 = (z0 - 1.0) / z1;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 2.0) {
		tmp = ((-2.0 * (t_1 - (z0 - 1.0))) / z1) - -1.0;
	} else {
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-2.0d0) / z1))
    t_1 = (z0 - 1.0d0) / z1
    if (t_0 <= 4d-15) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (t_0 <= 2.0d0) then
        tmp = (((-2.0d0) * (t_1 - (z0 - 1.0d0))) / z1) - (-1.0d0)
    else
        tmp = 1.0d0 + (((1.3333333333333333d0 * t_1) * 1.0d0) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = Math.exp((-2.0 / z1));
	double t_1 = (z0 - 1.0) / z1;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 2.0) {
		tmp = ((-2.0 * (t_1 - (z0 - 1.0))) / z1) - -1.0;
	} else {
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = math.exp((-2.0 / z1))
	t_1 = (z0 - 1.0) / z1
	tmp = 0
	if t_0 <= 4e-15:
		tmp = z0 - (-1.0 * 1.0)
	elif t_0 <= 2.0:
		tmp = ((-2.0 * (t_1 - (z0 - 1.0))) / z1) - -1.0
	else:
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	t_0 = exp(Float64(-2.0 / z1))
	t_1 = Float64(Float64(z0 - 1.0) / z1)
	tmp = 0.0
	if (t_0 <= 4e-15)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(Float64(-2.0 * Float64(t_1 - Float64(z0 - 1.0))) / z1) - -1.0);
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.3333333333333333 * t_1) * 1.0) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = exp((-2.0 / z1));
	t_1 = (z0 - 1.0) / z1;
	tmp = 0.0;
	if (t_0 <= 4e-15)
		tmp = z0 - (-1.0 * 1.0);
	elseif (t_0 <= 2.0)
		tmp = ((-2.0 * (t_1 - (z0 - 1.0))) / z1) - -1.0;
	else
		tmp = 1.0 + (((1.3333333333333333 * t_1) * 1.0) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-15], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 * N[(t$95$1 - N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision], N[(1.0 + N[(N[(N[(1.3333333333333333 * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 4.0000000000000003e-15

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 4.0000000000000003e-15 < (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 2

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

       1. lift--.f64 N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. sinh-+-cosh-rev N/A

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

       5. sinh-+-cosh-rev N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. pow-exp N/A

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

       9. lift--.f64 66.8%

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

       10. lift-+.f64 N/A

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

   11. Applied rewrites 66.8%

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

   if 2 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

       2. lower-/.f64 N/A

          \[\leadsto 1 + \frac{\left(\frac{4}{3} \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

       3. lower--.f64 57.9%

          \[\leadsto 1 + \frac{\left(1.3333333333333333 \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 57.9%

       \[\leadsto 1 + \frac{\left(1.3333333333333333 \cdot \frac{z0 - 1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0 - 1}{z1}\\ \mathbf{if}\;\frac{-2}{z1} \leq -20:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 0.0002:\\ \;\;\;\;1 + 2 \cdot t\_0\\ \mathbf{elif}\;\frac{-2}{z1} \leq 4 \cdot 10^{+96}:\\ \;\;\;\;z0 - \frac{z0 \cdot z0 - 1 \cdot 1}{z0 + 1} \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+150}:\\ \;\;\;\;1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2 \cdot t\_0 + -2}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ (- z0 1.0) z1)))
  (if (<= (/ -2.0 z1) -20.0)
    (- z0 (* -1.0 1.0))
    (if (<= (/ -2.0 z1) 0.0002)
      (+ 1.0 (* 2.0 t_0))
      (if (<= (/ -2.0 z1) 4e+96)
        (- z0 (* (/ (- (* z0 z0) (* 1.0 1.0)) (+ z0 1.0)) 1.0))
        (if (<= (/ -2.0 z1) 1e+150)
          (+ 1.0 (/ (* (/ -1.3333333333333333 z1) 1.0) (* z1 z1)))
          (+ 1.0 (/ (+ (* -2.0 t_0) -2.0) z1))))))))

C

double code(double z0, double z1) {
	double t_0 = (z0 - 1.0) / z1;
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (2.0 * t_0);
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0);
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z0 - 1.0d0) / z1
    if (((-2.0d0) / z1) <= (-20.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 0.0002d0) then
        tmp = 1.0d0 + (2.0d0 * t_0)
    else if (((-2.0d0) / z1) <= 4d+96) then
        tmp = z0 - ((((z0 * z0) - (1.0d0 * 1.0d0)) / (z0 + 1.0d0)) * 1.0d0)
    else if (((-2.0d0) / z1) <= 1d+150) then
        tmp = 1.0d0 + ((((-1.3333333333333333d0) / z1) * 1.0d0) / (z1 * z1))
    else
        tmp = 1.0d0 + ((((-2.0d0) * t_0) + (-2.0d0)) / z1)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = (z0 - 1.0) / z1;
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (2.0 * t_0);
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0);
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1);
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = (z0 - 1.0) / z1
	tmp = 0
	if (-2.0 / z1) <= -20.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 0.0002:
		tmp = 1.0 + (2.0 * t_0)
	elif (-2.0 / z1) <= 4e+96:
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0)
	elif (-2.0 / z1) <= 1e+150:
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1))
	else:
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1)
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(Float64(z0 - 1.0) / z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -20.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 0.0002)
		tmp = Float64(1.0 + Float64(2.0 * t_0));
	elseif (Float64(-2.0 / z1) <= 4e+96)
		tmp = Float64(z0 - Float64(Float64(Float64(Float64(z0 * z0) - Float64(1.0 * 1.0)) / Float64(z0 + 1.0)) * 1.0));
	elseif (Float64(-2.0 / z1) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.3333333333333333 / z1) * 1.0) / Float64(z1 * z1)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-2.0 * t_0) + -2.0) / z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = (z0 - 1.0) / z1;
	tmp = 0.0;
	if ((-2.0 / z1) <= -20.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 0.0002)
		tmp = 1.0 + (2.0 * t_0);
	elseif ((-2.0 / z1) <= 4e+96)
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0);
	elseif ((-2.0 / z1) <= 1e+150)
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	else
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -20.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 0.0002], N[(1.0 + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 4e+96], N[(z0 - N[(N[(N[(N[(z0 * z0), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(z0 + 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(N[(-1.3333333333333333 / z1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] + -2.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -20

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -20 < (/.f64 #s(literal -2 binary64) z1) < 2.0000000000000001e-4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   4. Applied rewrites 58.5%

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

   if 2.0000000000000001e-4 < (/.f64 #s(literal -2 binary64) z1) < 4.0000000000000002e96

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto z0 - \color{blue}{\left(z0 - 1\right)} \cdot 1 \]

      2. flip-- N/A

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

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

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

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

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

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

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

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

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

      7. lower-unsound-+.f64 33.7%

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

   6. Applied rewrites 33.7%

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

   if 4.0000000000000002e96 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999998e149

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\frac{\frac{-4}{3}}{z1} \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.0%

          \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 51.0%

       \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   if 9.9999999999999998e149 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in z0 around 0

       \[\leadsto 1 + \frac{-2 \cdot \frac{z0 - 1}{z1} + -2}{z1} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.2%

       \[\leadsto 1 + \frac{-2 \cdot \frac{z0 - 1}{z1} + -2}{z1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -20:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 0.0002:\\ \;\;\;\;1 + 2 \cdot \frac{z0 - 1}{z1}\\ \mathbf{elif}\;\frac{-2}{z1} \leq 4 \cdot 10^{+96}:\\ \;\;\;\;z0 - \frac{z0 \cdot z0 - 1 \cdot 1}{z0 + 1} \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+150}:\\ \;\;\;\;1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(z1 + z1\right) \cdot \left(z0 - 1\right)}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -20.0)
  (- z0 (* -1.0 1.0))
  (if (<= (/ -2.0 z1) 0.0002)
    (+ 1.0 (* 2.0 (/ (- z0 1.0) z1)))
    (if (<= (/ -2.0 z1) 4e+96)
      (- z0 (* (/ (- (* z0 z0) (* 1.0 1.0)) (+ z0 1.0)) 1.0))
      (if (<= (/ -2.0 z1) 1e+150)
        (+ 1.0 (/ (* (/ -1.3333333333333333 z1) 1.0) (* z1 z1)))
        (+ 1.0 (/ (* (+ z1 z1) (- z0 1.0)) (* z1 z1))))))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0);
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-20.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 0.0002d0) then
        tmp = 1.0d0 + (2.0d0 * ((z0 - 1.0d0) / z1))
    else if (((-2.0d0) / z1) <= 4d+96) then
        tmp = z0 - ((((z0 * z0) - (1.0d0 * 1.0d0)) / (z0 + 1.0d0)) * 1.0d0)
    else if (((-2.0d0) / z1) <= 1d+150) then
        tmp = 1.0d0 + ((((-1.3333333333333333d0) / z1) * 1.0d0) / (z1 * z1))
    else
        tmp = 1.0d0 + (((z1 + z1) * (z0 - 1.0d0)) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 0.0002) {
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0);
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -20.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 0.0002:
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1))
	elif (-2.0 / z1) <= 4e+96:
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0)
	elif (-2.0 / z1) <= 1e+150:
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1))
	else:
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -20.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 0.0002)
		tmp = Float64(1.0 + Float64(2.0 * Float64(Float64(z0 - 1.0) / z1)));
	elseif (Float64(-2.0 / z1) <= 4e+96)
		tmp = Float64(z0 - Float64(Float64(Float64(Float64(z0 * z0) - Float64(1.0 * 1.0)) / Float64(z0 + 1.0)) * 1.0));
	elseif (Float64(-2.0 / z1) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.3333333333333333 / z1) * 1.0) / Float64(z1 * z1)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(z1 + z1) * Float64(z0 - 1.0)) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -20.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 0.0002)
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	elseif ((-2.0 / z1) <= 4e+96)
		tmp = z0 - ((((z0 * z0) - (1.0 * 1.0)) / (z0 + 1.0)) * 1.0);
	elseif ((-2.0 / z1) <= 1e+150)
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	else
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -20.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 0.0002], N[(1.0 + N[(2.0 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 4e+96], N[(z0 - N[(N[(N[(N[(z0 * z0), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(z0 + 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(N[(-1.3333333333333333 / z1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(z1 + z1), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -20

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -20 < (/.f64 #s(literal -2 binary64) z1) < 2.0000000000000001e-4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   4. Applied rewrites 58.5%

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

   if 2.0000000000000001e-4 < (/.f64 #s(literal -2 binary64) z1) < 4.0000000000000002e96

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto z0 - \color{blue}{\left(z0 - 1\right)} \cdot 1 \]

      2. flip-- N/A

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

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

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

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

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

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

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

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

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

      7. lower-unsound-+.f64 33.7%

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

   6. Applied rewrites 33.7%

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

   if 4.0000000000000002e96 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999998e149

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\frac{\frac{-4}{3}}{z1} \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.0%

          \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 51.0%

       \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   if 9.9999999999999998e149 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   6. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. common-denominator N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. count-2 N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 52.1%

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

   8. Applied rewrites 52.1%

      \[\leadsto 1 + \frac{\left(z1 + z1\right) \cdot \left(z0 - 1\right)}{\color{blue}{z1 \cdot z1}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0 - 1}{z1}\\ \mathbf{if}\;\frac{-2}{z1} \leq -20:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 4 \cdot 10^{+96}:\\ \;\;\;\;\frac{-2 \cdot \left(t\_0 - \left(z0 - 1\right)\right)}{z1} - -1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+150}:\\ \;\;\;\;1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2 \cdot t\_0 + -2}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (/ (- z0 1.0) z1)))
  (if (<= (/ -2.0 z1) -20.0)
    (- z0 (* -1.0 1.0))
    (if (<= (/ -2.0 z1) 4e+96)
      (- (/ (* -2.0 (- t_0 (- z0 1.0))) z1) -1.0)
      (if (<= (/ -2.0 z1) 1e+150)
        (+ 1.0 (/ (* (/ -1.3333333333333333 z1) 1.0) (* z1 z1)))
        (+ 1.0 (/ (+ (* -2.0 t_0) -2.0) z1)))))))

C

double code(double z0, double z1) {
	double t_0 = (z0 - 1.0) / z1;
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = ((-2.0 * (t_0 - (z0 - 1.0))) / z1) - -1.0;
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z0 - 1.0d0) / z1
    if (((-2.0d0) / z1) <= (-20.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 4d+96) then
        tmp = (((-2.0d0) * (t_0 - (z0 - 1.0d0))) / z1) - (-1.0d0)
    else if (((-2.0d0) / z1) <= 1d+150) then
        tmp = 1.0d0 + ((((-1.3333333333333333d0) / z1) * 1.0d0) / (z1 * z1))
    else
        tmp = 1.0d0 + ((((-2.0d0) * t_0) + (-2.0d0)) / z1)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = (z0 - 1.0) / z1;
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = ((-2.0 * (t_0 - (z0 - 1.0))) / z1) - -1.0;
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1);
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = (z0 - 1.0) / z1
	tmp = 0
	if (-2.0 / z1) <= -20.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 4e+96:
		tmp = ((-2.0 * (t_0 - (z0 - 1.0))) / z1) - -1.0
	elif (-2.0 / z1) <= 1e+150:
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1))
	else:
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1)
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(Float64(z0 - 1.0) / z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -20.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 4e+96)
		tmp = Float64(Float64(Float64(-2.0 * Float64(t_0 - Float64(z0 - 1.0))) / z1) - -1.0);
	elseif (Float64(-2.0 / z1) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.3333333333333333 / z1) * 1.0) / Float64(z1 * z1)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-2.0 * t_0) + -2.0) / z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = (z0 - 1.0) / z1;
	tmp = 0.0;
	if ((-2.0 / z1) <= -20.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 4e+96)
		tmp = ((-2.0 * (t_0 - (z0 - 1.0))) / z1) - -1.0;
	elseif ((-2.0 / z1) <= 1e+150)
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	else
		tmp = 1.0 + (((-2.0 * t_0) + -2.0) / z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]}, If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -20.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 4e+96], N[(N[(N[(-2.0 * N[(t$95$0 - N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(N[(-1.3333333333333333 / z1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] + -2.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -20

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -20 < (/.f64 #s(literal -2 binary64) z1) < 4.0000000000000002e96

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

       1. lift--.f64 N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. sinh-+-cosh-rev N/A

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

       5. sinh-+-cosh-rev N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. pow-exp N/A

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

       9. lift--.f64 66.8%

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

       10. lift-+.f64 N/A

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

   11. Applied rewrites 66.8%

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

   if 4.0000000000000002e96 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999998e149

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\frac{\frac{-4}{3}}{z1} \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.0%

          \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 51.0%

       \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   if 9.9999999999999998e149 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in z0 around 0

       \[\leadsto 1 + \frac{-2 \cdot \frac{z0 - 1}{z1} + -2}{z1} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.2%

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

Alternative 15: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -40000:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 4 \cdot 10^{+96}:\\ \;\;\;\;1 + \frac{z0 \cdot \left(2 - 2 \cdot \frac{1}{z1}\right)}{z1}\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+150}:\\ \;\;\;\;1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2 \cdot \frac{z0 - 1}{z1} + -2}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -40000.0)
  (- z0 (* -1.0 1.0))
  (if (<= (/ -2.0 z1) 4e+96)
    (+ 1.0 (/ (* z0 (- 2.0 (* 2.0 (/ 1.0 z1)))) z1))
    (if (<= (/ -2.0 z1) 1e+150)
      (+ 1.0 (/ (* (/ -1.3333333333333333 z1) 1.0) (* z1 z1)))
      (+ 1.0 (/ (+ (* -2.0 (/ (- z0 1.0) z1)) -2.0) z1))))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -40000.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = 1.0 + ((z0 * (2.0 - (2.0 * (1.0 / z1)))) / z1);
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((-2.0 * ((z0 - 1.0) / z1)) + -2.0) / z1);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-40000.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 4d+96) then
        tmp = 1.0d0 + ((z0 * (2.0d0 - (2.0d0 * (1.0d0 / z1)))) / z1)
    else if (((-2.0d0) / z1) <= 1d+150) then
        tmp = 1.0d0 + ((((-1.3333333333333333d0) / z1) * 1.0d0) / (z1 * z1))
    else
        tmp = 1.0d0 + ((((-2.0d0) * ((z0 - 1.0d0) / z1)) + (-2.0d0)) / z1)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -40000.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 4e+96) {
		tmp = 1.0 + ((z0 * (2.0 - (2.0 * (1.0 / z1)))) / z1);
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((-2.0 * ((z0 - 1.0) / z1)) + -2.0) / z1);
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -40000.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 4e+96:
		tmp = 1.0 + ((z0 * (2.0 - (2.0 * (1.0 / z1)))) / z1)
	elif (-2.0 / z1) <= 1e+150:
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1))
	else:
		tmp = 1.0 + (((-2.0 * ((z0 - 1.0) / z1)) + -2.0) / z1)
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -40000.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 4e+96)
		tmp = Float64(1.0 + Float64(Float64(z0 * Float64(2.0 - Float64(2.0 * Float64(1.0 / z1)))) / z1));
	elseif (Float64(-2.0 / z1) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.3333333333333333 / z1) * 1.0) / Float64(z1 * z1)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-2.0 * Float64(Float64(z0 - 1.0) / z1)) + -2.0) / z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -40000.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 4e+96)
		tmp = 1.0 + ((z0 * (2.0 - (2.0 * (1.0 / z1)))) / z1);
	elseif ((-2.0 / z1) <= 1e+150)
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	else
		tmp = 1.0 + (((-2.0 * ((z0 - 1.0) / z1)) + -2.0) / z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -40000.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 4e+96], N[(1.0 + N[(N[(z0 * N[(2.0 - N[(2.0 * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(N[(-1.3333333333333333 / z1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(-2.0 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -4e4

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -4e4 < (/.f64 #s(literal -2 binary64) z1) < 4.0000000000000002e96

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in z0 around inf

       \[\leadsto 1 + \frac{z0 \cdot \left(2 - 2 \cdot \frac{1}{z1}\right)}{z1} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 61.5%

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

   12. Applied rewrites 61.5%

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

   if 4.0000000000000002e96 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999998e149

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\frac{\frac{-4}{3}}{z1} \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.0%

          \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 51.0%

       \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   if 9.9999999999999998e149 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in z0 around 0

       \[\leadsto 1 + \frac{-2 \cdot \frac{z0 - 1}{z1} + -2}{z1} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.2%

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

Alternative 16: 77.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -20:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+65}:\\ \;\;\;\;1 + 2 \cdot \frac{z0 - 1}{z1}\\ \mathbf{elif}\;\frac{-2}{z1} \leq 10^{+150}:\\ \;\;\;\;1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(z1 + z1\right) \cdot \left(z0 - 1\right)}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -20.0)
  (- z0 (* -1.0 1.0))
  (if (<= (/ -2.0 z1) 1e+65)
    (+ 1.0 (* 2.0 (/ (- z0 1.0) z1)))
    (if (<= (/ -2.0 z1) 1e+150)
      (+ 1.0 (/ (* (/ -1.3333333333333333 z1) 1.0) (* z1 z1)))
      (+ 1.0 (/ (* (+ z1 z1) (- z0 1.0)) (* z1 z1)))))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 1e+65) {
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-20.0d0)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (((-2.0d0) / z1) <= 1d+65) then
        tmp = 1.0d0 + (2.0d0 * ((z0 - 1.0d0) / z1))
    else if (((-2.0d0) / z1) <= 1d+150) then
        tmp = 1.0d0 + ((((-1.3333333333333333d0) / z1) * 1.0d0) / (z1 * z1))
    else
        tmp = 1.0d0 + (((z1 + z1) * (z0 - 1.0d0)) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -20.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if ((-2.0 / z1) <= 1e+65) {
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	} else if ((-2.0 / z1) <= 1e+150) {
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	} else {
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -20.0:
		tmp = z0 - (-1.0 * 1.0)
	elif (-2.0 / z1) <= 1e+65:
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1))
	elif (-2.0 / z1) <= 1e+150:
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1))
	else:
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -20.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (Float64(-2.0 / z1) <= 1e+65)
		tmp = Float64(1.0 + Float64(2.0 * Float64(Float64(z0 - 1.0) / z1)));
	elseif (Float64(-2.0 / z1) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.3333333333333333 / z1) * 1.0) / Float64(z1 * z1)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(z1 + z1) * Float64(z0 - 1.0)) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -20.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif ((-2.0 / z1) <= 1e+65)
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	elseif ((-2.0 / z1) <= 1e+150)
		tmp = 1.0 + (((-1.3333333333333333 / z1) * 1.0) / (z1 * z1));
	else
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -20.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+65], N[(1.0 + N[(2.0 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 / z1), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(N[(-1.3333333333333333 / z1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(z1 + z1), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -20

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -20 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999999e64

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   4. Applied rewrites 58.5%

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

   if 9.9999999999999999e64 < (/.f64 #s(literal -2 binary64) z1) < 9.9999999999999998e149

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - \frac{4}{3} \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

      5. lower-/.f64 51.4%

         \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   9. Applied rewrites 51.4%

      \[\leadsto 1 + \frac{\left(\left(2 + -2 \cdot z1\right) - 1.3333333333333333 \cdot \frac{1}{z1}\right) \cdot 1}{z1 \cdot z1} \]

   10. Taylor expanded in z1 around 0

       \[\leadsto 1 + \frac{\frac{\frac{-4}{3}}{z1} \cdot 1}{z1 \cdot z1} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.0%

          \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   12. Applied rewrites 51.0%

       \[\leadsto 1 + \frac{\frac{-1.3333333333333333}{z1} \cdot 1}{z1 \cdot z1} \]

   if 9.9999999999999998e149 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   6. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. common-denominator N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. count-2 N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 52.1%

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

   8. Applied rewrites 52.1%

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

Alternative 17: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{-2}{z1}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;1 + 2 \cdot \frac{z0}{z1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(z1 + z1\right) \cdot \left(z0 - 1\right)}{z1 \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (exp (/ -2.0 z1))))
  (if (<= t_0 0.0)
    (- z0 (* -1.0 1.0))
    (if (<= t_0 1.0)
      (+ 1.0 (* 2.0 (/ z0 z1)))
      (+ 1.0 (/ (* (+ z1 z1) (- z0 1.0)) (* z1 z1)))))))

C

double code(double z0, double z1) {
	double t_0 = exp((-2.0 / z1));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 1.0) {
		tmp = 1.0 + (2.0 * (z0 / z1));
	} else {
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((-2.0d0) / z1))
    if (t_0 <= 0.0d0) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else if (t_0 <= 1.0d0) then
        tmp = 1.0d0 + (2.0d0 * (z0 / z1))
    else
        tmp = 1.0d0 + (((z1 + z1) * (z0 - 1.0d0)) / (z1 * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double t_0 = Math.exp((-2.0 / z1));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else if (t_0 <= 1.0) {
		tmp = 1.0 + (2.0 * (z0 / z1));
	} else {
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = math.exp((-2.0 / z1))
	tmp = 0
	if t_0 <= 0.0:
		tmp = z0 - (-1.0 * 1.0)
	elif t_0 <= 1.0:
		tmp = 1.0 + (2.0 * (z0 / z1))
	else:
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1))
	return tmp

Julia

function code(z0, z1)
	t_0 = exp(Float64(-2.0 / z1))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(1.0 + Float64(2.0 * Float64(z0 / z1)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(z1 + z1) * Float64(z0 - 1.0)) / Float64(z1 * z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = exp((-2.0 / z1));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = z0 - (-1.0 * 1.0);
	elseif (t_0 <= 1.0)
		tmp = 1.0 + (2.0 * (z0 / z1));
	else
		tmp = 1.0 + (((z1 + z1) * (z0 - 1.0)) / (z1 * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(1.0 + N[(2.0 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(z1 + z1), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision] / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 0.0

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 0.0 < (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 1

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in z0 around inf

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

      1. lower-/.f64 58.0%

         \[\leadsto 1 + 2 \cdot \frac{z0}{z1} \]

   9. Applied rewrites 58.0%

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

   if 1 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   6. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. common-denominator N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. count-2 N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 52.1%

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

   8. Applied rewrites 52.1%

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

Alternative 18: 69.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{\frac{-2}{z1}} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{z0 - 1}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (exp (/ -2.0 z1)) 4e-15)
  (- z0 (* -1.0 1.0))
  (+ 1.0 (* 2.0 (/ (- z0 1.0) z1)))))

C

double code(double z0, double z1) {
	double tmp;
	if (exp((-2.0 / z1)) <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (exp(((-2.0d0) / z1)) <= 4d-15) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * ((z0 - 1.0d0) / z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if (Math.exp((-2.0 / z1)) <= 4e-15) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if math.exp((-2.0 / z1)) <= 4e-15:
		tmp = z0 - (-1.0 * 1.0)
	else:
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (exp(Float64(-2.0 / z1)) <= 4e-15)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(Float64(z0 - 1.0) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if (exp((-2.0 / z1)) <= 4e-15)
		tmp = z0 - (-1.0 * 1.0);
	else
		tmp = 1.0 + (2.0 * ((z0 - 1.0) / z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision], 4e-15], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 * N[(N[(z0 - 1.0), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 4.0000000000000003e-15

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 4.0000000000000003e-15 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   4. Applied rewrites 58.5%

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

Alternative 19: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{\frac{-2}{z1}} \leq 0:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{z0}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (exp (/ -2.0 z1)) 0.0)
  (- z0 (* -1.0 1.0))
  (+ 1.0 (* 2.0 (/ z0 z1)))))

C

double code(double z0, double z1) {
	double tmp;
	if (exp((-2.0 / z1)) <= 0.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (2.0 * (z0 / z1));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (exp(((-2.0d0) / z1)) <= 0.0d0) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (z0 / z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if (Math.exp((-2.0 / z1)) <= 0.0) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (2.0 * (z0 / z1));
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if math.exp((-2.0 / z1)) <= 0.0:
		tmp = z0 - (-1.0 * 1.0)
	else:
		tmp = 1.0 + (2.0 * (z0 / z1))
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (exp(Float64(-2.0 / z1)) <= 0.0)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(z0 / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if (exp((-2.0 / z1)) <= 0.0)
		tmp = z0 - (-1.0 * 1.0);
	else
		tmp = 1.0 + (2.0 * (z0 / z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[Exp[N[(-2.0 / z1), $MachinePrecision]], $MachinePrecision], 0.0], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 #s(literal -2 binary64) z1)) < 0.0

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if 0.0 < (exp.f64 (/.f64 #s(literal -2 binary64) z1))

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in z0 around inf

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

      1. lower-/.f64 58.0%

         \[\leadsto 1 + 2 \cdot \frac{z0}{z1} \]

   9. Applied rewrites 58.0%

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

Alternative 20: 50.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -3.5e-75) (- z0 (* -1.0 1.0)) (+ 1.0 (/ -2.0 z1))))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -3.5e-75) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (-2.0 / z1);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-3.5d-75)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else
        tmp = 1.0d0 + ((-2.0d0) / z1)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -3.5e-75) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0 + (-2.0 / z1);
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -3.5e-75:
		tmp = z0 - (-1.0 * 1.0)
	else:
		tmp = 1.0 + (-2.0 / z1)
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -3.5e-75)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	else
		tmp = Float64(1.0 + Float64(-2.0 / z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -3.5e-75)
		tmp = z0 - (-1.0 * 1.0);
	else
		tmp = 1.0 + (-2.0 / z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -3.5e-75], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-2.0 / z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -3.4999999999999999e-75

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -3.4999999999999999e-75 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 76.8%

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

   3. Applied rewrites 76.8%

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

   4. Taylor expanded in z1 around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 58.5%

         \[\leadsto 1 + 2 \cdot \frac{z0 - 1}{z1} \]

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in z0 around 0

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

      1. lower-/.f64 39.5%

         \[\leadsto 1 + \frac{-2}{z1} \]

   9. Applied rewrites 39.5%

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

Alternative 21: 50.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-2}{z1} \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;z0 - -1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= (/ -2.0 z1) -3.5e-75) (- z0 (* -1.0 1.0)) 1.0))

C

double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -3.5e-75) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((-2.0d0) / z1) <= (-3.5d-75)) then
        tmp = z0 - ((-1.0d0) * 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1) {
	double tmp;
	if ((-2.0 / z1) <= -3.5e-75) {
		tmp = z0 - (-1.0 * 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if (-2.0 / z1) <= -3.5e-75:
		tmp = z0 - (-1.0 * 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (Float64(-2.0 / z1) <= -3.5e-75)
		tmp = Float64(z0 - Float64(-1.0 * 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if ((-2.0 / z1) <= -3.5e-75)
		tmp = z0 - (-1.0 * 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[N[(-2.0 / z1), $MachinePrecision], -3.5e-75], N[(z0 - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal -2 binary64) z1) < -3.4999999999999999e-75

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around inf

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 27.8%

      \[\leadsto z0 - \left(z0 - 1\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto z0 - \color{blue}{-1} \cdot 1 \]

   if -3.4999999999999999e-75 < (/.f64 #s(literal -2 binary64) z1)

   1. Initial program 76.8%

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

   2. Taylor expanded in z1 around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

      4. mult-flip N/A

         \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

   6. Applied rewrites 57.8%

      \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

   7. Taylor expanded in z1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 66.8%

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in z1 around inf

       \[\leadsto 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 39.0%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 39.0% accurate, 123.0× speedup

Math

\[1 \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  1.0)

C

double code(double z0, double z1) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = 1.0d0
end function

Java

public static double code(double z0, double z1) {
	return 1.0;
}

Python

def code(z0, z1):
	return 1.0

Julia

function code(z0, z1)
	return 1.0
end

MATLAB

function tmp = code(z0, z1)
	tmp = 1.0;
end

Wolfram

code[z0_, z1_] := 1.0

Derivation

1. Initial program 76.8%

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

2. Taylor expanded in z1 around -inf

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

   1. lower-+.f64 N/A

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   3. lower-/.f64 N/A

      \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

4. Applied rewrites 71.0%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{z1}} \]

   2. lift-/.f64 N/A

      \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)}{\color{blue}{z1}} \]

   3. associate-*r/ N/A

      \[\leadsto 1 + \frac{-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)}{\color{blue}{z1}} \]

   4. mult-flip N/A

      \[\leadsto 1 + \left(-1 \cdot \left(-1 \cdot \frac{\frac{4}{3} \cdot \frac{z0 - 1}{z1} - 2 \cdot \left(z0 - 1\right)}{z1} - 2 \cdot \left(z0 - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{z1}} \]

6. Applied rewrites 57.8%

   \[\leadsto 1 + \frac{\left(\left(2 \cdot \left(z0 - 1\right)\right) \cdot z1 - \left(2 \cdot \left(z0 - 1\right) - \frac{z0 - 1}{z1} \cdot 1.3333333333333333\right)\right) \cdot 1}{\color{blue}{z1 \cdot z1}} \]

7. Taylor expanded in z1 around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 66.8%

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

9. Applied rewrites 66.8%

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

10. Taylor expanded in z1 around inf

    \[\leadsto 1 \]
11. Step-by-step derivation

12. Applied rewrites 39.0%

    \[\leadsto 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0 z1)
  :name "(- z0 (* (- z0 1) (exp (/ -2 z1))))"
  :precision binary64
  (- z0 (* (- z0 1.0) (exp (/ -2.0 z1)))))