(log (/ -27/64 (* (* (- 1 z0) (- 1 z0)) (- z0 1))))

Percentage Accurate: 79.1% → 99.8%

Time: 2.0s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (log (/ -0.421875 (* (* (- 1.0 z0) (- 1.0 z0)) (- z0 1.0)))))

C

double code(double z0) {
	return log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = log(((-0.421875d0) / (((1.0d0 - z0) * (1.0d0 - z0)) * (z0 - 1.0d0))))
end function

Java

public static double code(double z0) {
	return Math.log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))));
}

Python

def code(z0):
	return math.log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))))

Julia

function code(z0)
	return log(Float64(-0.421875 / Float64(Float64(Float64(1.0 - z0) * Float64(1.0 - z0)) * Float64(z0 - 1.0))))
end

MATLAB

function tmp = code(z0)
	tmp = log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))));
end

Wolfram

code[z0_] := N[Log[N[(-0.421875 / N[(N[(N[(1.0 - z0), $MachinePrecision] * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.1% accurate, 1.0× speedup

Math

\[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (log (/ -0.421875 (* (* (- 1.0 z0) (- 1.0 z0)) (- z0 1.0)))))

C

double code(double z0) {
	return log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = log(((-0.421875d0) / (((1.0d0 - z0) * (1.0d0 - z0)) * (z0 - 1.0d0))))
end function

Java

public static double code(double z0) {
	return Math.log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))));
}

Python

def code(z0):
	return math.log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))))

Julia

function code(z0)
	return log(Float64(-0.421875 / Float64(Float64(Float64(1.0 - z0) * Float64(1.0 - z0)) * Float64(z0 - 1.0))))
end

MATLAB

function tmp = code(z0)
	tmp = log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))));
end

Wolfram

code[z0_] := N[Log[N[(-0.421875 / N[(N[(N[(1.0 - z0), $MachinePrecision] * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\log \left(-z0\right) \cdot -2 - \log \left(-2.3703703703703702 \cdot z0\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= z0 -4e+102)
  (- (* (log (- z0)) -2.0) (log (* -2.3703703703703702 z0)))
  (log
   (/
    1.0
    (+
     2.3703703703703702
     (*
      z0
      (-
       (* z0 (+ 7.111111111111111 (* -2.3703703703703702 z0)))
       7.111111111111111)))))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -4e+102) {
		tmp = (log(-z0) * -2.0) - log((-2.3703703703703702 * z0));
	} else {
		tmp = log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-4d+102)) then
        tmp = (log(-z0) * (-2.0d0)) - log(((-2.3703703703703702d0) * z0))
    else
        tmp = log((1.0d0 / (2.3703703703703702d0 + (z0 * ((z0 * (7.111111111111111d0 + ((-2.3703703703703702d0) * z0))) - 7.111111111111111d0)))))
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -4e+102) {
		tmp = (Math.log(-z0) * -2.0) - Math.log((-2.3703703703703702 * z0));
	} else {
		tmp = Math.log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -4e+102:
		tmp = (math.log(-z0) * -2.0) - math.log((-2.3703703703703702 * z0))
	else:
		tmp = math.log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -4e+102)
		tmp = Float64(Float64(log(Float64(-z0)) * -2.0) - log(Float64(-2.3703703703703702 * z0)));
	else
		tmp = log(Float64(1.0 / Float64(2.3703703703703702 + Float64(z0 * Float64(Float64(z0 * Float64(7.111111111111111 + Float64(-2.3703703703703702 * z0))) - 7.111111111111111)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -4e+102)
		tmp = (log(-z0) * -2.0) - log((-2.3703703703703702 * z0));
	else
		tmp = log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -4e+102], N[(N[(N[Log[(-z0)], $MachinePrecision] * -2.0), $MachinePrecision] - N[Log[N[(-2.3703703703703702 * z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 / N[(2.3703703703703702 + N[(z0 * N[(N[(z0 * N[(7.111111111111111 + N[(-2.3703703703703702 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 7.111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -3.9999999999999999e102

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{\log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

      5. lower-/.f64 32.3%

         \[\leadsto \log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. +-commutative N/A

         \[\leadsto 3 \cdot \log \left(\frac{-1}{z0}\right) + \color{blue}{\log \frac{27}{64}} \]

      3. lift-*.f64 N/A

         \[\leadsto 3 \cdot \log \left(\frac{-1}{z0}\right) + \log \color{blue}{\frac{27}{64}} \]

      4. lift-log.f64 N/A

         \[\leadsto 3 \cdot \log \left(\frac{-1}{z0}\right) + \log \frac{27}{64} \]

      5. log-pow-rev N/A

         \[\leadsto \log \left({\left(\frac{-1}{z0}\right)}^{3}\right) + \log \color{blue}{\frac{27}{64}} \]

      6. lift-log.f64 N/A

         \[\leadsto \log \left({\left(\frac{-1}{z0}\right)}^{3}\right) + \log \frac{27}{64} \]

      7. sum-log N/A

         \[\leadsto \log \left({\left(\frac{-1}{z0}\right)}^{3} \cdot \frac{27}{64}\right) \]

      8. unpow3 N/A

         \[\leadsto \log \left(\left(\left(\frac{-1}{z0} \cdot \frac{-1}{z0}\right) \cdot \frac{-1}{z0}\right) \cdot \frac{27}{64}\right) \]

      9. associate-*l* N/A

         \[\leadsto \log \left(\left(\frac{-1}{z0} \cdot \frac{-1}{z0}\right) \cdot \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)\right) \]

      10. log-prod N/A

          \[\leadsto \log \left(\frac{-1}{z0} \cdot \frac{-1}{z0}\right) + \color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)} \]

      11. lower-unsound-+.f64 N/A

          \[\leadsto \log \left(\frac{-1}{z0} \cdot \frac{-1}{z0}\right) + \color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)} \]

   6. Applied rewrites 32.3%

      \[\leadsto 2 \cdot \left(-\log \left(-z0\right)\right) + \color{blue}{\log \left(\frac{-1}{z0} \cdot 0.421875\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(-\log \left(-z0\right)\right) + \color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)} \]

      2. add-flip N/A

         \[\leadsto 2 \cdot \left(-\log \left(-z0\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(-\log \left(-z0\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(-\log \left(-z0\right)\right) - \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)}\right)\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\log \left(-z0\right)\right)\right) - \left(\mathsf{neg}\left(\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)\right)\right) \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \log \left(-z0\right)\right)\right) - \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)}\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(2\right)\right) \cdot \log \left(-z0\right) - \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(2\right)\right) \cdot \log \left(-z0\right) - \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto -2 \cdot \log \left(-z0\right) - \left(\mathsf{neg}\left(\log \color{blue}{\left(\frac{-1}{z0} \cdot \frac{27}{64}\right)}\right)\right) \]

      10. lift-log.f64 N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \left(\mathsf{neg}\left(\log \left(\frac{-1}{z0} \cdot \frac{27}{64}\right)\right)\right) \]

      11. neg-log N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{1}{\frac{-1}{z0} \cdot \frac{27}{64}}\right) \]

      12. lower-log.f64 N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{1}{\frac{-1}{z0} \cdot \frac{27}{64}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{1}{\frac{-1}{z0} \cdot \frac{27}{64}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{1}{\frac{-1}{z0} \cdot \frac{27}{64}}\right) \]

      15. associate-*l/ N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{1}{\frac{-1 \cdot \frac{27}{64}}{z0}}\right) \]

      16. metadata-eval N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{1}{\frac{\frac{-27}{64}}{z0}}\right) \]

      17. div-flip-rev N/A

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{z0}{\frac{-27}{64}}\right) \]

      18. lower-/.f64 32.3%

          \[\leadsto -2 \cdot \log \left(-z0\right) - \log \left(\frac{z0}{-0.421875}\right) \]

   8. Applied rewrites 32.3%

      \[\leadsto -2 \cdot \log \left(-z0\right) - \color{blue}{\log \left(\frac{z0}{-0.421875}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -2 \cdot \log \left(-z0\right) - \log \color{blue}{\left(\frac{z0}{\frac{-27}{64}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \color{blue}{\left(\frac{z0}{\frac{-27}{64}}\right)} \]

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

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(\frac{\color{blue}{z0}}{\frac{-27}{64}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \color{blue}{\left(\frac{z0}{\frac{-27}{64}}\right)} \]

      5. lower-unsound-log.f64 32.3%

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(\frac{\color{blue}{z0}}{-0.421875}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(\frac{z0}{\frac{-27}{64}}\right) \]

      7. mult-flip N/A

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(z0 \cdot \frac{1}{\frac{-27}{64}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(z0 \cdot \frac{-64}{27}\right) \]

      9. *-commutative N/A

         \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(\frac{-64}{27} \cdot z0\right) \]

      10. lift-*.f64 32.3%

          \[\leadsto \log \left(-z0\right) \cdot -2 - \log \left(-2.3703703703703702 \cdot z0\right) \]

   10. Applied rewrites 32.3%

       \[\leadsto \log \left(-z0\right) \cdot -2 - \color{blue}{\log \left(-2.3703703703703702 \cdot z0\right)} \]

   if -3.9999999999999999e102 < z0

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      2. div-flip N/A

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{\frac{-27}{64}}}\right)} \]

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

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{\frac{-27}{64}}}\right)} \]

      4. lower-unsound-/.f64 79.1%

         \[\leadsto \log \left(\frac{1}{\color{blue}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{-0.421875}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}}{\frac{-27}{64}}}\right) \]

      6. *-commutative N/A

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(z0 - 1\right) \cdot \left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{\frac{-27}{64}}}\right) \]

      7. lower-*.f64 79.1%

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(z0 - 1\right) \cdot \left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{-0.421875}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{\frac{-27}{64}}}\right) \]

      9. sqr-neg-rev N/A

         \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}}{\frac{-27}{64}}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\color{blue}{\left(z0 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      12. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\color{blue}{\left(z0 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \color{blue}{\left(z0 - 1\right)}\right)}{\frac{-27}{64}}}\right) \]

      15. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \color{blue}{\left(z0 - 1\right)}\right)}{\frac{-27}{64}}}\right) \]

      16. lower-*.f64 79.1%

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(z0 - 1\right) \cdot \left(z0 - 1\right)\right)}}{-0.421875}}\right) \]

   3. Applied rewrites 79.1%

      \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \left(z0 - 1\right)\right)}{-0.421875}}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \log \left(\frac{1}{\color{blue}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + \color{blue}{z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \color{blue}{\frac{64}{9}}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right) \]

      6. lower-*.f64 79.1%

         \[\leadsto \log \left(\frac{1}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}\right) \]

   6. Applied rewrites 79.1%

      \[\leadsto \log \left(\frac{1}{\color{blue}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;-0.8630462173553428 + 3 \cdot \log \left(\frac{-1}{z0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= z0 -4e+102)
  (+ -0.8630462173553428 (* 3.0 (log (/ -1.0 z0))))
  (log
   (/
    1.0
    (+
     2.3703703703703702
     (*
      z0
      (-
       (* z0 (+ 7.111111111111111 (* -2.3703703703703702 z0)))
       7.111111111111111)))))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -4e+102) {
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	} else {
		tmp = log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-4d+102)) then
        tmp = (-0.8630462173553428d0) + (3.0d0 * log(((-1.0d0) / z0)))
    else
        tmp = log((1.0d0 / (2.3703703703703702d0 + (z0 * ((z0 * (7.111111111111111d0 + ((-2.3703703703703702d0) * z0))) - 7.111111111111111d0)))))
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -4e+102) {
		tmp = -0.8630462173553428 + (3.0 * Math.log((-1.0 / z0)));
	} else {
		tmp = Math.log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -4e+102:
		tmp = -0.8630462173553428 + (3.0 * math.log((-1.0 / z0)))
	else:
		tmp = math.log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -4e+102)
		tmp = Float64(-0.8630462173553428 + Float64(3.0 * log(Float64(-1.0 / z0))));
	else
		tmp = log(Float64(1.0 / Float64(2.3703703703703702 + Float64(z0 * Float64(Float64(z0 * Float64(7.111111111111111 + Float64(-2.3703703703703702 * z0))) - 7.111111111111111)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -4e+102)
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	else
		tmp = log((1.0 / (2.3703703703703702 + (z0 * ((z0 * (7.111111111111111 + (-2.3703703703703702 * z0))) - 7.111111111111111)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -4e+102], N[(-0.8630462173553428 + N[(3.0 * N[Log[N[(-1.0 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 / N[(2.3703703703703702 + N[(z0 * N[(N[(z0 * N[(7.111111111111111 + N[(-2.3703703703703702 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 7.111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -3.9999999999999999e102

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{\log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

      5. lower-/.f64 32.3%

         \[\leadsto \log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]

   5. Evaluated real constant 32.3%

      \[\leadsto -0.8630462173553428 + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

   if -3.9999999999999999e102 < z0

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      2. div-flip N/A

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{\frac{-27}{64}}}\right)} \]

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

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{\frac{-27}{64}}}\right)} \]

      4. lower-unsound-/.f64 79.1%

         \[\leadsto \log \left(\frac{1}{\color{blue}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{-0.421875}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}}{\frac{-27}{64}}}\right) \]

      6. *-commutative N/A

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(z0 - 1\right) \cdot \left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{\frac{-27}{64}}}\right) \]

      7. lower-*.f64 79.1%

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(z0 - 1\right) \cdot \left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{-0.421875}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{\frac{-27}{64}}}\right) \]

      9. sqr-neg-rev N/A

         \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}}{\frac{-27}{64}}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\color{blue}{\left(z0 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      12. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\color{blue}{\left(z0 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \color{blue}{\left(z0 - 1\right)}\right)}{\frac{-27}{64}}}\right) \]

      15. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \color{blue}{\left(z0 - 1\right)}\right)}{\frac{-27}{64}}}\right) \]

      16. lower-*.f64 79.1%

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(z0 - 1\right) \cdot \left(z0 - 1\right)\right)}}{-0.421875}}\right) \]

   3. Applied rewrites 79.1%

      \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \left(z0 - 1\right)\right)}{-0.421875}}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \log \left(\frac{1}{\color{blue}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + \color{blue}{z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \color{blue}{\frac{64}{9}}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right) \]

      6. lower-*.f64 79.1%

         \[\leadsto \log \left(\frac{1}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}\right) \]

   6. Applied rewrites 79.1%

      \[\leadsto \log \left(\frac{1}{\color{blue}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -4 \cdot 10^{+103}:\\ \;\;\;\;-0.8630462173553428 + 3 \cdot \log \left(\frac{-1}{z0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= z0 -4e+103)
  (+ -0.8630462173553428 (* 3.0 (log (/ -1.0 z0))))
  (log (/ -0.421875 (* (* (- 1.0 z0) (- 1.0 z0)) (- z0 1.0))))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -4e+103) {
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	} else {
		tmp = log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-4d+103)) then
        tmp = (-0.8630462173553428d0) + (3.0d0 * log(((-1.0d0) / z0)))
    else
        tmp = log(((-0.421875d0) / (((1.0d0 - z0) * (1.0d0 - z0)) * (z0 - 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -4e+103) {
		tmp = -0.8630462173553428 + (3.0 * Math.log((-1.0 / z0)));
	} else {
		tmp = Math.log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -4e+103:
		tmp = -0.8630462173553428 + (3.0 * math.log((-1.0 / z0)))
	else:
		tmp = math.log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -4e+103)
		tmp = Float64(-0.8630462173553428 + Float64(3.0 * log(Float64(-1.0 / z0))));
	else
		tmp = log(Float64(-0.421875 / Float64(Float64(Float64(1.0 - z0) * Float64(1.0 - z0)) * Float64(z0 - 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -4e+103)
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	else
		tmp = log((-0.421875 / (((1.0 - z0) * (1.0 - z0)) * (z0 - 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -4e+103], N[(-0.8630462173553428 + N[(3.0 * N[Log[N[(-1.0 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(-0.421875 / N[(N[(N[(1.0 - z0), $MachinePrecision] * N[(1.0 - z0), $MachinePrecision]), $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -4e103

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{\log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

      5. lower-/.f64 32.3%

         \[\leadsto \log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]

   5. Evaluated real constant 32.3%

      \[\leadsto -0.8630462173553428 + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

   if -4e103 < z0

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;-0.8630462173553428 + 3 \cdot \log \left(\frac{-1}{z0}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\left(\left(-2.3703703703703702 \cdot z0 - -7.111111111111111\right) \cdot z0 - 7.111111111111111\right) \cdot z0 - -2.3703703703703702\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= z0 -4e+102)
  (+ -0.8630462173553428 (* 3.0 (log (/ -1.0 z0))))
  (-
   (log
    (-
     (*
      (-
       (* (- (* -2.3703703703703702 z0) -7.111111111111111) z0)
       7.111111111111111)
      z0)
     -2.3703703703703702)))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -4e+102) {
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	} else {
		tmp = -log(((((((-2.3703703703703702 * z0) - -7.111111111111111) * z0) - 7.111111111111111) * z0) - -2.3703703703703702));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-4d+102)) then
        tmp = (-0.8630462173553428d0) + (3.0d0 * log(((-1.0d0) / z0)))
    else
        tmp = -log((((((((-2.3703703703703702d0) * z0) - (-7.111111111111111d0)) * z0) - 7.111111111111111d0) * z0) - (-2.3703703703703702d0)))
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -4e+102) {
		tmp = -0.8630462173553428 + (3.0 * Math.log((-1.0 / z0)));
	} else {
		tmp = -Math.log(((((((-2.3703703703703702 * z0) - -7.111111111111111) * z0) - 7.111111111111111) * z0) - -2.3703703703703702));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -4e+102:
		tmp = -0.8630462173553428 + (3.0 * math.log((-1.0 / z0)))
	else:
		tmp = -math.log(((((((-2.3703703703703702 * z0) - -7.111111111111111) * z0) - 7.111111111111111) * z0) - -2.3703703703703702))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -4e+102)
		tmp = Float64(-0.8630462173553428 + Float64(3.0 * log(Float64(-1.0 / z0))));
	else
		tmp = Float64(-log(Float64(Float64(Float64(Float64(Float64(Float64(-2.3703703703703702 * z0) - -7.111111111111111) * z0) - 7.111111111111111) * z0) - -2.3703703703703702)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -4e+102)
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	else
		tmp = -log(((((((-2.3703703703703702 * z0) - -7.111111111111111) * z0) - 7.111111111111111) * z0) - -2.3703703703703702));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -4e+102], N[(-0.8630462173553428 + N[(3.0 * N[Log[N[(-1.0 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N[(N[(N[(N[(N[(-2.3703703703703702 * z0), $MachinePrecision] - -7.111111111111111), $MachinePrecision] * z0), $MachinePrecision] - 7.111111111111111), $MachinePrecision] * z0), $MachinePrecision] - -2.3703703703703702), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z0 < -3.9999999999999999e102

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{\log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

      5. lower-/.f64 32.3%

         \[\leadsto \log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]

   5. Evaluated real constant 32.3%

      \[\leadsto -0.8630462173553428 + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

   if -3.9999999999999999e102 < z0

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      2. div-flip N/A

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{\frac{-27}{64}}}\right)} \]

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

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{\frac{-27}{64}}}\right)} \]

      4. lower-unsound-/.f64 79.1%

         \[\leadsto \log \left(\frac{1}{\color{blue}{\frac{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}{-0.421875}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}}{\frac{-27}{64}}}\right) \]

      6. *-commutative N/A

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(z0 - 1\right) \cdot \left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{\frac{-27}{64}}}\right) \]

      7. lower-*.f64 79.1%

         \[\leadsto \log \left(\frac{1}{\frac{\color{blue}{\left(z0 - 1\right) \cdot \left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{-0.421875}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right)}}{\frac{-27}{64}}}\right) \]

      9. sqr-neg-rev N/A

         \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}}{\frac{-27}{64}}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      11. sub-negate-rev N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\color{blue}{\left(z0 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      12. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\color{blue}{\left(z0 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right)\right)}{\frac{-27}{64}}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \color{blue}{\left(z0 - 1\right)}\right)}{\frac{-27}{64}}}\right) \]

      15. lift--.f64 N/A

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \color{blue}{\left(z0 - 1\right)}\right)}{\frac{-27}{64}}}\right) \]

      16. lower-*.f64 79.1%

          \[\leadsto \log \left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \color{blue}{\left(\left(z0 - 1\right) \cdot \left(z0 - 1\right)\right)}}{-0.421875}}\right) \]

   3. Applied rewrites 79.1%

      \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\left(z0 - 1\right) \cdot \left(\left(z0 - 1\right) \cdot \left(z0 - 1\right)\right)}{-0.421875}}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \log \left(\frac{1}{\color{blue}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + \color{blue}{z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \color{blue}{\frac{64}{9}}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right) \]

      6. lower-*.f64 79.1%

         \[\leadsto \log \left(\frac{1}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}\right) \]

   6. Applied rewrites 79.1%

      \[\leadsto \log \left(\frac{1}{\color{blue}{2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)}\right)} \]

      3. log-rec N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\log \left(\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)\right)\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\log \left(\frac{64}{27} + z0 \cdot \left(z0 \cdot \left(\frac{64}{9} + \frac{-64}{27} \cdot z0\right) - \frac{64}{9}\right)\right)} \]

      7. lower-unsound-log.f64 78.1%

         \[\leadsto -\color{blue}{\log \left(2.3703703703703702 + z0 \cdot \left(z0 \cdot \left(7.111111111111111 + -2.3703703703703702 \cdot z0\right) - 7.111111111111111\right)\right)} \]

   8. Applied rewrites 78.1%

      \[\leadsto \color{blue}{-\log \left(\left(\left(-2.3703703703703702 \cdot z0 - -7.111111111111111\right) \cdot z0 - 7.111111111111111\right) \cdot z0 - -2.3703703703703702\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -0.84:\\ \;\;\;\;-0.8630462173553428 + 3 \cdot \log \left(\frac{-1}{z0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - 0.8630462173553428\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= z0 -0.84)
  (+ -0.8630462173553428 (* 3.0 (log (/ -1.0 z0))))
  (- (* (- (* (- z0 -1.5) z0) -3.0) z0) 0.8630462173553428)))

C

double code(double z0) {
	double tmp;
	if (z0 <= -0.84) {
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	} else {
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-0.84d0)) then
        tmp = (-0.8630462173553428d0) + (3.0d0 * log(((-1.0d0) / z0)))
    else
        tmp = ((((z0 - (-1.5d0)) * z0) - (-3.0d0)) * z0) - 0.8630462173553428d0
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -0.84) {
		tmp = -0.8630462173553428 + (3.0 * Math.log((-1.0 / z0)));
	} else {
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428;
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -0.84:
		tmp = -0.8630462173553428 + (3.0 * math.log((-1.0 / z0)))
	else:
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -0.84)
		tmp = Float64(-0.8630462173553428 + Float64(3.0 * log(Float64(-1.0 / z0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -0.84)
		tmp = -0.8630462173553428 + (3.0 * log((-1.0 / z0)));
	else
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -0.84], N[(-0.8630462173553428 + N[(3.0 * N[Log[N[(-1.0 / z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(z0 - -1.5), $MachinePrecision] * z0), $MachinePrecision] - -3.0), $MachinePrecision] * z0), $MachinePrecision] - 0.8630462173553428), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -0.83999999999999997

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{\log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

      5. lower-/.f64 32.3%

         \[\leadsto \log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]

   5. Evaluated real constant 32.3%

      \[\leadsto -0.8630462173553428 + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

   if -0.83999999999999997 < z0

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{\frac{-27}{64}}{\color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}}{z0 - 1}\right)} \]

      5. frac-2neg N/A

         \[\leadsto \log \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\mathsf{neg}\left(\left(z0 - 1\right)\right)}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \log \left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\mathsf{neg}\left(\color{blue}{\left(z0 - 1\right)}\right)}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \log \left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\color{blue}{1 - z0}}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \log \left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\color{blue}{1 - z0}}\right) \]

      9. log-div N/A

         \[\leadsto \color{blue}{\log \left(\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)\right) - \log \left(1 - z0\right)} \]

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

          \[\leadsto \color{blue}{\log \left(\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)\right) - \log \left(1 - z0\right)} \]

   3. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\log \left(\frac{0.421875}{\left(z0 - 1\right) \cdot \left(z0 - 1\right)}\right) - \log \left(1 - z0\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\log \frac{27}{64} + z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0} \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \color{blue}{\left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \left(3 + \color{blue}{z0 \cdot \left(\frac{3}{2} + z0\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \left(3 + z0 \cdot \color{blue}{\left(\frac{3}{2} + z0\right)}\right) \]

      6. lower-+.f64 68.4%

         \[\leadsto \log 0.421875 + z0 \cdot \left(3 + z0 \cdot \left(1.5 + \color{blue}{z0}\right)\right) \]

   6. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\log 0.421875 + z0 \cdot \left(3 + z0 \cdot \left(1.5 + z0\right)\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) + \color{blue}{\log \frac{27}{64}} \]

      3. add-flip N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)} \]

      4. lift-log.f64 N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right) \]

      5. neg-log N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \left(\frac{1}{\frac{27}{64}}\right) \]

      6. metadata-eval N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \frac{64}{27} \]

      7. lift-log.f64 N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \frac{64}{27} \]

      8. lower--.f64 67.4%

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(1.5 + z0\right)\right) - \color{blue}{\log 2.3703703703703702} \]

      9. lift-*.f64 N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \color{blue}{\frac{64}{27}} \]

      10. *-commutative N/A

          \[\leadsto \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) \cdot z0 - \log \color{blue}{\frac{64}{27}} \]

      11. lower-*.f64 67.4%

          \[\leadsto \left(3 + z0 \cdot \left(1.5 + z0\right)\right) \cdot z0 - \log \color{blue}{2.3703703703703702} \]

      12. lift-+.f64 N/A

          \[\leadsto \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) \cdot z0 - \log \frac{64}{27} \]

      13. +-commutative N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) + 3\right) \cdot z0 - \log \frac{64}{27} \]

      14. add-flip N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) - \left(\mathsf{neg}\left(3\right)\right)\right) \cdot z0 - \log \frac{64}{27} \]

      15. metadata-eval N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) - -3\right) \cdot z0 - \log \frac{64}{27} \]

      16. lower--.f64 67.4%

          \[\leadsto \left(z0 \cdot \left(1.5 + z0\right) - -3\right) \cdot z0 - \log 2.3703703703703702 \]

      17. lift-*.f64 N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) - -3\right) \cdot z0 - \log \frac{64}{27} \]

      18. *-commutative N/A

          \[\leadsto \left(\left(\frac{3}{2} + z0\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      19. lower-*.f64 67.4%

          \[\leadsto \left(\left(1.5 + z0\right) \cdot z0 - -3\right) \cdot z0 - \log 2.3703703703703702 \]

      20. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{3}{2} + z0\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      21. +-commutative N/A

          \[\leadsto \left(\left(z0 + \frac{3}{2}\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      22. add-flip N/A

          \[\leadsto \left(\left(z0 - \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      23. lower--.f64 N/A

          \[\leadsto \left(\left(z0 - \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      24. metadata-eval 67.4%

          \[\leadsto \left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - \log 2.3703703703703702 \]

   8. Applied rewrites 67.4%

      \[\leadsto \left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - \color{blue}{\log 2.3703703703703702} \]

   9. Evaluated real constant 68.4%

      \[\leadsto \left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - 0.8630462173553428 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -5.4 \cdot 10^{+101}:\\ \;\;\;\;-0.8630462173553428\\ \mathbf{elif}\;z0 \leq -0.84:\\ \;\;\;\;-\log \left(\left(\left(z0 \cdot z0\right) \cdot \left(-z0\right)\right) \cdot 2.3703703703703702\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - 0.8630462173553428\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= z0 -5.4e+101)
  -0.8630462173553428
  (if (<= z0 -0.84)
    (- (log (* (* (* z0 z0) (- z0)) 2.3703703703703702)))
    (- (* (- (* (- z0 -1.5) z0) -3.0) z0) 0.8630462173553428))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -5.4e+101) {
		tmp = -0.8630462173553428;
	} else if (z0 <= -0.84) {
		tmp = -log((((z0 * z0) * -z0) * 2.3703703703703702));
	} else {
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-5.4d+101)) then
        tmp = -0.8630462173553428d0
    else if (z0 <= (-0.84d0)) then
        tmp = -log((((z0 * z0) * -z0) * 2.3703703703703702d0))
    else
        tmp = ((((z0 - (-1.5d0)) * z0) - (-3.0d0)) * z0) - 0.8630462173553428d0
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -5.4e+101) {
		tmp = -0.8630462173553428;
	} else if (z0 <= -0.84) {
		tmp = -Math.log((((z0 * z0) * -z0) * 2.3703703703703702));
	} else {
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428;
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -5.4e+101:
		tmp = -0.8630462173553428
	elif z0 <= -0.84:
		tmp = -math.log((((z0 * z0) * -z0) * 2.3703703703703702))
	else:
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -5.4e+101)
		tmp = -0.8630462173553428;
	elseif (z0 <= -0.84)
		tmp = Float64(-log(Float64(Float64(Float64(z0 * z0) * Float64(-z0)) * 2.3703703703703702)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -5.4e+101)
		tmp = -0.8630462173553428;
	elseif (z0 <= -0.84)
		tmp = -log((((z0 * z0) * -z0) * 2.3703703703703702));
	else
		tmp = ((((z0 - -1.5) * z0) - -3.0) * z0) - 0.8630462173553428;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -5.4e+101], -0.8630462173553428, If[LessEqual[z0, -0.84], (-N[Log[N[(N[(N[(z0 * z0), $MachinePrecision] * (-z0)), $MachinePrecision] * 2.3703703703703702), $MachinePrecision]], $MachinePrecision]), N[(N[(N[(N[(N[(z0 - -1.5), $MachinePrecision] * z0), $MachinePrecision] - -3.0), $MachinePrecision] * z0), $MachinePrecision] - 0.8630462173553428), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -5.4000000000000001e101

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\log \frac{27}{64} + z0 \cdot \left(3 + \frac{3}{2} \cdot z0\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0 \cdot \left(3 + \frac{3}{2} \cdot z0\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0} \cdot \left(3 + \frac{3}{2} \cdot z0\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \color{blue}{\left(3 + \frac{3}{2} \cdot z0\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \left(3 + \color{blue}{\frac{3}{2} \cdot z0}\right) \]

      5. lower-*.f64 67.3%

         \[\leadsto \log 0.421875 + z0 \cdot \left(3 + 1.5 \cdot \color{blue}{z0}\right) \]

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\log 0.421875 + z0 \cdot \left(3 + 1.5 \cdot z0\right)} \]

   5. Evaluated real constant 67.3%

      \[\leadsto -0.8630462173553428 + \color{blue}{z0} \cdot \left(3 + 1.5 \cdot z0\right) \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \frac{-1943407311442519}{2251799813685248} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.4%

      \[\leadsto -0.8630462173553428 \]

   if -5.4000000000000001e101 < z0 < -0.83999999999999997

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

   2. Taylor expanded in z0 around -inf

      \[\leadsto \color{blue}{\log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3} \cdot \log \left(\frac{-1}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

      4. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

      5. lower-/.f64 32.3%

         \[\leadsto \log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right) \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\log 0.421875 + 3 \cdot \log \left(\frac{-1}{z0}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{3 \cdot \log \left(\frac{-1}{z0}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + 3 \cdot \color{blue}{\log \left(\frac{-1}{z0}\right)} \]

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

         \[\leadsto \log \frac{27}{64} - \color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \log \left(\frac{-1}{z0}\right)} \]

      4. sub-negate-rev N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \log \left(\frac{-1}{z0}\right) - \log \frac{27}{64}\right)\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \log \left(\frac{-1}{z0}\right) - \log \frac{27}{64}\right) \]

      6. sub-flip N/A

         \[\leadsto -\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \log \left(\frac{-1}{z0}\right) + \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)\right) \]

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

         \[\leadsto -\left(\left(\mathsf{neg}\left(3 \cdot \log \left(\frac{-1}{z0}\right)\right)\right) + \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)\right) \]

      8. lift-log.f64 N/A

         \[\leadsto -\left(\left(\mathsf{neg}\left(3 \cdot \log \left(\frac{-1}{z0}\right)\right)\right) + \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)\right) \]

      9. log-pow-rev N/A

         \[\leadsto -\left(\left(\mathsf{neg}\left(\log \left({\left(\frac{-1}{z0}\right)}^{3}\right)\right)\right) + \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)\right) \]

      10. neg-log N/A

          \[\leadsto -\left(\log \left(\frac{1}{{\left(\frac{-1}{z0}\right)}^{3}}\right) + \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)\right) \]

      11. lift-log.f64 N/A

          \[\leadsto -\left(\log \left(\frac{1}{{\left(\frac{-1}{z0}\right)}^{3}}\right) + \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)\right) \]

      12. neg-log N/A

          \[\leadsto -\left(\log \left(\frac{1}{{\left(\frac{-1}{z0}\right)}^{3}}\right) + \log \left(\frac{1}{\frac{27}{64}}\right)\right) \]

      13. sum-log N/A

          \[\leadsto -\log \left(\frac{1}{{\left(\frac{-1}{z0}\right)}^{3}} \cdot \frac{1}{\frac{27}{64}}\right) \]

      14. lower-log.f64 N/A

          \[\leadsto -\log \left(\frac{1}{{\left(\frac{-1}{z0}\right)}^{3}} \cdot \frac{1}{\frac{27}{64}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto -\log \left(\frac{1}{{\left(\frac{-1}{z0}\right)}^{3}} \cdot \frac{1}{\frac{27}{64}}\right) \]

   6. Applied rewrites 11.5%

      \[\leadsto -\log \left(\left(\left(z0 \cdot z0\right) \cdot \left(-z0\right)\right) \cdot 2.3703703703703702\right) \]

   if -0.83999999999999997 < z0

   1. Initial program 79.1%

      \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{-27}{64}}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{\frac{-27}{64}}{\color{blue}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}}{z0 - 1}\right)} \]

      5. frac-2neg N/A

         \[\leadsto \log \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\mathsf{neg}\left(\left(z0 - 1\right)\right)}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \log \left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\mathsf{neg}\left(\color{blue}{\left(z0 - 1\right)}\right)}\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \log \left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\color{blue}{1 - z0}}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \log \left(\frac{\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)}{\color{blue}{1 - z0}}\right) \]

      9. log-div N/A

         \[\leadsto \color{blue}{\log \left(\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)\right) - \log \left(1 - z0\right)} \]

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

          \[\leadsto \color{blue}{\log \left(\mathsf{neg}\left(\frac{\frac{-27}{64}}{\left(1 - z0\right) \cdot \left(1 - z0\right)}\right)\right) - \log \left(1 - z0\right)} \]

   3. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\log \left(\frac{0.421875}{\left(z0 - 1\right) \cdot \left(z0 - 1\right)}\right) - \log \left(1 - z0\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\log \frac{27}{64} + z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0} \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \color{blue}{\left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \left(3 + \color{blue}{z0 \cdot \left(\frac{3}{2} + z0\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \log \frac{27}{64} + z0 \cdot \left(3 + z0 \cdot \color{blue}{\left(\frac{3}{2} + z0\right)}\right) \]

      6. lower-+.f64 68.4%

         \[\leadsto \log 0.421875 + z0 \cdot \left(3 + z0 \cdot \left(1.5 + \color{blue}{z0}\right)\right) \]

   6. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\log 0.421875 + z0 \cdot \left(3 + z0 \cdot \left(1.5 + z0\right)\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \log \frac{27}{64} + \color{blue}{z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) + \color{blue}{\log \frac{27}{64}} \]

      3. add-flip N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right)} \]

      4. lift-log.f64 N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \left(\mathsf{neg}\left(\log \frac{27}{64}\right)\right) \]

      5. neg-log N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \left(\frac{1}{\frac{27}{64}}\right) \]

      6. metadata-eval N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \frac{64}{27} \]

      7. lift-log.f64 N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \frac{64}{27} \]

      8. lower--.f64 67.4%

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(1.5 + z0\right)\right) - \color{blue}{\log 2.3703703703703702} \]

      9. lift-*.f64 N/A

         \[\leadsto z0 \cdot \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) - \log \color{blue}{\frac{64}{27}} \]

      10. *-commutative N/A

          \[\leadsto \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) \cdot z0 - \log \color{blue}{\frac{64}{27}} \]

      11. lower-*.f64 67.4%

          \[\leadsto \left(3 + z0 \cdot \left(1.5 + z0\right)\right) \cdot z0 - \log \color{blue}{2.3703703703703702} \]

      12. lift-+.f64 N/A

          \[\leadsto \left(3 + z0 \cdot \left(\frac{3}{2} + z0\right)\right) \cdot z0 - \log \frac{64}{27} \]

      13. +-commutative N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) + 3\right) \cdot z0 - \log \frac{64}{27} \]

      14. add-flip N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) - \left(\mathsf{neg}\left(3\right)\right)\right) \cdot z0 - \log \frac{64}{27} \]

      15. metadata-eval N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) - -3\right) \cdot z0 - \log \frac{64}{27} \]

      16. lower--.f64 67.4%

          \[\leadsto \left(z0 \cdot \left(1.5 + z0\right) - -3\right) \cdot z0 - \log 2.3703703703703702 \]

      17. lift-*.f64 N/A

          \[\leadsto \left(z0 \cdot \left(\frac{3}{2} + z0\right) - -3\right) \cdot z0 - \log \frac{64}{27} \]

      18. *-commutative N/A

          \[\leadsto \left(\left(\frac{3}{2} + z0\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      19. lower-*.f64 67.4%

          \[\leadsto \left(\left(1.5 + z0\right) \cdot z0 - -3\right) \cdot z0 - \log 2.3703703703703702 \]

      20. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{3}{2} + z0\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      21. +-commutative N/A

          \[\leadsto \left(\left(z0 + \frac{3}{2}\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      22. add-flip N/A

          \[\leadsto \left(\left(z0 - \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      23. lower--.f64 N/A

          \[\leadsto \left(\left(z0 - \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \cdot z0 - -3\right) \cdot z0 - \log \frac{64}{27} \]

      24. metadata-eval 67.4%

          \[\leadsto \left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - \log 2.3703703703703702 \]

   8. Applied rewrites 67.4%

      \[\leadsto \left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - \color{blue}{\log 2.3703703703703702} \]

   9. Evaluated real constant 68.4%

      \[\leadsto \left(\left(z0 - -1.5\right) \cdot z0 - -3\right) \cdot z0 - 0.8630462173553428 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.4% accurate, 131.0× speedup

Math

\[-0.8630462173553428 \]

FPCore

(FPCore (z0)
  :precision binary64
  -0.8630462173553428)

C

double code(double z0) {
	return -0.8630462173553428;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = -0.8630462173553428d0
end function

Java

public static double code(double z0) {
	return -0.8630462173553428;
}

Python

def code(z0):
	return -0.8630462173553428

Julia

function code(z0)
	return -0.8630462173553428
end

MATLAB

function tmp = code(z0)
	tmp = -0.8630462173553428;
end

Wolfram

code[z0_] := -0.8630462173553428

Derivation

1. Initial program 79.1%

   \[\log \left(\frac{-0.421875}{\left(\left(1 - z0\right) \cdot \left(1 - z0\right)\right) \cdot \left(z0 - 1\right)}\right) \]

2. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{\log \frac{27}{64} + z0 \cdot \left(3 + \frac{3}{2} \cdot z0\right)} \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \log \frac{27}{64} + \color{blue}{z0 \cdot \left(3 + \frac{3}{2} \cdot z0\right)} \]

   2. lower-log.f64 N/A

      \[\leadsto \log \frac{27}{64} + \color{blue}{z0} \cdot \left(3 + \frac{3}{2} \cdot z0\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \log \frac{27}{64} + z0 \cdot \color{blue}{\left(3 + \frac{3}{2} \cdot z0\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \log \frac{27}{64} + z0 \cdot \left(3 + \color{blue}{\frac{3}{2} \cdot z0}\right) \]

   5. lower-*.f64 67.3%

      \[\leadsto \log 0.421875 + z0 \cdot \left(3 + 1.5 \cdot \color{blue}{z0}\right) \]

4. Applied rewrites 67.3%

   \[\leadsto \color{blue}{\log 0.421875 + z0 \cdot \left(3 + 1.5 \cdot z0\right)} \]

5. Evaluated real constant 67.3%

   \[\leadsto -0.8630462173553428 + \color{blue}{z0} \cdot \left(3 + 1.5 \cdot z0\right) \]

6. Taylor expanded in z0 around 0

   \[\leadsto \frac{-1943407311442519}{2251799813685248} \]
7. Step-by-step derivation

8. Applied rewrites 70.4%

   \[\leadsto -0.8630462173553428 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(log (/ -27/64 (* (* (- 1 z0) (- 1 z0)) (- z0 1))))"
  :precision binary64
  (log (/ -0.421875 (* (* (- 1.0 z0) (- 1.0 z0)) (- z0 1.0)))))