(/ (sqrt PI) (- (/ (sqrt (- (- 1 z1) z1)) (* (exp (* z1 z1)) z1)) (* (- -1 z0) (sqrt PI))))

Percentage Accurate: 99.3% → 99.5%

Time: 2.5s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

\[\frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/
 1.772453850905516
 (-
  (/ (sqrt (- (- 1.0 z1) z1)) (* (exp (* z1 z1)) z1))
  (* (- -1.0 z0) 1.772453850905516))))

C

double code(double z1, double z0) {
	return 1.772453850905516 / ((sqrt(((1.0 - z1) - z1)) / (exp((z1 * z1)) * z1)) - ((-1.0 - z0) * 1.772453850905516));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = 1.772453850905516d0 / ((sqrt(((1.0d0 - z1) - z1)) / (exp((z1 * z1)) * z1)) - (((-1.0d0) - z0) * 1.772453850905516d0))
end function

Java

public static double code(double z1, double z0) {
	return 1.772453850905516 / ((Math.sqrt(((1.0 - z1) - z1)) / (Math.exp((z1 * z1)) * z1)) - ((-1.0 - z0) * 1.772453850905516));
}

Python

def code(z1, z0):
	return 1.772453850905516 / ((math.sqrt(((1.0 - z1) - z1)) / (math.exp((z1 * z1)) * z1)) - ((-1.0 - z0) * 1.772453850905516))

Julia

function code(z1, z0)
	return Float64(1.772453850905516 / Float64(Float64(sqrt(Float64(Float64(1.0 - z1) - z1)) / Float64(exp(Float64(z1 * z1)) * z1)) - Float64(Float64(-1.0 - z0) * 1.772453850905516)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.772453850905516 / ((sqrt(((1.0 - z1) - z1)) / (exp((z1 * z1)) * z1)) - ((-1.0 - z0) * 1.772453850905516));
end

Wolfram

code[z1_, z0_] := N[(1.772453850905516 / N[(N[(N[Sqrt[N[(N[(1.0 - z1), $MachinePrecision] - z1), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[N[(z1 * z1), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 - z0), $MachinePrecision] * 1.772453850905516), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.0%

   \[\leadsto \frac{\color{blue}{1.772453850905516}}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \sqrt{\pi}} \]

3. Evaluated real constant 99.5%

   \[\leadsto \frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \color{blue}{1.772453850905516}} \]
4. Add Preprocessing

Alternative 2: 98.6% accurate, 4.6× speedup

Math

\[\frac{1}{\frac{1}{z1} \cdot 0.5641895835477563 - \left(-1 - z0\right) \cdot 1} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 1.0 (- (* (/ 1.0 z1) 0.5641895835477563) (* (- -1.0 z0) 1.0))))

C

double code(double z1, double z0) {
	return 1.0 / (((1.0 / z1) * 0.5641895835477563) - ((-1.0 - 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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = 1.0d0 / (((1.0d0 / z1) * 0.5641895835477563d0) - (((-1.0d0) - z0) * 1.0d0))
end function

Java

public static double code(double z1, double z0) {
	return 1.0 / (((1.0 / z1) * 0.5641895835477563) - ((-1.0 - z0) * 1.0));
}

Python

def code(z1, z0):
	return 1.0 / (((1.0 / z1) * 0.5641895835477563) - ((-1.0 - z0) * 1.0))

Julia

function code(z1, z0)
	return Float64(1.0 / Float64(Float64(Float64(1.0 / z1) * 0.5641895835477563) - Float64(Float64(-1.0 - z0) * 1.0)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.0 / (((1.0 / z1) * 0.5641895835477563) - ((-1.0 - z0) * 1.0));
end

Wolfram

code[z1_, z0_] := N[(1.0 / N[(N[(N[(1.0 / z1), $MachinePrecision] * 0.5641895835477563), $MachinePrecision] - N[(N[(-1.0 - z0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in z1 around 0

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

   1. lower-/.f64 98.3%

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

4. Applied rewrites 98.3%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 98.4%

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.4%

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

6. Applied rewrites 98.4%

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

7. Evaluated real constant 97.7%

   \[\leadsto \frac{1}{\frac{\frac{1}{z1} - \color{blue}{1.772453850905516} \cdot \left(-1 - z0\right)}{\sqrt{\pi}}} \]

8. Evaluated real constant 98.5%

   \[\leadsto \frac{1}{\frac{\frac{1}{z1} - 1.772453850905516 \cdot \left(-1 - z0\right)}{\color{blue}{1.772453850905516}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{z1} - \frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}{\frac{7982422502469483}{4503599627370496}}}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{z1} - \frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}}{\frac{7982422502469483}{4503599627370496}}} \]

   3. div-sub N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{z1}}{\frac{7982422502469483}{4503599627370496}} - \frac{\frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}{\frac{7982422502469483}{4503599627370496}}}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{z1}}{\frac{7982422502469483}{4503599627370496}} - \frac{\frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}{\frac{7982422502469483}{4503599627370496}}}} \]

   5. mult-flip N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z1} \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} - \frac{\frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}{\frac{7982422502469483}{4503599627370496}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z1} \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} - \frac{\frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}{\frac{7982422502469483}{4503599627370496}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{1}{z1} \cdot \color{blue}{\frac{4503599627370496}{7982422502469483}} - \frac{\frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}{\frac{7982422502469483}{4503599627370496}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{z1} \cdot \frac{4503599627370496}{7982422502469483} - \frac{\color{blue}{\frac{7982422502469483}{4503599627370496} \cdot \left(-1 - z0\right)}}{\frac{7982422502469483}{4503599627370496}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\frac{1}{z1} \cdot \frac{4503599627370496}{7982422502469483} - \frac{\color{blue}{\left(-1 - z0\right) \cdot \frac{7982422502469483}{4503599627370496}}}{\frac{7982422502469483}{4503599627370496}}} \]

   10. associate-/l* N/A

       \[\leadsto \frac{1}{\frac{1}{z1} \cdot \frac{4503599627370496}{7982422502469483} - \color{blue}{\left(-1 - z0\right) \cdot \frac{\frac{7982422502469483}{4503599627370496}}{\frac{7982422502469483}{4503599627370496}}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{z1} \cdot \frac{4503599627370496}{7982422502469483} - \left(-1 - z0\right) \cdot \color{blue}{1}} \]

   12. lower-*.f64 98.6%

       \[\leadsto \frac{1}{\frac{1}{z1} \cdot 0.5641895835477563 - \color{blue}{\left(-1 - z0\right) \cdot 1}} \]

10. Applied rewrites 98.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{z1} \cdot 0.5641895835477563 - \left(-1 - z0\right) \cdot 1}} \]
11. Add Preprocessing

Alternative 3: 98.5% accurate, 5.3× speedup

Math

\[\frac{1.772453850905516}{\frac{1}{z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 1.772453850905516 (- (/ 1.0 z1) (* (- -1.0 z0) 1.772453850905516))))

C

double code(double z1, double z0) {
	return 1.772453850905516 / ((1.0 / z1) - ((-1.0 - z0) * 1.772453850905516));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = 1.772453850905516d0 / ((1.0d0 / z1) - (((-1.0d0) - z0) * 1.772453850905516d0))
end function

Java

public static double code(double z1, double z0) {
	return 1.772453850905516 / ((1.0 / z1) - ((-1.0 - z0) * 1.772453850905516));
}

Python

def code(z1, z0):
	return 1.772453850905516 / ((1.0 / z1) - ((-1.0 - z0) * 1.772453850905516))

Julia

function code(z1, z0)
	return Float64(1.772453850905516 / Float64(Float64(1.0 / z1) - Float64(Float64(-1.0 - z0) * 1.772453850905516)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.772453850905516 / ((1.0 / z1) - ((-1.0 - z0) * 1.772453850905516));
end

Wolfram

code[z1_, z0_] := N[(1.772453850905516 / N[(N[(1.0 / z1), $MachinePrecision] - N[(N[(-1.0 - z0), $MachinePrecision] * 1.772453850905516), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.0%

   \[\leadsto \frac{\color{blue}{1.772453850905516}}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \sqrt{\pi}} \]

3. Evaluated real constant 99.5%

   \[\leadsto \frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \color{blue}{1.772453850905516}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \frac{1.772453850905516}{\color{blue}{\frac{1 + z1 \cdot \left(\frac{-3}{2} \cdot z1 - 1\right)}{z1}} - \left(-1 - z0\right) \cdot 1.772453850905516} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{7982422502469483}{4503599627370496}}{\frac{1 + z1 \cdot \left(\frac{-3}{2} \cdot z1 - 1\right)}{\color{blue}{z1}} - \left(-1 - z0\right) \cdot \frac{7982422502469483}{4503599627370496}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{7982422502469483}{4503599627370496}}{\frac{1 + z1 \cdot \left(\frac{-3}{2} \cdot z1 - 1\right)}{z1} - \left(-1 - z0\right) \cdot \frac{7982422502469483}{4503599627370496}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{7982422502469483}{4503599627370496}}{\frac{1 + z1 \cdot \left(\frac{-3}{2} \cdot z1 - 1\right)}{z1} - \left(-1 - z0\right) \cdot \frac{7982422502469483}{4503599627370496}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{7982422502469483}{4503599627370496}}{\frac{1 + z1 \cdot \left(\frac{-3}{2} \cdot z1 - 1\right)}{z1} - \left(-1 - z0\right) \cdot \frac{7982422502469483}{4503599627370496}} \]

   5. lower-*.f64 73.4%

      \[\leadsto \frac{1.772453850905516}{\frac{1 + z1 \cdot \left(-1.5 \cdot z1 - 1\right)}{z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]

6. Applied rewrites 73.4%

   \[\leadsto \frac{1.772453850905516}{\color{blue}{\frac{1 + z1 \cdot \left(-1.5 \cdot z1 - 1\right)}{z1}} - \left(-1 - z0\right) \cdot 1.772453850905516} \]

7. Taylor expanded in z1 around 0

   \[\leadsto \frac{1.772453850905516}{\frac{1 + z1 \cdot -1}{z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]
8. Step-by-step derivation

9. Applied rewrites 85.0%

   \[\leadsto \frac{1.772453850905516}{\frac{1 + z1 \cdot -1}{z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]

10. Taylor expanded in z1 around 0

    \[\leadsto \frac{1.772453850905516}{\frac{1}{z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]
11. Step-by-step derivation

12. Applied rewrites 98.5%

    \[\leadsto \frac{1.772453850905516}{\frac{1}{z1} - \left(-1 - z0\right) \cdot 1.772453850905516} \]
13. Add Preprocessing

Alternative 4: 68.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -6 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{z0}\\ \mathbf{elif}\;z0 \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot z0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z0}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= z0 -6e+73)
  (/ 1.0 z0)
  (if (<= z0 1.15e+46)
    (*
     z1
     (+
      1.772453850905516
      (* 1.772453850905516 (* z1 (+ 1.0 (* -1.772453850905516 z0))))))
    (/ 1.0 z0))))

C

double code(double z1, double z0) {
	double tmp;
	if (z0 <= -6e+73) {
		tmp = 1.0 / z0;
	} else if (z0 <= 1.15e+46) {
		tmp = z1 * (1.772453850905516 + (1.772453850905516 * (z1 * (1.0 + (-1.772453850905516 * z0)))));
	} else {
		tmp = 1.0 / 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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-6d+73)) then
        tmp = 1.0d0 / z0
    else if (z0 <= 1.15d+46) then
        tmp = z1 * (1.772453850905516d0 + (1.772453850905516d0 * (z1 * (1.0d0 + ((-1.772453850905516d0) * z0)))))
    else
        tmp = 1.0d0 / z0
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0) {
	double tmp;
	if (z0 <= -6e+73) {
		tmp = 1.0 / z0;
	} else if (z0 <= 1.15e+46) {
		tmp = z1 * (1.772453850905516 + (1.772453850905516 * (z1 * (1.0 + (-1.772453850905516 * z0)))));
	} else {
		tmp = 1.0 / z0;
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if z0 <= -6e+73:
		tmp = 1.0 / z0
	elif z0 <= 1.15e+46:
		tmp = z1 * (1.772453850905516 + (1.772453850905516 * (z1 * (1.0 + (-1.772453850905516 * z0)))))
	else:
		tmp = 1.0 / z0
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (z0 <= -6e+73)
		tmp = Float64(1.0 / z0);
	elseif (z0 <= 1.15e+46)
		tmp = Float64(z1 * Float64(1.772453850905516 + Float64(1.772453850905516 * Float64(z1 * Float64(1.0 + Float64(-1.772453850905516 * z0))))));
	else
		tmp = Float64(1.0 / z0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (z0 <= -6e+73)
		tmp = 1.0 / z0;
	elseif (z0 <= 1.15e+46)
		tmp = z1 * (1.772453850905516 + (1.772453850905516 * (z1 * (1.0 + (-1.772453850905516 * z0)))));
	else
		tmp = 1.0 / z0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[z0, -6e+73], N[(1.0 / z0), $MachinePrecision], If[LessEqual[z0, 1.15e+46], N[(z1 * N[(1.772453850905516 + N[(1.772453850905516 * N[(z1 * N[(1.0 + N[(-1.772453850905516 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / z0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -6.0000000000000002e73 or 1.15e46 < z0

   1. Initial program 99.3%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-/.f64 34.3%

         \[\leadsto \frac{1}{\color{blue}{z0}} \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\frac{1}{z0}} \]

   if -6.0000000000000002e73 < z0 < 1.15e46

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.0%

      \[\leadsto \frac{\color{blue}{1.772453850905516}}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \sqrt{\pi}} \]

   3. Evaluated real constant 99.5%

      \[\leadsto \frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \color{blue}{1.772453850905516}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 \cdot \color{blue}{\left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \color{blue}{\frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \color{blue}{\left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \color{blue}{\left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \color{blue}{\frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \color{blue}{\left(1 + z0\right)}\right)\right)\right) \]

      7. lower-+.f64 49.8%

         \[\leadsto z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot \left(1 + \color{blue}{z0}\right)\right)\right)\right) \]

   6. Applied rewrites 49.8%

      \[\leadsto \color{blue}{z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot \left(1 + z0\right)\right)\right)\right)} \]

   7. Taylor expanded in z0 around inf

      \[\leadsto z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \color{blue}{z0}\right)\right)\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 50.0%

         \[\leadsto z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot z0\right)\right)\right) \]

   9. Applied rewrites 50.0%

      \[\leadsto z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot \color{blue}{z0}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -6 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{z0}\\ \mathbf{elif}\;z0 \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;z1 \cdot \left(1.772453850905516 + -1.3691388026842775 \cdot z1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z0}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= z0 -6e+73)
  (/ 1.0 z0)
  (if (<= z0 1.15e+46)
    (* z1 (+ 1.772453850905516 (* -1.3691388026842775 z1)))
    (/ 1.0 z0))))

C

double code(double z1, double z0) {
	double tmp;
	if (z0 <= -6e+73) {
		tmp = 1.0 / z0;
	} else if (z0 <= 1.15e+46) {
		tmp = z1 * (1.772453850905516 + (-1.3691388026842775 * z1));
	} else {
		tmp = 1.0 / 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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-6d+73)) then
        tmp = 1.0d0 / z0
    else if (z0 <= 1.15d+46) then
        tmp = z1 * (1.772453850905516d0 + ((-1.3691388026842775d0) * z1))
    else
        tmp = 1.0d0 / z0
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0) {
	double tmp;
	if (z0 <= -6e+73) {
		tmp = 1.0 / z0;
	} else if (z0 <= 1.15e+46) {
		tmp = z1 * (1.772453850905516 + (-1.3691388026842775 * z1));
	} else {
		tmp = 1.0 / z0;
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if z0 <= -6e+73:
		tmp = 1.0 / z0
	elif z0 <= 1.15e+46:
		tmp = z1 * (1.772453850905516 + (-1.3691388026842775 * z1))
	else:
		tmp = 1.0 / z0
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (z0 <= -6e+73)
		tmp = Float64(1.0 / z0);
	elseif (z0 <= 1.15e+46)
		tmp = Float64(z1 * Float64(1.772453850905516 + Float64(-1.3691388026842775 * z1)));
	else
		tmp = Float64(1.0 / z0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (z0 <= -6e+73)
		tmp = 1.0 / z0;
	elseif (z0 <= 1.15e+46)
		tmp = z1 * (1.772453850905516 + (-1.3691388026842775 * z1));
	else
		tmp = 1.0 / z0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[z0, -6e+73], N[(1.0 / z0), $MachinePrecision], If[LessEqual[z0, 1.15e+46], N[(z1 * N[(1.772453850905516 + N[(-1.3691388026842775 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / z0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -6.0000000000000002e73 or 1.15e46 < z0

   1. Initial program 99.3%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-/.f64 34.3%

         \[\leadsto \frac{1}{\color{blue}{z0}} \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\frac{1}{z0}} \]

   if -6.0000000000000002e73 < z0 < 1.15e46

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.0%

      \[\leadsto \frac{\color{blue}{1.772453850905516}}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \sqrt{\pi}} \]

   3. Evaluated real constant 99.5%

      \[\leadsto \frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \color{blue}{1.772453850905516}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 \cdot \color{blue}{\left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \color{blue}{\frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \color{blue}{\left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \color{blue}{\left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \color{blue}{\frac{-7982422502469483}{4503599627370496} \cdot \left(1 + z0\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z1 \cdot \left(\frac{7982422502469483}{4503599627370496} + \frac{7982422502469483}{4503599627370496} \cdot \left(z1 \cdot \left(1 + \frac{-7982422502469483}{4503599627370496} \cdot \color{blue}{\left(1 + z0\right)}\right)\right)\right) \]

      7. lower-+.f64 49.8%

         \[\leadsto z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot \left(1 + \color{blue}{z0}\right)\right)\right)\right) \]

   6. Applied rewrites 49.8%

      \[\leadsto \color{blue}{z1 \cdot \left(1.772453850905516 + 1.772453850905516 \cdot \left(z1 \cdot \left(1 + -1.772453850905516 \cdot \left(1 + z0\right)\right)\right)\right)} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto z1 \cdot \left(1.772453850905516 + \frac{-27769434000295737506075571713721}{20282409603651670423947251286016} \cdot \color{blue}{z1}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 50.0%

         \[\leadsto z1 \cdot \left(1.772453850905516 + -1.3691388026842775 \cdot z1\right) \]

   9. Applied rewrites 50.0%

      \[\leadsto z1 \cdot \left(1.772453850905516 + -1.3691388026842775 \cdot \color{blue}{z1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -6 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{z0}\\ \mathbf{elif}\;z0 \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;1.772453850905516 \cdot z1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z0}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= z0 -6e+73)
  (/ 1.0 z0)
  (if (<= z0 1.15e+46) (* 1.772453850905516 z1) (/ 1.0 z0))))

C

double code(double z1, double z0) {
	double tmp;
	if (z0 <= -6e+73) {
		tmp = 1.0 / z0;
	} else if (z0 <= 1.15e+46) {
		tmp = 1.772453850905516 * z1;
	} else {
		tmp = 1.0 / 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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (z0 <= (-6d+73)) then
        tmp = 1.0d0 / z0
    else if (z0 <= 1.15d+46) then
        tmp = 1.772453850905516d0 * z1
    else
        tmp = 1.0d0 / z0
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0) {
	double tmp;
	if (z0 <= -6e+73) {
		tmp = 1.0 / z0;
	} else if (z0 <= 1.15e+46) {
		tmp = 1.772453850905516 * z1;
	} else {
		tmp = 1.0 / z0;
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if z0 <= -6e+73:
		tmp = 1.0 / z0
	elif z0 <= 1.15e+46:
		tmp = 1.772453850905516 * z1
	else:
		tmp = 1.0 / z0
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (z0 <= -6e+73)
		tmp = Float64(1.0 / z0);
	elseif (z0 <= 1.15e+46)
		tmp = Float64(1.772453850905516 * z1);
	else
		tmp = Float64(1.0 / z0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (z0 <= -6e+73)
		tmp = 1.0 / z0;
	elseif (z0 <= 1.15e+46)
		tmp = 1.772453850905516 * z1;
	else
		tmp = 1.0 / z0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[z0, -6e+73], N[(1.0 / z0), $MachinePrecision], If[LessEqual[z0, 1.15e+46], N[(1.772453850905516 * z1), $MachinePrecision], N[(1.0 / z0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -6.0000000000000002e73 or 1.15e46 < z0

   1. Initial program 99.3%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-/.f64 34.3%

         \[\leadsto \frac{1}{\color{blue}{z0}} \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\frac{1}{z0}} \]

   if -6.0000000000000002e73 < z0 < 1.15e46

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.0%

      \[\leadsto \frac{\color{blue}{1.772453850905516}}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \sqrt{\pi}} \]

   3. Evaluated real constant 99.5%

      \[\leadsto \frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \color{blue}{1.772453850905516}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{7982422502469483}{4503599627370496} \cdot z1} \]
   5. Step-by-step derivation

      1. lower-*.f64 49.8%

         \[\leadsto 1.772453850905516 \cdot \color{blue}{z1} \]

   6. Applied rewrites 49.8%

      \[\leadsto \color{blue}{1.772453850905516 \cdot z1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 49.8% accurate, 30.0× speedup

Math

\[1.772453850905516 \cdot z1 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* 1.772453850905516 z1))

C

double code(double z1, double z0) {
	return 1.772453850905516 * 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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = 1.772453850905516d0 * z1
end function

Java

public static double code(double z1, double z0) {
	return 1.772453850905516 * z1;
}

Python

def code(z1, z0):
	return 1.772453850905516 * z1

Julia

function code(z1, z0)
	return Float64(1.772453850905516 * z1)
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.772453850905516 * z1;
end

Wolfram

code[z1_, z0_] := N[(1.772453850905516 * z1), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.0%

   \[\leadsto \frac{\color{blue}{1.772453850905516}}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \sqrt{\pi}} \]

3. Evaluated real constant 99.5%

   \[\leadsto \frac{1.772453850905516}{\frac{\sqrt{\left(1 - z1\right) - z1}}{e^{z1 \cdot z1} \cdot z1} - \left(-1 - z0\right) \cdot \color{blue}{1.772453850905516}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{7982422502469483}{4503599627370496} \cdot z1} \]
5. Step-by-step derivation

   1. lower-*.f64 49.8%

      \[\leadsto 1.772453850905516 \cdot \color{blue}{z1} \]

6. Applied rewrites 49.8%

   \[\leadsto \color{blue}{1.772453850905516 \cdot z1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(/ (sqrt PI) (- (/ (sqrt (- (- 1 z1) z1)) (* (exp (* z1 z1)) z1)) (* (- -1 z0) (sqrt PI))))"
  :precision binary64
  (/ (sqrt PI) (- (/ (sqrt (- (- 1.0 z1) z1)) (* (exp (* z1 z1)) z1)) (* (- -1.0 z0) (sqrt PI)))))