Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 10

Speedup: 1.1×

Specification

Math

\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

FPCore

(FPCore (x)
  :precision binary64
  (-
 (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
 x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

FPCore

(FPCore (x)
  :precision binary64
  (-
 (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
 x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x \]

FPCore

(FPCore (x)
  :precision binary64
  (- (/ (fma 0.27061 x 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0)) x))

C

double code(double x) {
	return (fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x;
}

Julia

function code(x)
	return Float64(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x)
end

Wolfram

code[x_] := N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   5. lower-fma.f64 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   10. lower-fma.f64 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

   12. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

   15. lower-fma.f64 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x \cdot x, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.5)
  (- (/ 6.039053782637804 x) x)
  (if (<= x 600.0)
    (fma
     (fma -1.7950336306565942 x 1.900161040244073)
     (* x x)
     (fma -3.0191289437 x 2.30753))
    (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 600.0) {
		tmp = fma(fma(-1.7950336306565942, x, 1.900161040244073), (x * x), fma(-3.0191289437, x, 2.30753));
	} else {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	elseif (x <= 600.0)
		tmp = fma(fma(-1.7950336306565942, x, 1.900161040244073), Float64(x * x), fma(-3.0191289437, x, 2.30753));
	else
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 600.0], N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(-3.0191289437 * x + 2.30753), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   3. Step-by-step derivation

      1. lower-/.f64 51.1%

         \[\leadsto \frac{6.039053782637804}{\color{blue}{x}} - x \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -2.5 < x < 600

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \color{blue}{\frac{30191289437}{10000000000}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \]

      6. lower-*.f64 52.9%

         \[\leadsto 2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437\right) \]

   4. Applied rewrites 52.9%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \color{blue}{\frac{230753}{100000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \cdot x + \frac{230753}{100000} \]

      5. lower-fma.f64 52.9%

         \[\leadsto \mathsf{fma}\left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437, \color{blue}{x}, 2.30753\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      15. metadata-eval 52.9%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right) \]

   6. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), \color{blue}{x}, 2.30753\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x, \frac{-30191289437}{10000000000}\right) \cdot x + \color{blue}{\frac{230753}{100000}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x, \frac{-30191289437}{10000000000}\right) + \frac{230753}{100000} \]

      3. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot x + \frac{-30191289437}{10000000000}\right) + \frac{230753}{100000} \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot x\right) \cdot x + \frac{-30191289437}{10000000000} \cdot x\right) + \frac{230753}{100000} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot x\right) \cdot x + \frac{-30191289437}{10000000000} \cdot x\right) + \frac{230753}{100000} \]

      6. associate-+l+ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot x\right) \cdot x + \color{blue}{\left(\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot x\right) \cdot x + \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot x\right) \cdot x + \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot \left(x \cdot x\right) + \left(\color{blue}{\frac{230753}{100000}} + \frac{-30191289437}{10000000000} \cdot x\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot \left(x \cdot x\right) + \left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \]

      11. lower-unsound-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right) \cdot \left(x \cdot x\right) + \left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), \color{blue}{x \cdot x}, \frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \]

      13. lower-unsound-*.f64 52.9%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x \cdot \color{blue}{x}, 2.30753 + -3.0191289437 \cdot x\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x \cdot x, \frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x \cdot x, \frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x \cdot x, \frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right) \]

      17. lower-fma.f64 52.9%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x \cdot x, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right) \]

   8. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), \color{blue}{x \cdot x}, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right) \]

   if 600 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{\color{blue}{x}} - x \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      4. lower-/.f64 50.0%

         \[\leadsto \frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x} - x \]

   6. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      2. sub-flip N/A

         \[\leadsto \frac{\frac{27061}{4481} + \left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right)}{x} - x \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) + \frac{27061}{4481}}{x} - x \]

      4. add-flip N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      8. mult-flip-rev N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\frac{1651231776}{20079361}}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1651231776}{20079361}\right)}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1651231776}{20079361}\right)}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{-1651231776}{20079361}}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      12. metadata-eval 50.0%

          \[\leadsto \frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x \]

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.5)
  (- (/ 6.039053782637804 x) x)
  (if (<= x 600.0)
    (fma
     (fma
      (fma -1.7950336306565942 x 1.900161040244073)
      x
      -3.0191289437)
     x
     2.30753)
    (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 600.0) {
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -3.0191289437), x, 2.30753);
	} else {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	elseif (x <= 600.0)
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -3.0191289437), x, 2.30753);
	else
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 600.0], N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   3. Step-by-step derivation

      1. lower-/.f64 51.1%

         \[\leadsto \frac{6.039053782637804}{\color{blue}{x}} - x \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -2.5 < x < 600

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \color{blue}{\frac{30191289437}{10000000000}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \]

      6. lower-*.f64 52.9%

         \[\leadsto 2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437\right) \]

   4. Applied rewrites 52.9%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \color{blue}{\frac{230753}{100000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \cdot x + \frac{230753}{100000} \]

      5. lower-fma.f64 52.9%

         \[\leadsto \mathsf{fma}\left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437, \color{blue}{x}, 2.30753\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}, x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-179503363065659419717}{100000000000000000000}, x, \frac{1900161040244073}{1000000000000000}\right), x, \mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      15. metadata-eval 52.9%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right) \]

   6. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), \color{blue}{x}, 2.30753\right) \]

   if 600 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{\color{blue}{x}} - x \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      4. lower-/.f64 50.0%

         \[\leadsto \frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x} - x \]

   6. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      2. sub-flip N/A

         \[\leadsto \frac{\frac{27061}{4481} + \left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right)}{x} - x \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) + \frac{27061}{4481}}{x} - x \]

      4. add-flip N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      8. mult-flip-rev N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\frac{1651231776}{20079361}}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1651231776}{20079361}\right)}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1651231776}{20079361}\right)}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{-1651231776}{20079361}}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      12. metadata-eval 50.0%

          \[\leadsto \frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x \]

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.6)
  (- (/ 6.039053782637804 x) x)
  (if (<= x 75000.0)
    (- (fma (fma 1.900161040244073 x -2.0191289437) x 2.30753) x)
    (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 75000.0) {
		tmp = fma(fma(1.900161040244073, x, -2.0191289437), x, 2.30753) - x;
	} else {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	elseif (x <= 75000.0)
		tmp = Float64(fma(fma(1.900161040244073, x, -2.0191289437), x, 2.30753) - x);
	else
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.6], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 75000.0], N[(N[(N[(1.900161040244073 * x + -2.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000001

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   3. Step-by-step derivation

      1. lower-/.f64 51.1%

         \[\leadsto \frac{6.039053782637804}{\color{blue}{x}} - x \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -2.6000000000000001 < x < 75000

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)\right)} - x \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)}\right) - x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + x \cdot \color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)}\right) - x \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\frac{20191289437}{10000000000}}\right)\right) - x \]

      4. lower-*.f64 51.5%

         \[\leadsto \left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right) - x \]

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)}\right) - x \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) + \color{blue}{\frac{230753}{100000}}\right) - x \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right) - x \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) \cdot x + \frac{230753}{100000}\right) - x \]

      5. lower-fma.f64 51.5%

         \[\leadsto \mathsf{fma}\left(1.900161040244073 \cdot x - 2.0191289437, \color{blue}{x}, 2.30753\right) - x \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right) - x \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) - x \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) - x \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000}, x, \mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) - x \]

      10. metadata-eval 51.5%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x \]

   6. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), \color{blue}{x}, 2.30753\right) - x \]

   if 75000 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{\color{blue}{x}} - x \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      4. lower-/.f64 50.0%

         \[\leadsto \frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x} - x \]

   6. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x \]

      2. sub-flip N/A

         \[\leadsto \frac{\frac{27061}{4481} + \left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right)}{x} - x \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) + \frac{27061}{4481}}{x} - x \]

      4. add-flip N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1651231776}{20079361} \cdot \frac{1}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      8. mult-flip-rev N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\frac{1651231776}{20079361}}{x}\right)\right) - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1651231776}{20079361}\right)}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1651231776}{20079361}\right)}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{-1651231776}{20079361}}{x} - \left(\mathsf{neg}\left(\frac{27061}{4481}\right)\right)}{x} - x \]

      12. metadata-eval 50.0%

          \[\leadsto \frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x \]

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{6.039053782637804}{x} - x\\ \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (- (/ 6.039053782637804 x) x)))
  (if (<= x -2.6)
    t_0
    (if (<= x 75000.0)
      (- (fma (fma 1.900161040244073 x -2.0191289437) x 2.30753) x)
      t_0))))

C

double code(double x) {
	double t_0 = (6.039053782637804 / x) - x;
	double tmp;
	if (x <= -2.6) {
		tmp = t_0;
	} else if (x <= 75000.0) {
		tmp = fma(fma(1.900161040244073, x, -2.0191289437), x, 2.30753) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(6.039053782637804 / x) - x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = t_0;
	elseif (x <= 75000.0)
		tmp = Float64(fma(fma(1.900161040244073, x, -2.0191289437), x, 2.30753) - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -2.6], t$95$0, If[LessEqual[x, 75000.0], N[(N[(N[(1.900161040244073 * x + -2.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001 or 75000 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   3. Step-by-step derivation

      1. lower-/.f64 51.1%

         \[\leadsto \frac{6.039053782637804}{\color{blue}{x}} - x \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -2.6000000000000001 < x < 75000

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)\right)} - x \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)}\right) - x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + x \cdot \color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)}\right) - x \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\frac{20191289437}{10000000000}}\right)\right) - x \]

      4. lower-*.f64 51.5%

         \[\leadsto \left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right) - x \]

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)}\right) - x \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) + \color{blue}{\frac{230753}{100000}}\right) - x \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right) - x \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) \cdot x + \frac{230753}{100000}\right) - x \]

      5. lower-fma.f64 51.5%

         \[\leadsto \mathsf{fma}\left(1.900161040244073 \cdot x - 2.0191289437, \color{blue}{x}, 2.30753\right) - x \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right) - x \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) - x \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) - x \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000}, x, \mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) - x \]

      10. metadata-eval 51.5%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x \]

   6. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), \color{blue}{x}, 2.30753\right) - x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{6.039053782637804}{x} - x\\ \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (- (/ 6.039053782637804 x) x)))
  (if (<= x -2.6)
    t_0
    (if (<= x 75000.0)
      (fma (fma 1.900161040244073 x -3.0191289437) x 2.30753)
      t_0))))

C

double code(double x) {
	double t_0 = (6.039053782637804 / x) - x;
	double tmp;
	if (x <= -2.6) {
		tmp = t_0;
	} else if (x <= 75000.0) {
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(6.039053782637804 / x) - x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = t_0;
	elseif (x <= 75000.0)
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -2.6], t$95$0, If[LessEqual[x, 75000.0], N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001 or 75000 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   3. Step-by-step derivation

      1. lower-/.f64 51.1%

         \[\leadsto \frac{6.039053782637804}{\color{blue}{x}} - x \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -2.6000000000000001 < x < 75000

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\frac{30191289437}{10000000000}}\right) \]

      4. lower-*.f64 51.6%

         \[\leadsto 2.30753 + x \cdot \left(1.900161040244073 \cdot x - 3.0191289437\right) \]

   6. Applied rewrites 51.6%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(1.900161040244073 \cdot x - 3.0191289437\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) + \color{blue}{\frac{230753}{100000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) \cdot x + \frac{230753}{100000} \]

      5. lower-fma.f64 51.6%

         \[\leadsto \mathsf{fma}\left(1.900161040244073 \cdot x - 3.0191289437, \color{blue}{x}, 2.30753\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right), x, \frac{230753}{100000}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \frac{-30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      10. lower-fma.f64 51.6%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right) \]

   8. Applied rewrites 51.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), \color{blue}{x}, 2.30753\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{6.039053782637804}{x} - x\\ \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(x, -3.0191289437, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (- (/ 6.039053782637804 x) x)))
  (if (<= x -2.8)
    t_0
    (if (<= x 980.0) (fma x -3.0191289437 2.30753) t_0))))

C

double code(double x) {
	double t_0 = (6.039053782637804 / x) - x;
	double tmp;
	if (x <= -2.8) {
		tmp = t_0;
	} else if (x <= 980.0) {
		tmp = fma(x, -3.0191289437, 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(6.039053782637804 / x) - x)
	tmp = 0.0
	if (x <= -2.8)
		tmp = t_0;
	elseif (x <= 980.0)
		tmp = fma(x, -3.0191289437, 2.30753);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -2.8], t$95$0, If[LessEqual[x, 980.0], N[(x * -3.0191289437 + 2.30753), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999998 or 980 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   3. Step-by-step derivation

      1. lower-/.f64 51.1%

         \[\leadsto \frac{6.039053782637804}{\color{blue}{x}} - x \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -2.7999999999999998 < x < 980

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

      2. lower-*.f64 58.4%

         \[\leadsto 2.30753 + -3.0191289437 \cdot \color{blue}{x} \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \frac{-30191289437}{10000000000} \cdot x + \color{blue}{\frac{230753}{100000}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \frac{-30191289437}{10000000000} + \frac{230753}{100000} \]

      5. lower-fma.f64 58.4%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{-3.0191289437}, 2.30753\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{-3.0191289437}, 2.30753\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(x, -3.0191289437, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -1.0)
  (- x)
  (if (<= x 980.0) (fma x -3.0191289437 2.30753) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -x;
	} else if (x <= 980.0) {
		tmp = fma(x, -3.0191289437, 2.30753);
	} else {
		tmp = -x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-x);
	elseif (x <= 980.0)
		tmp = fma(x, -3.0191289437, 2.30753);
	else
		tmp = Float64(-x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], (-x), If[LessEqual[x, 980.0], N[(x * -3.0191289437 + 2.30753), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 980 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      3. lower-neg.f64 51.3%

         \[\leadsto -x \]

   6. Applied rewrites 51.3%

      \[\leadsto -x \]

   if -1 < x < 980

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

      2. lower-*.f64 58.4%

         \[\leadsto 2.30753 + -3.0191289437 \cdot \color{blue}{x} \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \frac{-30191289437}{10000000000} \cdot x + \color{blue}{\frac{230753}{100000}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \frac{-30191289437}{10000000000} + \frac{230753}{100000} \]

      5. lower-fma.f64 58.4%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{-3.0191289437}, 2.30753\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{-3.0191289437}, 2.30753\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1750000:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -1750000.0) (- x) (if (<= x 75000.0) 2.30753 (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1750000.0) {
		tmp = -x;
	} else if (x <= 75000.0) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1750000.0d0)) then
        tmp = -x
    else if (x <= 75000.0d0) then
        tmp = 2.30753d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1750000.0) {
		tmp = -x;
	} else if (x <= 75000.0) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1750000.0:
		tmp = -x
	elif x <= 75000.0:
		tmp = 2.30753
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1750000.0)
		tmp = Float64(-x);
	elseif (x <= 75000.0)
		tmp = 2.30753;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1750000.0)
		tmp = -x;
	elseif (x <= 75000.0)
		tmp = 2.30753;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1750000.0], (-x), If[LessEqual[x, 75000.0], 2.30753, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e6 or 75000 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      3. lower-neg.f64 51.3%

         \[\leadsto -x \]

   6. Applied rewrites 51.3%

      \[\leadsto -x \]

   if -1.75e6 < x < 75000

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. lower-fma.f64 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

      10. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

      15. lower-fma.f64 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

      2. lower-*.f64 58.4%

         \[\leadsto 2.30753 + -3.0191289437 \cdot \color{blue}{x} \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{230753}{100000} \]
   8. Step-by-step derivation

   9. Applied rewrites 50.6%

      \[\leadsto 2.30753 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 50.6% accurate, 24.2× speedup

Math

\[2.30753 \]

FPCore

(FPCore (x)
  :precision binary64
  2.30753)

C

double code(double x) {
	return 2.30753;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 2.30753d0
end function

Java

public static double code(double x) {
	return 2.30753;
}

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

function tmp = code(x)
	tmp = 2.30753;
end

Wolfram

code[x_] := 2.30753

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   5. lower-fma.f64 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   10. lower-fma.f64 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

   12. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

   15. lower-fma.f64 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

   2. lower-*.f64 58.4%

      \[\leadsto 2.30753 + -3.0191289437 \cdot \color{blue}{x} \]

6. Applied rewrites 58.4%

   \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{230753}{100000} \]
8. Step-by-step derivation

9. Applied rewrites 50.6%

   \[\leadsto 2.30753 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))