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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 12

Speedup: N/A×

Specification

Math

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

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

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

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(x * 0.99229 + N[(N[(x * x), $MachinePrecision] * 0.04481 + 1.0), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. associate-+l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x} - x\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (/ (fma x 0.27061 2.30753) (* (fma 0.04481 x 0.99229) x)) x)
   (if (<= x 2.5)
     (-
      (fma
       (fma (fma -1.7950336306565942 x 1.900161040244073) x -2.0191289437)
       x
       2.30753)
      x)
     (- (/ 6.039053782637804 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (fma(x, 0.27061, 2.30753) / (fma(0.04481, x, 0.99229) * x)) - x;
	} else if (x <= 2.5) {
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -2.0191289437), x, 2.30753) - x;
	} else {
		tmp = (6.039053782637804 / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(fma(x, 0.27061, 2.30753) / Float64(fma(0.04481, x, 0.99229) * x)) - x);
	elseif (x <= 2.5)
		tmp = Float64(fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -2.0191289437), x, 2.30753) - x);
	else
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 2.5], N[(N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x + -2.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] - x), $MachinePrecision], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right)}} - x \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{{x}^{2} \cdot \frac{4481}{100000} + {x}^{2} \cdot \left(\frac{99229}{100000} \cdot \frac{1}{x}\right)}} - x \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\frac{4481}{100000} \cdot {x}^{2}} + {x}^{2} \cdot \left(\frac{99229}{100000} \cdot \frac{1}{x}\right)} - x \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} + \color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}}} - x \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} + \color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot {x}^{2}\right)}} - x \]

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

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) \cdot \left(\frac{1}{x} \cdot {x}^{2}\right)}} - x \]

      6. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) \cdot \frac{1}{x}\right) \cdot {x}^{2}}} - x \]

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

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \frac{1}{x}\right)\right)} \cdot {x}^{2}} - x \]

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

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}\right)\right)}} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \left(\frac{99229}{100000} \cdot \frac{1}{x}\right)}\right)\right)} - x \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}}\right)\right)} - x \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{99229}{100000} \cdot 1}{x}} \cdot {x}^{2}\right)\right)} - x \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{99229}{100000}}}{x} \cdot {x}^{2}\right)\right)} - x \]

      13. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{99229}{100000} \cdot {x}^{2}}{x}}\right)\right)} - x \]

      14. associate-/l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\frac{99229}{100000} \cdot \frac{{x}^{2}}{x}}\right)\right)} - x \]

      15. *-lft-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \frac{\color{blue}{1 \cdot {x}^{2}}}{x}\right)\right)} - x \]

      16. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot {x}^{2}\right)}\right)\right)} - x \]

      17. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} - x \]

      18. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \color{blue}{\left(\left(\frac{1}{x} \cdot x\right) \cdot x\right)}\right)\right)} - x \]

      19. lft-mult-inverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \left(\color{blue}{1} \cdot x\right)\right)\right)} - x \]

      20. *-lft-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{99229}{100000} \cdot \color{blue}{x}\right)\right)} - x \]

      21. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{4481}{100000} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{99229}{100000}}\right)\right)} - x \]

   9. Applied rewrites 98.9%

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

   if -1.05000000000000004 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]

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

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x \]

      4. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right) - x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000}\right) - x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right)} - x \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \color{blue}{\frac{20191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) - x \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) - x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1, x, \frac{230753}{100000}\right) - x \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-20191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) - x \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-20191289437}{10000000000}}, x, \frac{230753}{100000}\right) - x \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \frac{-20191289437}{10000000000}\right)}, x, \frac{230753}{100000}\right) - x \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, x, \frac{-20191289437}{10000000000}\right), x, \frac{230753}{100000}\right) - x \]

      14. lower-fma.f64 99.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right)}, x, -2.0191289437\right), x, 2.30753\right) - x \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right)} - x \]

   if 2.5 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x} - x\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.8)
   (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x)
   (if (<= x 2.5)
     (-
      (fma
       (fma (fma -1.7950336306565942 x 1.900161040244073) x -2.0191289437)
       x
       2.30753)
      x)
     (- (/ 6.039053782637804 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -4.8) {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	} else if (x <= 2.5) {
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -2.0191289437), x, 2.30753) - x;
	} else {
		tmp = (6.039053782637804 / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.8)
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	elseif (x <= 2.5)
		tmp = Float64(fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -2.0191289437), x, 2.30753) - x);
	else
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.8], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 2.5], N[(N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x + -2.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] - x), $MachinePrecision], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999982

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 98.7%

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

   if -4.79999999999999982 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]

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

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x \]

      4. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right) - x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000}\right) - x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right)} - x \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \color{blue}{\frac{20191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) - x \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) - x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1, x, \frac{230753}{100000}\right) - x \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-20191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) - x \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-20191289437}{10000000000}}, x, \frac{230753}{100000}\right) - x \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \frac{-20191289437}{10000000000}\right)}, x, \frac{230753}{100000}\right) - x \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, x, \frac{-20191289437}{10000000000}\right), x, \frac{230753}{100000}\right) - x \]

      14. lower-fma.f64 99.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right)}, x, -2.0191289437\right), x, 2.30753\right) - x \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right)} - x \]

   if 2.5 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.5)))
   (- (/ 6.039053782637804 x) x)
   (-
    (fma
     (fma (fma -1.7950336306565942 x 1.900161040244073) x -2.0191289437)
     x
     2.30753)
    x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.5)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -2.0191289437), x, 2.30753) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.5))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = Float64(fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -2.0191289437), x, 2.30753) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x + -2.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.5 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.05000000000000004 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]

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

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x \]

      4. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right) - x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000}\right) - x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right)} - x \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \color{blue}{\frac{20191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) - x \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) - x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1, x, \frac{230753}{100000}\right) - x \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-20191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) - x \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-20191289437}{10000000000}}, x, \frac{230753}{100000}\right) - x \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, x, \frac{-20191289437}{10000000000}\right)}, x, \frac{230753}{100000}\right) - x \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, x, \frac{-20191289437}{10000000000}\right), x, \frac{230753}{100000}\right) - x \]

      14. lower-fma.f64 99.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right)}, x, -2.0191289437\right), x, 2.30753\right) - x \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right)} - x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -2.0191289437\right), x, 2.30753\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.2× speedup

Math

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

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. Add Preprocessing
3. 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

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

4. Applied rewrites 100.0%

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

Alternative 6: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.5)))
   (- (/ 6.039053782637804 x) x)
   (fma
    (fma (fma -1.7950336306565942 x 1.900161040244073) x -3.0191289437)
    x
    2.30753)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.5)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -3.0191289437), x, 2.30753);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.5))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -3.0191289437), x, 2.30753);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.5 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.05000000000000004 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \color{blue}{\frac{30191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot 1, x, \frac{230753}{100000}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-30191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.6)))
   (- (/ 6.039053782637804 x) x)
   (- (fma (fma 1.900161040244073 x -2.0191289437) x 2.30753) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.6)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(fma(1.900161040244073, x, -2.0191289437), x, 2.30753) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.6))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = Float64(fma(fma(1.900161040244073, x, -2.0191289437), x, 2.30753) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.6]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(1.900161040244073 * x + -2.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.6000000000000001 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.05000000000000004 < x < 1.6000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\frac{20191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) - x \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) - x \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-20191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) - x \]

      10. metadata-eval N/A

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

      11. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.6)))
   (- (/ 6.039053782637804 x) x)
   (fma (fma 1.900161040244073 x -3.0191289437) x 2.30753)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.6)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.6))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.6]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.6000000000000001 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.05000000000000004 < x < 1.6000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\frac{30191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (- x)
   (fma (fma 1.900161040244073 x -3.0191289437) x 2.30753)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\frac{30191289437}{10000000000} \cdot 1}, x, \frac{230753}{100000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot 1}, x, \frac{230753}{100000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000}} \cdot 1, x, \frac{230753}{100000}\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15))) (- x) (fma -3.0191289437 x 2.30753)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = fma(-3.0191289437, x, 2.30753);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = fma(-3.0191289437, x, 2.30753);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), N[(-3.0191289437 * x + 2.30753), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.6) (not (<= x 1.15))) (- x) 2.30753))

C

double code(double x) {
	double tmp;
	if ((x <= -3.6) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	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 <= (-3.6d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = -x
    else
        tmp = 2.30753d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -3.6) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.6) or not (x <= 1.15):
		tmp = -x
	else:
		tmp = 2.30753
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.6) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = 2.30753;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -3.6) || ~((x <= 1.15)))
		tmp = -x;
	else
		tmp = 2.30753;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.6], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), 2.30753]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 or 1.1499999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -3.60000000000000009 < x < 1.1499999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 98.3%

      \[\leadsto \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.1% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ 2.30753 \end{array} \]

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. associate-+l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 53.0%

   \[\leadsto \color{blue}{2.30753} \]

10. Final simplification 53.0%

    \[\leadsto 2.30753 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(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))