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

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 11

Speedup: 0.7×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)\\ \frac{2.30753}{t\_0} - \mathsf{fma}\left(\frac{-0.27061}{t\_0}, x, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.04481 x 0.99229) x 1.0)))
   (- (/ 2.30753 t_0) (fma (/ -0.27061 t_0) x x))))

C

double code(double x) {
	double t_0 = fma(fma(0.04481, x, 0.99229), x, 1.0);
	return (2.30753 / t_0) - fma((-0.27061 / t_0), x, x);
}

Julia

function code(x)
	t_0 = fma(fma(0.04481, x, 0.99229), x, 1.0)
	return Float64(Float64(2.30753 / t_0) - fma(Float64(-0.27061 / t_0), x, x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]}, N[(N[(2.30753 / t$95$0), $MachinePrecision] - N[(N[(-0.27061 / t$95$0), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. add-flip N/A

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

   6. div-sub N/A

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

   7. associate-+l- N/A

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

   8. lower--.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× 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. 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. remove-double-neg N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-in N/A

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

   12. remove-double-neg N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 100.0

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 100.0

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

3. Applied rewrites 100.0%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 2.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right), x, -3.0191289437\right), x, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x)
   (if (<= x 2.55)
     (fma
      (fma (fma -1.7950336306565942 x 1.900161040244073) x -3.0191289437)
      x
      2.30753)
     (- (/ 6.039053782637804 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	} else if (x <= 2.55) {
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -3.0191289437), x, 2.30753);
	} else {
		tmp = (6.039053782637804 / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	elseif (x <= 2.55)
		tmp = fma(fma(fma(-1.7950336306565942, x, 1.900161040244073), x, -3.0191289437), x, 2.30753);
	else
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 2.55], N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $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. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.0

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

   4. Applied rewrites 49.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 49.0

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

   6. Applied rewrites 49.0%

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

   if -1.05000000000000004 < x < 2.5499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 53.8

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

   4. Applied rewrites 53.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.8

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 53.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 53.8

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

   6. Applied rewrites 53.8%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 50.2

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

   4. Applied rewrites 50.2%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 1.900161040244073, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x)
   (if (<= x 1.6)
     (fma (* x x) 1.900161040244073 (fma -3.0191289437 x 2.30753))
     (- (/ 6.039053782637804 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	} else if (x <= 1.6) {
		tmp = fma((x * x), 1.900161040244073, fma(-3.0191289437, x, 2.30753));
	} else {
		tmp = (6.039053782637804 / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	elseif (x <= 1.6)
		tmp = fma(Float64(x * x), 1.900161040244073, fma(-3.0191289437, x, 2.30753));
	else
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(x * x), $MachinePrecision] * 1.900161040244073 + N[(-3.0191289437 * x + 2.30753), $MachinePrecision]), $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. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.0

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

   4. Applied rewrites 49.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 49.0

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

   6. Applied rewrites 49.0%

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

   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. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 52.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if 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. Taylor expanded in x around inf

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

      1. lower-/.f64 50.2

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

   4. Applied rewrites 50.2%

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

Alternative 5: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (/ (- (/ -82.23527511657367 x) -6.039053782637804) x) x)
   (if (<= x 1.6)
     (fma (fma 1.900161040244073 x -3.0191289437) x 2.30753)
     (- (/ 6.039053782637804 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (((-82.23527511657367 / x) - -6.039053782637804) / x) - x;
	} else if (x <= 1.6) {
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	} else {
		tmp = (6.039053782637804 / x) - x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(Float64(Float64(-82.23527511657367 / x) - -6.039053782637804) / x) - x);
	elseif (x <= 1.6)
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	else
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(N[(N[(-82.23527511657367 / x), $MachinePrecision] - -6.039053782637804), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $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. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.0

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

   4. Applied rewrites 49.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 49.0

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

   6. Applied rewrites 49.0%

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

   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. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 52.5

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if 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. Taylor expanded in x around inf

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

      1. lower-/.f64 50.2

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

   4. Applied rewrites 50.2%

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

Alternative 6: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{6.039053782637804}{x} - x\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 6.039053782637804 x) x)))
   (if (<= x -1.05)
     t_0
     (if (<= x 1.6)
       (fma (fma 1.900161040244073 x -3.0191289437) x 2.30753)
       t_0))))

C

double code(double x) {
	double t_0 = (6.039053782637804 / x) - x;
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.6) {
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(6.039053782637804 / x) - x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.6)
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.6], N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -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. Taylor expanded in x around inf

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

      1. lower-/.f64 50.2

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

   4. Applied rewrites 50.2%

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

   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. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 52.5

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 52.5

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

   6. Applied rewrites 52.5%

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

Alternative 7: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{6.039053782637804}{x} - x\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.9:\\ \;\;\;\;\mathsf{fma}\left(x, -3.0191289437, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 6.039053782637804 x) x)))
   (if (<= x -1.05) t_0 (if (<= x 2.9) (fma x -3.0191289437 2.30753) t_0))))

C

double code(double x) {
	double t_0 = (6.039053782637804 / x) - x;
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 2.9) {
		tmp = fma(x, -3.0191289437, 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(6.039053782637804 / x) - x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 2.9)
		tmp = fma(x, -3.0191289437, 2.30753);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 2.9], N[(x * -3.0191289437 + 2.30753), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.89999999999999991 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 50.2

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

   4. Applied rewrites 50.2%

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

   if -1.05000000000000004 < x < 2.89999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 8: 98.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (- x) (if (<= x 1.15) (fma x -3.0191289437 2.30753) (- x))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.05], (-x), If[LessEqual[x, 1.15], N[(x * -3.0191289437 + 2.30753), $MachinePrecision], (-x)]]

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. Taylor expanded in x around inf

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

      1. lower-*.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 50.3

         \[\leadsto -x \]

   6. Applied rewrites 50.3%

      \[\leadsto -x \]

   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. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 9: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. add-flip N/A

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

   6. div-sub N/A

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

   7. associate-+l- N/A

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

   8. lower--.f64 N/A

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.6%

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

Alternative 10: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (- x) (if (<= x 1.16) 2.30753 (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -x;
	} else if (x <= 1.16) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = -x
    else if (x <= 1.16d0) then
        tmp = 2.30753d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -x;
	} else if (x <= 1.16) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = -x
	elif x <= 1.16:
		tmp = 2.30753
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(-x);
	elseif (x <= 1.16)
		tmp = 2.30753;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = -x;
	elseif (x <= 1.16)
		tmp = 2.30753;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], (-x), If[LessEqual[x, 1.16], 2.30753, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.15999999999999992 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 50.3

         \[\leadsto -x \]

   6. Applied rewrites 50.3%

      \[\leadsto -x \]

   if -1.05000000000000004 < x < 1.15999999999999992

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.6%

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

Alternative 11: 51.6% accurate, 24.0× 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. Taylor expanded in x around 0

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

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{2.30753} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(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))