Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 94.0% → 98.5%

Time: 11.3s

Alternatives: 15

Speedup: 1.1×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 0.5\right) \cdot \log x\\ \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - 0.91893853320467}{t\_0}, t\_0, \frac{0.083333333333333}{x}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x 0.5) (log x))))
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (fma (- 1.0 (/ (- x 0.91893853320467) t_0)) t_0 (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 0.5) * log(x);
	return fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), fma((1.0 - ((x - 0.91893853320467) / t_0)), t_0, (0.083333333333333 / x)));
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 0.5) * log(x))
	return fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), fma(Float64(1.0 - Float64(Float64(x - 0.91893853320467) / t_0)), t_0, Float64(0.083333333333333 / x)))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(1.0 - N[(N[(x - 0.91893853320467), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   5. div-add N/A

      \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

3. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\log x \cdot \left(x - \frac{1}{2}\right) - x\right) - \frac{-91893853320467}{100000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\left(\left(\log x \cdot \left(x - \frac{1}{2}\right) - x\right) - \frac{-91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\left(\left(\log x \cdot \left(x - \frac{1}{2}\right) - x\right) - \frac{-91893853320467}{100000000000000}\right)} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \left(\color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) - x\right)} - \frac{-91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   5. associate--l- N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) - \left(x + \frac{-91893853320467}{100000000000000}\right)\right)} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \left(\log x \cdot \left(x - \frac{1}{2}\right) - \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right)}\right)\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   7. sub-flip N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \left(\log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \left(\log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   9. sub-to-mult N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\left(1 - \frac{x - \frac{91893853320467}{100000000000000}}{\log x \cdot \left(x - \frac{1}{2}\right)}\right) \cdot \left(\log x \cdot \left(x - \frac{1}{2}\right)\right)} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\mathsf{fma}\left(1 - \frac{x - \frac{91893853320467}{100000000000000}}{\log x \cdot \left(x - \frac{1}{2}\right)}, \log x \cdot \left(x - \frac{1}{2}\right), \frac{\frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \mathsf{fma}\left(\color{blue}{1 - \frac{x - \frac{91893853320467}{100000000000000}}{\log x \cdot \left(x - \frac{1}{2}\right)}}, \log x \cdot \left(x - \frac{1}{2}\right), \frac{\frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   12. lower-/.f64 98.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(1 - \color{blue}{\frac{x - 0.91893853320467}{\log x \cdot \left(x - 0.5\right)}}, \log x \cdot \left(x - 0.5\right), \frac{0.083333333333333}{x}\right)\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - \frac{91893853320467}{100000000000000}}{\color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)}}, \log x \cdot \left(x - \frac{1}{2}\right), \frac{\frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - \frac{91893853320467}{100000000000000}}{\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x}}, \log x \cdot \left(x - \frac{1}{2}\right), \frac{\frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   15. lift-*.f64 98.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - 0.91893853320467}{\color{blue}{\left(x - 0.5\right) \cdot \log x}}, \log x \cdot \left(x - 0.5\right), \frac{0.083333333333333}{x}\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - \frac{91893853320467}{100000000000000}}{\left(x - \frac{1}{2}\right) \cdot \log x}, \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - \frac{91893853320467}{100000000000000}}{\left(x - \frac{1}{2}\right) \cdot \log x}, \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   18. lift-*.f64 98.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(1 - \frac{x - 0.91893853320467}{\left(x - 0.5\right) \cdot \log x}, \color{blue}{\left(x - 0.5\right) \cdot \log x}, \frac{0.083333333333333}{x}\right)\right) \]

5. Applied rewrites 98.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \color{blue}{\mathsf{fma}\left(1 - \frac{x - 0.91893853320467}{\left(x - 0.5\right) \cdot \log x}, \left(x - 0.5\right) \cdot \log x, \frac{0.083333333333333}{x}\right)}\right) \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma
  (fma z (- y -0.0007936500793651) -0.0027777777777778)
  (/ z x)
  (+
   (/ 0.083333333333333 x)
   (- (- (* (log x) (- x 0.5)) x) -0.91893853320467))))

C

double code(double x, double y, double z) {
	return fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), ((0.083333333333333 / x) + (((log(x) * (x - 0.5)) - x) - -0.91893853320467)));
}

Julia

function code(x, y, z)
	return fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(Float64(0.083333333333333 / x) + Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) - -0.91893853320467)))
end

Wolfram

code[x_, y_, z_] := N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] - -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   5. div-add N/A

      \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

3. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]
4. Add Preprocessing

Alternative 3: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma
  (- x 0.5)
  (log x)
  (-
   (-
    (/
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)
     x)
    x)
   -0.91893853320467)))

C

double code(double x, double y, double z) {
	return fma((x - 0.5), log(x), (((fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) - x) - -0.91893853320467));
}

Julia

function code(x, y, z)
	return fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) - x) - -0.91893853320467))
end

Wolfram

code[x_, y_, z_] := N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision] - -0.91893853320467), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   5. div-add N/A

      \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

3. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

5. Applied rewrites 94.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)} \]
6. Add Preprocessing

Alternative 4: 93.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma
  (- x 0.5)
  (log x)
  (-
   (- (/ (fma (* z (+ 0.0007936500793651 y)) z 0.083333333333333) x) x)
   -0.91893853320467)))

C

double code(double x, double y, double z) {
	return fma((x - 0.5), log(x), (((fma((z * (0.0007936500793651 + y)), z, 0.083333333333333) / x) - x) - -0.91893853320467));
}

Julia

function code(x, y, z)
	return fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(fma(Float64(z * Float64(0.0007936500793651 + y)), z, 0.083333333333333) / x) - x) - -0.91893853320467))
end

Wolfram

code[x_, y_, z_] := N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision] - -0.91893853320467), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   5. div-add N/A

      \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

3. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

5. Applied rewrites 94.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} - x\right) - \frac{-91893853320467}{100000000000000}\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\mathsf{fma}\left(z \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} - x\right) - \frac{-91893853320467}{100000000000000}\right) \]

   2. lower-+.f64 93.5

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(z \cdot \left(0.0007936500793651 + \color{blue}{y}\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right) \]

8. Applied rewrites 93.5%

   \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(0.0007936500793651 + y\right)}, z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right) \]
9. Add Preprocessing

Alternative 5: 91.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.043:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.043)
   (/
    (/ 1.0 x)
    (/
     1.0
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)))
   (fma
    (- x 0.5)
    (log x)
    (- (- (/ (fma (* y z) z 0.083333333333333) x) x) -0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.043) {
		tmp = (1.0 / x) / (1.0 / fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333));
	} else {
		tmp = fma((x - 0.5), log(x), (((fma((y * z), z, 0.083333333333333) / x) - x) - -0.91893853320467));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.043)
		tmp = Float64(Float64(1.0 / x) / Float64(1.0 / fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(fma(Float64(y * z), z, 0.083333333333333) / x) - x) - -0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.043], N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 / N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision] - -0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.042999999999999997

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.6

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. Applied rewrites 63.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - \frac{-7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y - \frac{-7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(y - \frac{-7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \color{blue}{\frac{83333333333333}{1000000000000000}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x}{z \cdot \mathsf{fma}\left(y - \frac{-7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{x}{z \cdot \left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{x}{z \cdot \left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{x}{z \cdot \left(\left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}} \]

      8. add-flip N/A

         \[\leadsto \frac{1}{\frac{x}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{x}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) + \frac{83333333333333}{1000000000000000}}} \]

      10. sub-flip N/A

          \[\leadsto \frac{1}{\frac{x}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

   8. Applied rewrites 63.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}} \]

   if 0.042999999999999997 < x

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\mathsf{fma}\left(\color{blue}{y \cdot z}, z, \frac{83333333333333}{1000000000000000}\right)}{x} - x\right) - \frac{-91893853320467}{100000000000000}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 82.0

         \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(y \cdot \color{blue}{z}, z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right) \]

   8. Applied rewrites 82.0%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\color{blue}{y \cdot z}, z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(12.000000000000048, \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z, 1\right) \cdot 0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} - x\right) - -0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+149)
     (/
      (*
       (fma
        12.000000000000048
        (* (fma (- y -0.0007936500793651) z -0.0027777777777778) z)
        1.0)
       0.083333333333333)
      x)
     (if (<= t_0 5e+300)
       (fma
        (- x 0.5)
        (log x)
        (- (- (/ 0.083333333333333 x) x) -0.91893853320467))
       (fma
        (fma z (- y -0.0007936500793651) -0.0027777777777778)
        (/ z x)
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+149) {
		tmp = (fma(12.000000000000048, (fma((y - -0.0007936500793651), z, -0.0027777777777778) * z), 1.0) * 0.083333333333333) / x;
	} else if (t_0 <= 5e+300) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) - x) - -0.91893853320467));
	} else {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -1e+149)
		tmp = Float64(Float64(fma(12.000000000000048, Float64(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778) * z), 1.0) * 0.083333333333333) / x);
	elseif (t_0 <= 5e+300)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) - x) - -0.91893853320467));
	else
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+149], N[(N[(N[(12.000000000000048 * N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 1.0), $MachinePrecision] * 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e+300], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision] - -0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000005e149

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.6

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      7. add-flip N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      17. sum-to-mult-rev N/A

          \[\leadsto \frac{\left(1 + \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{\frac{83333333333333}{1000000000000000}}\right) \cdot \frac{83333333333333}{1000000000000000}}{x} \]

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\mathsf{fma}\left(12.000000000000048, \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z, 1\right) \cdot 0.083333333333333}{x} \]

   if -1.00000000000000005e149 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 5.00000000000000026e300

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} - x\right) - \frac{-91893853320467}{100000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 57.1%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\color{blue}{0.083333333333333}}{x} - x\right) - -0.91893853320467\right) \]

   if 5.00000000000000026e300 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 65.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{\color{blue}{x}}\right) \]

   6. Applied rewrites 65.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \color{blue}{\frac{0.083333333333333}{x}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3400000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3400000000.0)
   (/
    (-
     (* (fma (- y -0.0007936500793651) z -0.0027777777777778) z)
     -0.083333333333333)
    x)
   (fma (- x 0.5) (log x) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3400000000.0) {
		tmp = ((fma((y - -0.0007936500793651), z, -0.0027777777777778) * z) - -0.083333333333333) / x;
	} else {
		tmp = fma((x - 0.5), log(x), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3400000000.0)
		tmp = Float64(Float64(Float64(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778) * z) - -0.083333333333333) / x);
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3400000000.0], N[(N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4e9

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.6

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      7. add-flip N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      17. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

   6. Applied rewrites 63.6%

      \[\leadsto \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]

   if 3.4e9 < x

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{-1 \cdot x}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 35.9

         \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, -1 \cdot \color{blue}{x}\right) \]

   8. Applied rewrites 35.9%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3400000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3400000000.0)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (fma (- x 0.5) (log x) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3400000000.0) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma((x - 0.5), log(x), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3400000000.0)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3400000000.0], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4e9

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.6

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      8. add-flip N/A

         \[\leadsto \frac{\left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      15. lift-fma.f64 63.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

      16. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      18. lower-fma.f64 63.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   6. Applied rewrites 63.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]

   if 3.4e9 < x

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}}{x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot \frac{z}{x}} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - x\right) - -0.91893853320467\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{-1 \cdot x}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 35.9

         \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, -1 \cdot \color{blue}{x}\right) \]

   8. Applied rewrites 35.9%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (fma
   (fma (- y -0.0007936500793651) z -0.0027777777777778)
   z
   0.083333333333333)
  x))

C

double code(double x, double y, double z) {
	return fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.6

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   2. +-commutative N/A

      \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   7. +-commutative N/A

      \[\leadsto \frac{\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   8. add-flip N/A

      \[\leadsto \frac{\left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   12. sub-flip-reverse N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   15. lift-fma.f64 63.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   16. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   17. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   18. lower-fma.f64 63.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

6. Applied rewrites 63.6%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]
7. Add Preprocessing

Alternative 10: 63.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y \cdot z\right) \cdot z - -0.083333333333333}{x}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0035:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (* (* y z) z) -0.083333333333333) x)))
   (if (<= y -6.2e+72)
     t_0
     (if (<= y 0.0035)
       (/
        (-
         (* (fma 0.0007936500793651 z -0.0027777777777778) z)
         -0.083333333333333)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((y * z) * z) - -0.083333333333333) / x;
	double tmp;
	if (y <= -6.2e+72) {
		tmp = t_0;
	} else if (y <= 0.0035) {
		tmp = ((fma(0.0007936500793651, z, -0.0027777777777778) * z) - -0.083333333333333) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y * z) * z) - -0.083333333333333) / x)
	tmp = 0.0
	if (y <= -6.2e+72)
		tmp = t_0;
	elseif (y <= 0.0035)
		tmp = Float64(Float64(Float64(fma(0.0007936500793651, z, -0.0027777777777778) * z) - -0.083333333333333) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -6.2e+72], t$95$0, If[LessEqual[y, 0.0035], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999977e72 or 0.00350000000000000007 < y

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.6

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      7. add-flip N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      17. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

   6. Applied rewrites 63.6%

      \[\leadsto \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\left(y \cdot z\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x} \]
   8. Step-by-step derivation

      1. lower-*.f64 50.4

         \[\leadsto \frac{\left(y \cdot z\right) \cdot z - -0.083333333333333}{x} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\left(y \cdot z\right) \cdot z - -0.083333333333333}{x} \]

   if -6.19999999999999977e72 < y < 0.00350000000000000007

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.6

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      7. add-flip N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      17. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

   6. Applied rewrites 63.6%

      \[\leadsto \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - -0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (* (* z (+ 0.0007936500793651 y)) z) -0.083333333333333) x))

C

double code(double x, double y, double z) {
	return (((z * (0.0007936500793651 + y)) * z) - -0.083333333333333) / 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((z * (0.0007936500793651d0 + y)) * z) - (-0.083333333333333d0)) / x
end function

Java

public static double code(double x, double y, double z) {
	return (((z * (0.0007936500793651 + y)) * z) - -0.083333333333333) / x;
}

Python

def code(x, y, z):
	return (((z * (0.0007936500793651 + y)) * z) - -0.083333333333333) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) * z) - -0.083333333333333) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((z * (0.0007936500793651 + y)) * z) - -0.083333333333333) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.6

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   7. add-flip N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   11. sub-flip-reverse N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

   15. *-commutative N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   17. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   18. add-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

   19. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

6. Applied rewrites 63.6%

   \[\leadsto \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]

7. Taylor expanded in z around inf

   \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x} \]

   2. lower-+.f64 63.0

      \[\leadsto \frac{\left(z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - -0.083333333333333}{x} \]

9. Applied rewrites 63.0%

   \[\leadsto \frac{\left(z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - -0.083333333333333}{x} \]
10. Add Preprocessing

Alternative 12: 50.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(y \cdot z\right) \cdot z - -0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (- (* (* y z) z) -0.083333333333333) x))

C

double code(double x, double y, double z) {
	return (((y * z) * z) - -0.083333333333333) / 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((y * z) * z) - (-0.083333333333333d0)) / x
end function

Java

public static double code(double x, double y, double z) {
	return (((y * z) * z) - -0.083333333333333) / x;
}

Python

def code(x, y, z):
	return (((y * z) * z) - -0.083333333333333) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(y * z) * z) - -0.083333333333333) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((y * z) * z) - -0.083333333333333) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.6

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   7. add-flip N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   11. sub-flip-reverse N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

   15. *-commutative N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   17. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   18. add-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

   19. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

6. Applied rewrites 63.6%

   \[\leadsto \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]

7. Taylor expanded in y around inf

   \[\leadsto \frac{\left(y \cdot z\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x} \]
8. Step-by-step derivation

   1. lower-*.f64 50.4

      \[\leadsto \frac{\left(y \cdot z\right) \cdot z - -0.083333333333333}{x} \]

9. Applied rewrites 50.4%

   \[\leadsto \frac{\left(y \cdot z\right) \cdot z - -0.083333333333333}{x} \]
10. Add Preprocessing

Alternative 13: 29.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{-0.0027777777777778 \cdot z - -0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (* -0.0027777777777778 z) -0.083333333333333) x))

C

double code(double x, double y, double z) {
	return ((-0.0027777777777778 * z) - -0.083333333333333) / 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((-0.0027777777777778d0) * z) - (-0.083333333333333d0)) / x
end function

Java

public static double code(double x, double y, double z) {
	return ((-0.0027777777777778 * z) - -0.083333333333333) / x;
}

Python

def code(x, y, z):
	return ((-0.0027777777777778 * z) - -0.083333333333333) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(-0.0027777777777778 * z) - -0.083333333333333) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((-0.0027777777777778 * z) - -0.083333333333333) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(-0.0027777777777778 * z), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.6

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

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

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   7. add-flip N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   11. sub-flip-reverse N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)}{x} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} - \left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

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

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right)}{x} \]

   15. *-commutative N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   17. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   18. add-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

   19. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z - \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

6. Applied rewrites 63.6%

   \[\leadsto \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z - -0.083333333333333}{x} \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z - \frac{-83333333333333}{1000000000000000}}{x} \]
8. Step-by-step derivation

9. Applied rewrites 29.3%

   \[\leadsto \frac{-0.0027777777777778 \cdot z - -0.083333333333333}{x} \]
10. Add Preprocessing

Alternative 14: 23.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 12.000000000000048} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (* x 12.000000000000048)))

C

double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (x * 12.000000000000048d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Python

def code(x, y, z):
	return 1.0 / (x * 12.000000000000048)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(x * 12.000000000000048))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (x * 12.000000000000048);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.6

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

6. Applied rewrites 63.5%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right) \cdot z}{x}, x \cdot 12.000000000000048, 1\right)}{\color{blue}{x \cdot 12.000000000000048}} \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{1}{\color{blue}{x} \cdot \frac{1000000000000000}{83333333333333}} \]
8. Step-by-step derivation

9. Applied rewrites 23.7%

   \[\leadsto \frac{1}{\color{blue}{x} \cdot 12.000000000000048} \]
10. Add Preprocessing

Alternative 15: 23.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.6

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 23.7%

   \[\leadsto \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))