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

Percentage Accurate: 93.9% → 98.7%

Time: 6.2s

Alternatives: 17

Speedup: 1.0×

• 31.7% 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: 93.9% 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: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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 98.6%

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

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

   7. add-flip-rev N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 98.7

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

9. Applied rewrites 98.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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 98.6%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.6

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift--.f64 N/A

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

   7. associate-+l- N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 3: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(-x\right) - -0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{x} \cdot \mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.2e+196)
   (+
    (fma (- x 0.5) (log x) (- (- x) -0.91893853320467))
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (fma
    z
    (* (/ 1.0 x) (fma 0.0007936500793651 z -0.0027777777777778))
    (-
     (fma (log x) (- x 0.5) (- 0.91893853320467 x))
     (/ -0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e+196) {
		tmp = fma((x - 0.5), log(x), (-x - -0.91893853320467)) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = fma(z, ((1.0 / x) * fma(0.0007936500793651, z, -0.0027777777777778)), (fma(log(x), (x - 0.5), (0.91893853320467 - x)) - (-0.083333333333333 / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.2e+196)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(-x) - -0.91893853320467)) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = fma(z, Float64(Float64(1.0 / x) * fma(0.0007936500793651, z, -0.0027777777777778)), Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) - Float64(-0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.2e+196], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[((-x) - -0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(1.0 / x), $MachinePrecision] * N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999993e196

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval 94.0

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

   3. Applied rewrites 94.0%

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

   if 3.19999999999999993e196 < x

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 98.6%

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

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

      7. add-flip-rev N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 80.4%

       \[\leadsto \mathsf{fma}\left(z, \frac{1}{x} \cdot \mathsf{fma}\left(\color{blue}{0.0007936500793651}, z, -0.0027777777777778\right), \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(-x\right) - -0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.2e+196)
   (+
    (fma (- x 0.5) (log x) (- (- x) -0.91893853320467))
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (fma
    z
    (/ (- (* 0.0007936500793651 z) 0.0027777777777778) x)
    (-
     (fma (log x) (- x 0.5) (- 0.91893853320467 x))
     (/ -0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e+196) {
		tmp = fma((x - 0.5), log(x), (-x - -0.91893853320467)) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = fma(z, (((0.0007936500793651 * z) - 0.0027777777777778) / x), (fma(log(x), (x - 0.5), (0.91893853320467 - x)) - (-0.083333333333333 / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.2e+196)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(-x) - -0.91893853320467)) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = fma(z, Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) / x), Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) - Float64(-0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.2e+196], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[((-x) - -0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999993e196

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval 94.0

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

   3. Applied rewrites 94.0%

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

   if 3.19999999999999993e196 < x

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 98.6%

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

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

      7. add-flip-rev N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 80.4

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

   10. Applied rewrites 80.4%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - \left(\left(x - 0.91893853320467\right) - \left(x - 0.5\right) \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.2e+196)
   (-
    (/
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)
     x)
    (- (- x 0.91893853320467) (* (- x 0.5) (log x))))
   (fma
    z
    (/ (- (* 0.0007936500793651 z) 0.0027777777777778) x)
    (-
     (fma (log x) (- x 0.5) (- 0.91893853320467 x))
     (/ -0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e+196) {
		tmp = (fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) - ((x - 0.91893853320467) - ((x - 0.5) * log(x)));
	} else {
		tmp = fma(z, (((0.0007936500793651 * z) - 0.0027777777777778) / x), (fma(log(x), (x - 0.5), (0.91893853320467 - x)) - (-0.083333333333333 / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.2e+196)
		tmp = Float64(Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) - Float64(Float64(x - 0.91893853320467) - Float64(Float64(x - 0.5) * log(x))));
	else
		tmp = fma(z, Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) / x), Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) - Float64(-0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.2e+196], N[(N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999993e196

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 98.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. div-add-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 93.9%

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

   if 3.19999999999999993e196 < x

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 98.6%

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

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

      7. add-flip-rev N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 80.4

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

   10. Applied rewrites 80.4%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{+196}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - \left(\left(x - 0.91893853320467\right) - \left(x - 0.5\right) \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.3e+196)
   (-
    (/
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)
     x)
    (- (- x 0.91893853320467) (* (- x 0.5) (log x))))
   (+
    (fma (log x) (- x 0.5) (- 0.91893853320467 x))
    (/ (fma -0.0027777777777778 z 0.083333333333333) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.3e+196) {
		tmp = (fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) - ((x - 0.91893853320467) - ((x - 0.5) * log(x)));
	} else {
		tmp = fma(log(x), (x - 0.5), (0.91893853320467 - x)) + (fma(-0.0027777777777778, z, 0.083333333333333) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.3e+196)
		tmp = Float64(Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) - Float64(Float64(x - 0.91893853320467) - Float64(Float64(x - 0.5) * log(x))));
	else
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) + Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.3e+196], N[(N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.30000000000000012e196

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 98.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. div-add-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 93.9%

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

   if 4.30000000000000012e196 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.5%

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

   6. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{+196}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) - \left(-0.91893853320467 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.3e+196)
   (-
    (- (* (log x) (- x 0.5)) x)
    (-
     -0.91893853320467
     (/
      (fma
       (fma z (- y -0.0007936500793651) -0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (+
    (fma (log x) (- x 0.5) (- 0.91893853320467 x))
    (/ (fma -0.0027777777777778 z 0.083333333333333) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.3e+196) {
		tmp = ((log(x) * (x - 0.5)) - x) - (-0.91893853320467 - (fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = fma(log(x), (x - 0.5), (0.91893853320467 - x)) + (fma(-0.0027777777777778, z, 0.083333333333333) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.3e+196)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) - Float64(-0.91893853320467 - Float64(fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) + Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.3e+196], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] - N[(-0.91893853320467 - N[(N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.30000000000000012e196

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 93.9

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

   3. Applied rewrites 93.9%

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

   if 4.30000000000000012e196 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.5%

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

   6. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;t\_0 + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \leq -4 \cdot 10^{+30}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (<=
        (+
         t_0
         (/
          (+
           (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
           0.083333333333333)
          x))
        -4e+30)
     (* (* (/ (- y -0.0007936500793651) x) z) z)
     (+
      t_0
      (/
       (+
        (* (- (* 0.0007936500793651 z) 0.0027777777777778) z)
        0.083333333333333)
       x)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if ((t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)) <= -4e+30) {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	} else {
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0
    if ((t_0 + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)) <= (-4d+30)) then
        tmp = (((y - (-0.0007936500793651d0)) / x) * z) * z
    else
        tmp = t_0 + (((((0.0007936500793651d0 * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * Math.log(x)) - x) + 0.91893853320467;
	double tmp;
	if ((t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)) <= -4e+30) {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	} else {
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	tmp = 0
	if (t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)) <= -4e+30:
		tmp = (((y - -0.0007936500793651) / x) * z) * z
	else:
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)) <= -4e+30)
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	tmp = 0.0;
	if ((t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)) <= -4e+30)
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	else
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -4e+30], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(t$95$0 + N[(N[(N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 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)) < -4.0000000000000001e30

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \color{blue}{{z}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot {z}^{\color{blue}{2}} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \left(z \cdot \color{blue}{z}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      7. lower-*.f64 44.1

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

      8. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      12. mult-flip-rev N/A

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

      13. div-add-rev N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if -4.0000000000000001e30 < (+.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 93.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 78.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\color{blue}{0.0007936500793651} \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -4e+33)
     t_0
     (if (<= t_1 1e+51)
       (+
        (fma (log x) (- x 0.5) (- 0.91893853320467 x))
        (/ (fma -0.0027777777777778 z 0.083333333333333) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -4e+33) {
		tmp = t_0;
	} else if (t_1 <= 1e+51) {
		tmp = fma(log(x), (x - 0.5), (0.91893853320467 - x)) + (fma(-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 - -0.0007936500793651) / x) * z) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -4e+33)
		tmp = t_0;
	elseif (t_1 <= 1e+51)
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) + Float64(fma(-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 - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+33], t$95$0, If[LessEqual[t$95$1, 1e+51], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -3.9999999999999998e33 or 1e51 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \color{blue}{{z}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot {z}^{\color{blue}{2}} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \left(z \cdot \color{blue}{z}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      7. lower-*.f64 44.1

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

      8. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      12. mult-flip-rev N/A

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

      13. div-add-rev N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if -3.9999999999999998e33 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1e51

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.5%

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

   6. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -4e+33)
     t_0
     (if (<= t_1 1e+51)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ 0.083333333333333 x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -4e+33) {
		tmp = t_0;
	} else if (t_1 <= 1e+51) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((y - (-0.0007936500793651d0)) / x) * z) * z
    t_1 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_1 <= (-4d+33)) then
        tmp = t_0
    else if (t_1 <= 1d+51) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -4e+33) {
		tmp = t_0;
	} else if (t_1 <= 1e+51) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y - -0.0007936500793651) / x) * z) * z
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_1 <= -4e+33:
		tmp = t_0
	elif t_1 <= 1e+51:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -4e+33)
		tmp = t_0;
	elseif (t_1 <= 1e+51)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_1 <= -4e+33)
		tmp = t_0;
	elseif (t_1 <= 1e+51)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+33], t$95$0, If[LessEqual[t$95$1, 1e+51], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -3.9999999999999998e33 or 1e51 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \color{blue}{{z}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot {z}^{\color{blue}{2}} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \left(z \cdot \color{blue}{z}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      7. lower-*.f64 44.1

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

      8. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      12. mult-flip-rev N/A

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

      13. div-add-rev N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if -3.9999999999999998e33 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1e51

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 56.8%

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

Alternative 11: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.55e+78)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.55e+78) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.55e+78)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.55e+78], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.55000000000000016e78

   1. Initial program 93.9%

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

   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 63.2

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

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 N/A

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

      12. metadata-eval 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. add-flip 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 63.2

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

      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 63.2

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

   6. Applied rewrites 63.2%

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

   if 2.55000000000000016e78 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.0

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 35.0

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      8. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      10. remove-double-neg 35.0

          \[\leadsto \left(\log x - 1\right) \cdot x \]

   6. Applied rewrites 35.0%

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

Alternative 12: 74.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{if}\;x \leq 4.4 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z)))
   (if (<= x 4.4e-163)
     t_0
     (if (<= x 1.1e-37)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       (if (<= x 2.55e+78) t_0 (* (- (log x) 1.0) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double tmp;
	if (x <= 4.4e-163) {
		tmp = t_0;
	} else if (x <= 1.1e-37) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else if (x <= 2.55e+78) {
		tmp = t_0;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	tmp = 0.0
	if (x <= 4.4e-163)
		tmp = t_0;
	elseif (x <= 1.1e-37)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	elseif (x <= 2.55e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 4.4e-163], t$95$0, If[LessEqual[x, 1.1e-37], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.55e+78], t$95$0, N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.40000000000000022e-163 or 1.10000000000000001e-37 < x < 2.55000000000000016e78

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \color{blue}{{z}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot {z}^{\color{blue}{2}} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \left(z \cdot \color{blue}{z}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      7. lower-*.f64 44.1

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

      8. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      12. mult-flip-rev N/A

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

      13. div-add-rev N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if 4.40000000000000022e-163 < x < 1.10000000000000001e-37

   1. Initial program 93.9%

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

   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 63.2

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

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 N/A

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

      12. metadata-eval 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. add-flip 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 63.2

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

      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 63.2

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 46.7%

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

   if 2.55000000000000016e78 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.0

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 35.0

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      8. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      10. remove-double-neg 35.0

          \[\leadsto \left(\log x - 1\right) \cdot x \]

   6. Applied rewrites 35.0%

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

Alternative 13: 70.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{if}\;x \leq 6.8 \cdot 10^{-209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z)))
   (if (<= x 6.8e-209)
     t_0
     (if (<= x 2.6e-15)
       (/ (fma (* y z) z 0.083333333333333) x)
       (if (<= x 2.55e+78) t_0 (* (- (log x) 1.0) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double tmp;
	if (x <= 6.8e-209) {
		tmp = t_0;
	} else if (x <= 2.6e-15) {
		tmp = fma((y * z), z, 0.083333333333333) / x;
	} else if (x <= 2.55e+78) {
		tmp = t_0;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	tmp = 0.0
	if (x <= 6.8e-209)
		tmp = t_0;
	elseif (x <= 2.6e-15)
		tmp = Float64(fma(Float64(y * z), z, 0.083333333333333) / x);
	elseif (x <= 2.55e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 6.8e-209], t$95$0, If[LessEqual[x, 2.6e-15], N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.55e+78], t$95$0, N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.79999999999999976e-209 or 2.60000000000000004e-15 < x < 2.55000000000000016e78

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \color{blue}{{z}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot {z}^{\color{blue}{2}} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \left(z \cdot \color{blue}{z}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      7. lower-*.f64 44.1

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

      8. lift-fma.f64 N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      12. mult-flip-rev N/A

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

      13. div-add-rev N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if 6.79999999999999976e-209 < x < 2.60000000000000004e-15

   1. Initial program 93.9%

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

   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 63.2

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

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 N/A

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

      12. metadata-eval 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. add-flip 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 63.2

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

      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 63.2

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 50.0

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

   9. Applied rewrites 50.0%

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

   if 2.55000000000000016e78 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.0

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 35.0

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      8. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      10. remove-double-neg 35.0

          \[\leadsto \left(\log x - 1\right) \cdot x \]

   6. Applied rewrites 35.0%

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

Alternative 14: 66.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.4e+18)
   (/ (fma (* y z) z 0.083333333333333) x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.4e+18) {
		tmp = fma((y * z), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.4e+18)
		tmp = Float64(fma(Float64(y * z), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.4e+18], N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e18

   1. Initial program 93.9%

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

   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 63.2

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

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 N/A

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

      12. metadata-eval 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. add-flip 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 63.2

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

      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 63.2

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 50.0

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

   9. Applied rewrites 50.0%

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

   if 2.4e18 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.0

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 35.0

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot \color{blue}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - 1\right) \cdot x \]

      8. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - 1\right) \cdot x \]

      10. remove-double-neg 35.0

          \[\leadsto \left(\log x - 1\right) \cdot x \]

   6. Applied rewrites 35.0%

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

Alternative 15: 50.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fma((y * z), z, 0.083333333333333) / x;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

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 63.2

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

4. Applied rewrites 63.2%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
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. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. add-flip N/A

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

   10. metadata-eval N/A

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

   11. lift--.f64 N/A

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

   12. metadata-eval 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. add-flip 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 63.2

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

   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 63.2

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

6. Applied rewrites 63.2%

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

7. Taylor expanded in y around inf

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

   1. lower-*.f64 50.0

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

9. Applied rewrites 50.0%

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

Alternative 16: 29.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

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 63.2

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

4. Applied rewrites 63.2%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
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. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. add-flip N/A

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

   10. metadata-eval N/A

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

   11. lift--.f64 N/A

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

   12. metadata-eval 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. add-flip 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 63.2

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

   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 63.2

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

6. Applied rewrites 63.2%

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

7. Taylor expanded in z around 0

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

9. Applied rewrites 29.0%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x} \]
10. Add Preprocessing

Alternative 17: 23.2% 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 93.9%

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

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 63.2

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

4. Applied rewrites 63.2%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 23.2%

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

Reproduce

herbie shell --seed 2025156 
(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)))