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

Percentage Accurate: 94.1% → 98.8%

Time: 10.3s

Alternatives: 26

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% 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: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{1}{x \cdot 12.000000000000048}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (fma
   (fma z (- y -0.0007936500793651) -0.0027777777777778)
   (/ z x)
   (/ 1.0 (* x 12.000000000000048)))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (1.0 / (x * 12.000000000000048)));
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(1.0 / Float64(x * 12.000000000000048))))
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[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. add-flip N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 98.7

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

3. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 98.8

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{1}{x \cdot \color{blue}{12.000000000000048}}\right) \]

5. Applied rewrites 98.8%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \color{blue}{\frac{1}{x \cdot 12.000000000000048}}\right) \]
6. Add Preprocessing

Alternative 2: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(fma(z, 0.0007936500793651, fma(y, z, -0.0027777777777778)), Float64(z / x), Float64(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[(z * 0.0007936500793651 + N[(y * z + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. add-flip N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 98.7

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

3. Applied rewrites 98.7%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. metadata-eval N/A

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

   4. add-flip N/A

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

   5. distribute-rgt-in N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-fma.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(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[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. add-flip N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 98.7

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

3. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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 98.7%

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

Alternative 5: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{if}\;y \leq -0.00078:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, \frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x - 0.5\right) - x\right) - -0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (fma (* y z) (/ z x) (/ 0.083333333333333 x)))))
   (if (<= y -0.00078)
     t_0
     (if (<= y 8e-8)
       (fma
        z
        (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
        (+
         (/ 0.083333333333333 x)
         (- (- (* (log x) (- x 0.5)) x) -0.91893853320467)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma((y * z), (z / x), (0.083333333333333 / x));
	double tmp;
	if (y <= -0.00078) {
		tmp = t_0;
	} else if (y <= 8e-8) {
		tmp = fma(z, (fma(0.0007936500793651, z, -0.0027777777777778) / x), ((0.083333333333333 / x) + (((log(x) * (x - 0.5)) - x) - -0.91893853320467)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(y * z), Float64(z / x), Float64(0.083333333333333 / x)))
	tmp = 0.0
	if (y <= -0.00078)
		tmp = t_0;
	elseif (y <= 8e-8)
		tmp = fma(z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), Float64(Float64(0.083333333333333 / x) + Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) - -0.91893853320467)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00078], t$95$0, If[LessEqual[y, 8e-8], N[(z * N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] - -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.79999999999999986e-4 or 8.0000000000000002e-8 < y

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 84.7

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

   6. Applied rewrites 84.7%

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

   if -7.79999999999999986e-4 < y < 8.0000000000000002e-8

   1. Initial program 94.1%

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

   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 78.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. lift-/.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 80.9%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

3. Applied rewrites 97.9%

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

Alternative 7: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 10^{+271}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, \mathsf{fma}\left(y, z, -0.0027777777777778\right)\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      1e+271)
   (-
    (* (log x) (- x 0.5))
    (-
     (- x 0.91893853320467)
     (/
      (fma
       (fma z (- y -0.0007936500793651) -0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (+
    (+ (* -0.5 (log x)) 0.91893853320467)
    (fma
     (fma z 0.0007936500793651 (fma y z -0.0027777777777778))
     (/ z x)
     (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+271) {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + fma(fma(z, 0.0007936500793651, fma(y, z, -0.0027777777777778)), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+271)
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(-0.5 * log(x)) + 0.91893853320467) + fma(fma(z, 0.0007936500793651, fma(y, z, -0.0027777777777778)), Float64(z / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 1e+271], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * 0.0007936500793651 + N[(y * z + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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)) < 9.99999999999999953e270

   1. Initial program 94.1%

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

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

      4. associate-+l- N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 94.1

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

   3. Applied rewrites 94.1%

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

   if 9.99999999999999953e270 < (+.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 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.7

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

   3. Applied rewrites 98.7%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 64.9

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

   8. Applied rewrites 64.9%

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

Alternative 8: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right)\\ \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 10^{+271}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(t\_0, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \mathsf{fma}\left(t\_0, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (- y -0.0007936500793651) -0.0027777777777778)))
   (if (<=
        (+
         (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
         0.083333333333333)
        1e+271)
     (-
      (* (log x) (- x 0.5))
      (- (- x 0.91893853320467) (/ (fma t_0 z 0.083333333333333) x)))
     (+
      (+ (* -0.5 (log x)) 0.91893853320467)
      (fma t_0 (/ z x) (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (y - -0.0007936500793651), -0.0027777777777778);
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+271) {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (fma(t_0, z, 0.083333333333333) / x));
	} else {
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + fma(t_0, (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+271)
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(fma(t_0, z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(-0.5 * log(x)) + 0.91893853320467) + fma(t_0, Float64(z / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 1e+271], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(t$95$0 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(t$95$0 * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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)) < 9.99999999999999953e270

   1. Initial program 94.1%

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

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

      4. associate-+l- N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 94.1

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

   3. Applied rewrites 94.1%

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

   if 9.99999999999999953e270 < (+.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 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 64.9

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

   6. Applied rewrites 64.9%

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

Alternative 9: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+180}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - \left(-0.91893853320467 - \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+69)
     (+
      (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
      (fma (* y z) (/ z x) (/ 0.083333333333333 x)))
     (if (<= t_0 1e+180)
       (-
        (/
         (fma
          (fma 0.0007936500793651 z -0.0027777777777778)
          z
          0.083333333333333)
         x)
        (- -0.91893853320467 (- (* (log x) (- x 0.5)) x)))
       (+
        (+ (* -0.5 (log x)) 0.91893853320467)
        (fma
         (fma z (- y -0.0007936500793651) -0.0027777777777778)
         (/ z x)
         (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+69) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma((y * z), (z / x), (0.083333333333333 / x));
	} else if (t_0 <= 1e+180) {
		tmp = (fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) - (-0.91893853320467 - ((log(x) * (x - 0.5)) - x));
	} else {
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+69)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(y * z), Float64(z / x), Float64(0.083333333333333 / x)));
	elseif (t_0 <= 1e+180)
		tmp = Float64(Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) - Float64(-0.91893853320467 - Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	else
		tmp = Float64(Float64(Float64(-0.5 * log(x)) + 0.91893853320467) + fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+69], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+180], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(-0.91893853320467 - N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)) < -1.0000000000000001e69

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 84.7

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

   6. Applied rewrites 84.7%

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

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1e180

   1. Initial program 94.1%

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

   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 78.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. add-flip N/A

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

   6. Applied rewrites 78.5%

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

   if 1e180 < (+.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 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 64.9

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

   6. Applied rewrites 64.9%

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

Alternative 10: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x \cdot \left(x - 0.5\right)\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;t\_0 - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+180}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - \left(-0.91893853320467 - \left(t\_0 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log x + 0.91893853320467\right) + \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log x) (- x 0.5)))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+69)
     (- t_0 (- (- x 0.91893853320467) (/ (fma (* y z) z 0.083333333333333) x)))
     (if (<= t_1 1e+180)
       (-
        (/
         (fma
          (fma 0.0007936500793651 z -0.0027777777777778)
          z
          0.083333333333333)
         x)
        (- -0.91893853320467 (- t_0 x)))
       (+
        (+ (* -0.5 (log x)) 0.91893853320467)
        (fma
         (fma z (- y -0.0007936500793651) -0.0027777777777778)
         (/ z x)
         (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) * (x - 0.5);
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = t_0 - ((x - 0.91893853320467) - (fma((y * z), z, 0.083333333333333) / x));
	} else if (t_1 <= 1e+180) {
		tmp = (fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) - (-0.91893853320467 - (t_0 - x));
	} else {
		tmp = ((-0.5 * log(x)) + 0.91893853320467) + fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) * Float64(x - 0.5))
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = Float64(t_0 - Float64(Float64(x - 0.91893853320467) - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	elseif (t_1 <= 1e+180)
		tmp = Float64(Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) - Float64(-0.91893853320467 - Float64(t_0 - x)));
	else
		tmp = Float64(Float64(Float64(-0.5 * log(x)) + 0.91893853320467) + fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $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, -1e+69], N[(t$95$0 - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+180], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(-0.91893853320467 - N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)) < -1.0000000000000001e69

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1e180

   1. Initial program 94.1%

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

   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 78.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. add-flip N/A

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

   6. Applied rewrites 78.5%

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

   if 1e180 < (+.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 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 64.9

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

   6. Applied rewrites 64.9%

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

Alternative 11: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x \cdot \left(x - 0.5\right)\\ \mathbf{if}\;y + 0.0007936500793651 \leq -0.002:\\ \;\;\;\;t\_0 - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - \left(-0.91893853320467 - \left(t\_0 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(y \cdot z\right) \cdot z + 0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log x) (- x 0.5))))
   (if (<= (+ y 0.0007936500793651) -0.002)
     (- t_0 (- (- x 0.91893853320467) (/ (fma (* y z) z 0.083333333333333) x)))
     (if (<= (+ y 0.0007936500793651) 5e+38)
       (-
        (/
         (fma
          (fma 0.0007936500793651 z -0.0027777777777778)
          z
          0.083333333333333)
         x)
        (- -0.91893853320467 (- t_0 x)))
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ (+ (* (* y z) z) 0.083333333333333) x))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) * (x - 0.5);
	double tmp;
	if ((y + 0.0007936500793651) <= -0.002) {
		tmp = t_0 - ((x - 0.91893853320467) - (fma((y * z), z, 0.083333333333333) / x));
	} else if ((y + 0.0007936500793651) <= 5e+38) {
		tmp = (fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) - (-0.91893853320467 - (t_0 - x));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((y * z) * z) + 0.083333333333333) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) * Float64(x - 0.5))
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -0.002)
		tmp = Float64(t_0 - Float64(Float64(x - 0.91893853320467) - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	elseif (Float64(y + 0.0007936500793651) <= 5e+38)
		tmp = Float64(Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) - Float64(-0.91893853320467 - Float64(t_0 - x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(y * z) * z) + 0.083333333333333) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -0.002], N[(t$95$0 - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 5e+38], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(-0.91893853320467 - N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -2e-3

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if -2e-3 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 4.9999999999999997e38

   1. Initial program 94.1%

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

   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 78.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. add-flip N/A

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

   6. Applied rewrites 78.5%

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

   if 4.9999999999999997e38 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

Alternative 12: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(y \cdot z\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e+40)
   (/
    1.0
    (/
     x
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)))
   (if (<= z 1.3e+29)
     (+
      (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
      (/ (+ (* (* y z) z) 0.083333333333333) x))
     (* (pow z 2.0) (fma 0.0007936500793651 (/ 1.0 x) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+40) {
		tmp = 1.0 / (x / fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333));
	} else if (z <= 1.3e+29) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((y * z) * z) + 0.083333333333333) / x);
	} else {
		tmp = pow(z, 2.0) * fma(0.0007936500793651, (1.0 / x), (y / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e+40)
		tmp = Float64(1.0 / Float64(x / fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)));
	elseif (z <= 1.3e+29)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(y * z) * z) + 0.083333333333333) / x));
	else
		tmp = Float64((z ^ 2.0) * fma(0.0007936500793651, Float64(1.0 / x), Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e+40], N[(1.0 / N[(x / N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e+29], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Power[z, 2.0], $MachinePrecision] * N[(0.0007936500793651 * N[(1.0 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.39999999999999989e40

   1. Initial program 94.1%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if -3.39999999999999989e40 < z < 1.3e29

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

   if 1.3e29 < z

   1. Initial program 94.1%

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

   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 41.9

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

   4. Applied rewrites 41.9%

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

Alternative 13: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+29}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) - \left(-0.91893853320467 - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e+40)
   (/
    1.0
    (/
     x
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)))
   (if (<= z 1.3e+29)
     (-
      (- (* (log x) (- x 0.5)) x)
      (- -0.91893853320467 (/ (fma (* y z) z 0.083333333333333) x)))
     (* (pow z 2.0) (fma 0.0007936500793651 (/ 1.0 x) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+40) {
		tmp = 1.0 / (x / fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333));
	} else if (z <= 1.3e+29) {
		tmp = ((log(x) * (x - 0.5)) - x) - (-0.91893853320467 - (fma((y * z), z, 0.083333333333333) / x));
	} else {
		tmp = pow(z, 2.0) * fma(0.0007936500793651, (1.0 / x), (y / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e+40)
		tmp = Float64(1.0 / Float64(x / fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)));
	elseif (z <= 1.3e+29)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) - Float64(-0.91893853320467 - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	else
		tmp = Float64((z ^ 2.0) * fma(0.0007936500793651, Float64(1.0 / x), Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e+40], N[(1.0 / N[(x / N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e+29], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] - N[(-0.91893853320467 - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[z, 2.0], $MachinePrecision] * N[(0.0007936500793651 * N[(1.0 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.39999999999999989e40

   1. Initial program 94.1%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if -3.39999999999999989e40 < z < 1.3e29

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if 1.3e29 < z

   1. Initial program 94.1%

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

   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 41.9

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

   4. Applied rewrites 41.9%

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

Alternative 14: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-31}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(\mathsf{fma}\left(y, 1260.0011340009878, 1\right) \cdot \left(0.0007936500793651 \cdot z\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.95e-31)
   (/
    (+
     0.083333333333333
     (*
      z
      (-
       (* (fma y 1260.0011340009878 1.0) (* 0.0007936500793651 z))
       0.0027777777777778)))
    x)
   (-
    (* (log x) (- x 0.5))
    (- (- x 0.91893853320467) (/ (fma (* y z) z 0.083333333333333) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.95e-31) {
		tmp = (0.083333333333333 + (z * ((fma(y, 1260.0011340009878, 1.0) * (0.0007936500793651 * z)) - 0.0027777777777778))) / x;
	} else {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (fma((y * z), z, 0.083333333333333) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.95e-31)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(fma(y, 1260.0011340009878, 1.0) * Float64(0.0007936500793651 * z)) - 0.0027777777777778))) / x);
	else
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.95e-31], N[(N[(0.083333333333333 + N[(z * N[(N[(N[(y * 1260.0011340009878 + 1.0), $MachinePrecision] * N[(0.0007936500793651 * z), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000001e-31

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. mult-flip N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 63.1

          \[\leadsto \frac{0.083333333333333 + z \cdot \left(\mathsf{fma}\left(y, 1260.0011340009878, 1\right) \cdot \left(0.0007936500793651 \cdot z\right) - 0.0027777777777778\right)}{x} \]

   6. Applied rewrites 63.1%

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(\mathsf{fma}\left(y, 1260.0011340009878, 1\right) \cdot \left(0.0007936500793651 \cdot z\right) - 0.0027777777777778\right)}{x} \]

   if 1.9500000000000001e-31 < x

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 82.4

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

   6. Applied rewrites 82.4%

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

Alternative 15: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-31}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(\mathsf{fma}\left(y, 1260.0011340009878, 1\right) \cdot \left(0.0007936500793651 \cdot z\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) - \left(-0.91893853320467 - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.95e-31)
   (/
    (+
     0.083333333333333
     (*
      z
      (-
       (* (fma y 1260.0011340009878 1.0) (* 0.0007936500793651 z))
       0.0027777777777778)))
    x)
   (-
    (- (* (log x) (- x 0.5)) x)
    (- -0.91893853320467 (/ (fma (* y z) z 0.083333333333333) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.95e-31) {
		tmp = (0.083333333333333 + (z * ((fma(y, 1260.0011340009878, 1.0) * (0.0007936500793651 * z)) - 0.0027777777777778))) / x;
	} else {
		tmp = ((log(x) * (x - 0.5)) - x) - (-0.91893853320467 - (fma((y * z), z, 0.083333333333333) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.95e-31)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(fma(y, 1260.0011340009878, 1.0) * Float64(0.0007936500793651 * z)) - 0.0027777777777778))) / x);
	else
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) - Float64(-0.91893853320467 - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.95e-31], N[(N[(0.083333333333333 + N[(z * N[(N[(N[(y * 1260.0011340009878 + 1.0), $MachinePrecision] * N[(0.0007936500793651 * z), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] - N[(-0.91893853320467 - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000001e-31

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. mult-flip N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 63.1

          \[\leadsto \frac{0.083333333333333 + z \cdot \left(\mathsf{fma}\left(y, 1260.0011340009878, 1\right) \cdot \left(0.0007936500793651 \cdot z\right) - 0.0027777777777778\right)}{x} \]

   6. Applied rewrites 63.1%

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(\mathsf{fma}\left(y, 1260.0011340009878, 1\right) \cdot \left(0.0007936500793651 \cdot z\right) - 0.0027777777777778\right)}{x} \]

   if 1.9500000000000001e-31 < x

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 82.4

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

   6. Applied rewrites 82.4%

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

Alternative 16: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\left(0.083333333333333 + z \cdot \left(\left(z \cdot \mathsf{fma}\left(y, 1260.0011340009878, 1\right)\right) \cdot 0.0007936500793651 - 0.0027777777777778\right)\right) \cdot \frac{1}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+31}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+69)
     (*
      (+
       0.083333333333333
       (*
        z
        (-
         (* (* z (fma y 1260.0011340009878 1.0)) 0.0007936500793651)
         0.0027777777777778)))
      (/ 1.0 x))
     (if (<= t_0 1e+31)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ 0.083333333333333 x))
       (/
        1.0
        (/
         x
         (fma
          (fma (- y -0.0007936500793651) z -0.0027777777777778)
          z
          0.083333333333333)))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+69) {
		tmp = (0.083333333333333 + (z * (((z * fma(y, 1260.0011340009878, 1.0)) * 0.0007936500793651) - 0.0027777777777778))) * (1.0 / x);
	} else if (t_0 <= 1e+31) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else {
		tmp = 1.0 / (x / fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+69)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(Float64(z * fma(y, 1260.0011340009878, 1.0)) * 0.0007936500793651) - 0.0027777777777778))) * Float64(1.0 / x));
	elseif (t_0 <= 1e+31)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(1.0 / Float64(x / fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+69], N[(N[(0.083333333333333 + N[(z * N[(N[(N[(z * N[(y * 1260.0011340009878 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+31], 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], N[(1.0 / N[(x / N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)) < -1.0000000000000001e69

   1. Initial program 94.1%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. mult-flip N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 63.1

          \[\leadsto \left(0.083333333333333 + z \cdot \left(\left(z \cdot \mathsf{fma}\left(y, 1260.0011340009878, 1\right)\right) \cdot 0.0007936500793651 - 0.0027777777777778\right)\right) \cdot \frac{1}{x} \]

   8. Applied rewrites 63.1%

      \[\leadsto \left(0.083333333333333 + z \cdot \left(\left(z \cdot \mathsf{fma}\left(y, 1260.0011340009878, 1\right)\right) \cdot 0.0007936500793651 - 0.0027777777777778\right)\right) \cdot \frac{1}{x} \]

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.9999999999999996e30

   1. Initial program 94.1%

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

   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 57.5%

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

   if 9.9999999999999996e30 < (+.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 94.1%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

Alternative 17: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+31}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(t\_0, z, 0.083333333333333\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y -0.0007936500793651) z -0.0027777777777778))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+69)
     (/ (* z t_0) x)
     (if (<= t_1 1e+31)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ 0.083333333333333 x))
       (/ 1.0 (/ x (fma t_0 z 0.083333333333333)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y - -0.0007936500793651), z, -0.0027777777777778);
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 1e+31) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else {
		tmp = 1.0 / (x / fma(t_0, z, 0.083333333333333));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 1e+31)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(1.0 / Float64(x / fma(t_0, z, 0.083333333333333)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $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, -1e+69], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 1e+31], 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], N[(1.0 / N[(x / N[(t$95$0 * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)) < -1.0000000000000001e69

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sum-to-mult N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 42.1%

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

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.9999999999999996e30

   1. Initial program 94.1%

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

   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 57.5%

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

   if 9.9999999999999996e30 < (+.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 94.1%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

Alternative 18: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\frac{z \cdot t\_0}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+31}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) - \left(-0.91893853320467 - \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(t\_0, z, 0.083333333333333\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y -0.0007936500793651) z -0.0027777777777778))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+69)
     (/ (* z t_0) x)
     (if (<= t_1 1e+31)
       (-
        (- (* (log x) (- x 0.5)) x)
        (- -0.91893853320467 (/ 0.083333333333333 x)))
       (/ 1.0 (/ x (fma t_0 z 0.083333333333333)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y - -0.0007936500793651), z, -0.0027777777777778);
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = (z * t_0) / x;
	} else if (t_1 <= 1e+31) {
		tmp = ((log(x) * (x - 0.5)) - x) - (-0.91893853320467 - (0.083333333333333 / x));
	} else {
		tmp = 1.0 / (x / fma(t_0, z, 0.083333333333333));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = Float64(Float64(z * t_0) / x);
	elseif (t_1 <= 1e+31)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) - Float64(-0.91893853320467 - Float64(0.083333333333333 / x)));
	else
		tmp = Float64(1.0 / Float64(x / fma(t_0, z, 0.083333333333333)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $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, -1e+69], N[(N[(z * t$95$0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 1e+31], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] - N[(-0.91893853320467 - N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x / N[(t$95$0 * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)) < -1.0000000000000001e69

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sum-to-mult N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 42.1%

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

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.9999999999999996e30

   1. Initial program 94.1%

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

   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 57.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) - \left(\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) - \left(\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - x\right) - \left(\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - x\right) - \left(\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - x\right) - \left(\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - x\right) - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{91893853320467}{100000000000000}\right)\right) - \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      10. metadata-eval 57.5

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

   6. Applied rewrites 57.5%

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

   if 9.9999999999999996e30 < (+.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 94.1%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

Alternative 19: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+99)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* x (- (* -1.0 (log (/ 1.0 x))) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+99) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = x * ((-1.0 * log((1.0 / x))) - 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+99)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(x * Float64(Float64(-1.0 * log(Float64(1.0 / x))) - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+99], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(-1.0 * N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999992e99

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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. lower-fma.f64 63.3

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 63.3

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

   6. Applied rewrites 63.3%

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

   if 1.69999999999999992e99 < x

   1. Initial program 94.1%

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

   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%

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

Alternative 20: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+99)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* 1.0 (- (* (log x) (- x 0.5)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+99) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = 1.0 * ((log(x) * (x - 0.5)) - x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+99)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(1.0 * Float64(Float64(log(x) * Float64(x - 0.5)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+99], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(1.0 * N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999992e99

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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. lower-fma.f64 63.3

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 63.3

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

   6. Applied rewrites 63.3%

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

   if 1.69999999999999992e99 < x

   1. Initial program 94.1%

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

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

      4. sum-to-mult N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 93.9%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.2%

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

Alternative 21: 63.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

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 63.3

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

4. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
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. lower-fma.f64 63.3

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

   6. lift--.f64 N/A

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

   7. sub-flip N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. add-flip N/A

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

   15. metadata-eval N/A

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

   16. lift--.f64 63.3

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

6. Applied rewrites 63.3%

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

Alternative 22: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z (fma (- y -0.0007936500793651) z -0.0027777777777778)) x))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+69)
     t_0
     (if (<= t_1 50.0)
       (/
        (+
         0.083333333333333
         (* z (- (* 0.0007936500793651 z) 0.0027777777777778)))
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * fma((y - -0.0007936500793651), z, -0.0027777777777778)) / x;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = t_0;
	} else if (t_1 <= 50.0) {
		tmp = (0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778)) / x)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = t_0;
	elseif (t_1 <= 50.0)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $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, -1e+69], t$95$0, If[LessEqual[t$95$1, 50.0], N[(N[(0.083333333333333 + N[(z * N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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)) < -1.0000000000000001e69 or 50 < (+.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 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sum-to-mult N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 42.1%

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

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 50

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 47.0

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

   7. Applied rewrites 47.0%

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

Alternative 23: 62.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z (fma (- y -0.0007936500793651) z -0.0027777777777778)) x))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+69)
     t_0
     (if (<= t_1 0.2)
       (/ (+ 0.083333333333333 (* -0.0027777777777778 z)) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * fma((y - -0.0007936500793651), z, -0.0027777777777778)) / x;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = t_0;
	} else if (t_1 <= 0.2) {
		tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778)) / x)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = t_0;
	elseif (t_1 <= 0.2)
		tmp = Float64(Float64(0.083333333333333 + Float64(-0.0027777777777778 * z)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $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, -1e+69], t$95$0, If[LessEqual[t$95$1, 0.2], N[(N[(0.083333333333333 + N[(-0.0027777777777778 * z), $MachinePrecision]), $MachinePrecision] / x), $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)) < -1.0000000000000001e69 or 0.20000000000000001 < (+.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 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sum-to-mult N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 42.1%

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

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.20000000000000001

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 29.3

         \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

   7. Applied rewrites 29.3%

      \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 35.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-56}:\\ \;\;\;\;\frac{-29.99999999999964 \cdot \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4e-56)
   (/
    (*
     -29.99999999999964
     (fma (- y -0.0007936500793651) z -0.0027777777777778))
    x)
   (/ (+ 0.083333333333333 (* -0.0027777777777778 z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4e-56) {
		tmp = (-29.99999999999964 * fma((y - -0.0007936500793651), z, -0.0027777777777778)) / x;
	} else {
		tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4e-56)
		tmp = Float64(Float64(-29.99999999999964 * fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778)) / x);
	else
		tmp = Float64(Float64(0.083333333333333 + Float64(-0.0027777777777778 * z)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4e-56], N[(N[(-29.99999999999964 * N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.083333333333333 + N[(-0.0027777777777778 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0000000000000002e-56

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sum-to-mult N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

   7. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{-416666666666665}{13888888888889} \cdot \mathsf{fma}\left(y - \frac{-7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right)}{x} \]
   8. Step-by-step derivation

   9. Applied rewrites 31.1%

      \[\leadsto \frac{-29.99999999999964 \cdot \mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x} \]

   if -4.0000000000000002e-56 < z

   1. Initial program 94.1%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 29.3

         \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

   7. Applied rewrites 29.3%

      \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 29.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (0.083333333333333 + (-0.0027777777777778 * z)) / 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 + ((-0.0027777777777778d0) * z)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

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 63.3

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

4. Applied rewrites 63.3%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 29.3

      \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

7. Applied rewrites 29.3%

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

Alternative 26: 23.7% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

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 63.3

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

4. Applied rewrites 63.3%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 23.7%

   \[\leadsto \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))