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

Percentage Accurate: 93.7% → 95.0%

Time: 38.7s

Alternatives: 10

Speedup: 1.0×

• 68.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (+
 (+ (- (* (- x 1/2) (log x)) x) 91893853320467/100000000000000)
 (/
  (+
   (*
    (-
     (* (+ y 7936500793651/10000000000000000) z)
     13888888888889/5000000000000000)
    z)
   83333333333333/1000000000000000)
  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 - 1/2), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (+
 (+ (- (* (- x 1/2) (log x)) x) 91893853320467/100000000000000)
 (/
  (+
   (*
    (-
     (* (+ y 7936500793651/10000000000000000) z)
     13888888888889/5000000000000000)
    z)
   83333333333333/1000000000000000)
  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 - 1/2), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\\ \mathbf{if}\;x \leq 14499999999999998720272738994373587179286645250166295040728605556879842875184063487747636105150035135758427588878949939476246172727771136:\\ \;\;\;\;t\_0 + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \mathsf{134\_z0z1z2z3z4}\left(1, \left(\frac{z}{x}\right), \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}, \left(\frac{-1}{x}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0
        (+
         (- (* (- x 1/2) (log x)) x)
         91893853320467/100000000000000)))
  (if (<=
       x
       14499999999999998720272738994373587179286645250166295040728605556879842875184063487747636105150035135758427588878949939476246172727771136)
    (+
     t_0
     (/
      (+
       (*
        (-
         (* (+ y 7936500793651/10000000000000000) z)
         13888888888889/5000000000000000)
        z)
       83333333333333/1000000000000000)
      x))
    (+
     t_0
     (134-z0z1z2z3z4
      1
      (/ z x)
      (-
       (* z (- y -7936500793651/10000000000000000))
       13888888888889/5000000000000000)
      83333333333333/1000000000000000
      (/ -1 x))))))

Derivation

1. Split input into 2 regimes

2. if x < 1.4499999999999999e136

   1. Initial program 93.7%

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

   if 1.4499999999999999e136 < x

   1. Initial program 93.7%

      \[\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   2. 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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. frac-2neg N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. div-sub N/A

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

      14. remove-double-neg N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. distribute-neg-frac2 N/A

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

   3. Applied rewrites 78.5%

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(1, \left(\frac{z}{x}\right), \left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}, \left(\frac{-1}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1449999999999999992460678196288629153054020930222251832223387052753289011020174947123066913981252566629499979477761704400760581292442970358410575888164338643648836751836661491932116579591574117208504300150387150920068806070988869032355037184:\\ \;\;\;\;\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\log x \cdot \left(x - \frac{1}{2}\right) - x}{\frac{91893853320467}{100000000000000}}\right) \cdot \frac{91893853320467}{100000000000000} + \frac{\frac{83333333333333}{1000000000000000}}{x}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<=
     x
     1449999999999999992460678196288629153054020930222251832223387052753289011020174947123066913981252566629499979477761704400760581292442970358410575888164338643648836751836661491932116579591574117208504300150387150920068806070988869032355037184)
  (+
   (+ (- (* (- x 1/2) (log x)) x) 91893853320467/100000000000000)
   (/
    (+
     (*
      (-
       (* (+ y 7936500793651/10000000000000000) z)
       13888888888889/5000000000000000)
      z)
     83333333333333/1000000000000000)
    x))
  (+
   (*
    (+
     1
     (/ (- (* (log x) (- x 1/2)) x) 91893853320467/100000000000000))
    91893853320467/100000000000000)
   (/ 83333333333333/1000000000000000 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.45e+240) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((1.0 + (((log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.45d+240) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = ((1.0d0 + (((log(x) * (x - 0.5d0)) - x) / 0.91893853320467d0)) * 0.91893853320467d0) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.45e+240) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((1.0 + (((Math.log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.45e+240:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = ((1.0 + (((math.log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.45e+240)
		tmp = 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));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.45e+240)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	else
		tmp = ((1.0 + (((log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1449999999999999992460678196288629153054020930222251832223387052753289011020174947123066913981252566629499979477761704400760581292442970358410575888164338643648836751836661491932116579591574117208504300150387150920068806070988869032355037184], N[(N[(N[(N[(N[(x - 1/2), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1 + N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 1/2), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / 91893853320467/100000000000000), $MachinePrecision]), $MachinePrecision] * 91893853320467/100000000000000), $MachinePrecision] + N[(83333333333333/1000000000000000 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e240

   1. Initial program 93.7%

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

   if 1.45e240 < x

   1. Initial program 93.7%

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

   2. 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 57.4%

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

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(1 + \frac{\log x \cdot \left(x - \frac{1}{2}\right) - x}{\frac{91893853320467}{100000000000000}}\right) \cdot \frac{91893853320467}{100000000000000}} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 4200000000000000291073617592167745876312847934514011129242422538996295031764355890434553731035472059531777210246775006765987958916798452501690195630869952371250925690567080429988001482765576850862064528907126853598896138331579566803386368:\\ \;\;\;\;\left(\left(\log x - 1\right) \cdot x - \frac{-91893853320467}{100000000000000}\right) - \frac{\frac{-83333333333333}{1000000000000000} - \left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\log x \cdot \left(x - \frac{1}{2}\right) - x}{\frac{91893853320467}{100000000000000}}\right) \cdot \frac{91893853320467}{100000000000000} + \frac{\frac{83333333333333}{1000000000000000}}{x}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<=
     x
     4200000000000000291073617592167745876312847934514011129242422538996295031764355890434553731035472059531777210246775006765987958916798452501690195630869952371250925690567080429988001482765576850862064528907126853598896138331579566803386368)
  (-
   (- (* (- (log x) 1) x) -91893853320467/100000000000000)
   (/
    (-
     -83333333333333/1000000000000000
     (*
      (-
       (* (- y -7936500793651/10000000000000000) z)
       13888888888889/5000000000000000)
      z))
    x))
  (+
   (*
    (+
     1
     (/ (- (* (log x) (- x 1/2)) x) 91893853320467/100000000000000))
    91893853320467/100000000000000)
   (/ 83333333333333/1000000000000000 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+237) {
		tmp = (((log(x) - 1.0) * x) - -0.91893853320467) - ((-0.083333333333333 - ((((y - -0.0007936500793651) * z) - 0.0027777777777778) * z)) / x);
	} else {
		tmp = ((1.0 + (((log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.2d+237) then
        tmp = (((log(x) - 1.0d0) * x) - (-0.91893853320467d0)) - (((-0.083333333333333d0) - ((((y - (-0.0007936500793651d0)) * z) - 0.0027777777777778d0) * z)) / x)
    else
        tmp = ((1.0d0 + (((log(x) * (x - 0.5d0)) - x) / 0.91893853320467d0)) * 0.91893853320467d0) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+237) {
		tmp = (((Math.log(x) - 1.0) * x) - -0.91893853320467) - ((-0.083333333333333 - ((((y - -0.0007936500793651) * z) - 0.0027777777777778) * z)) / x);
	} else {
		tmp = ((1.0 + (((Math.log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.2e+237:
		tmp = (((math.log(x) - 1.0) * x) - -0.91893853320467) - ((-0.083333333333333 - ((((y - -0.0007936500793651) * z) - 0.0027777777777778) * z)) / x)
	else:
		tmp = ((1.0 + (((math.log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.2e+237)
		tmp = Float64(Float64(Float64(Float64(log(x) - 1.0) * x) - -0.91893853320467) - Float64(Float64(-0.083333333333333 - Float64(Float64(Float64(Float64(y - -0.0007936500793651) * z) - 0.0027777777777778) * z)) / x));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.2e+237)
		tmp = (((log(x) - 1.0) * x) - -0.91893853320467) - ((-0.083333333333333 - ((((y - -0.0007936500793651) * z) - 0.0027777777777778) * z)) / x);
	else
		tmp = ((1.0 + (((log(x) * (x - 0.5)) - x) / 0.91893853320467)) * 0.91893853320467) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4200000000000000291073617592167745876312847934514011129242422538996295031764355890434553731035472059531777210246775006765987958916798452501690195630869952371250925690567080429988001482765576850862064528907126853598896138331579566803386368], N[(N[(N[(N[(N[Log[x], $MachinePrecision] - 1), $MachinePrecision] * x), $MachinePrecision] - -91893853320467/100000000000000), $MachinePrecision] - N[(N[(-83333333333333/1000000000000000 - N[(N[(N[(N[(y - -7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1 + N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 1/2), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / 91893853320467/100000000000000), $MachinePrecision]), $MachinePrecision] * 91893853320467/100000000000000), $MachinePrecision] + N[(83333333333333/1000000000000000 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000003e237

   1. Initial program 93.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 92.8%

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

   4. Applied rewrites 92.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   6. Applied rewrites 92.8%

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

   if 4.2000000000000003e237 < x

   1. Initial program 93.7%

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

   2. 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 57.4%

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

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(1 + \frac{\log x \cdot \left(x - \frac{1}{2}\right) - x}{\frac{91893853320467}{100000000000000}}\right) \cdot \frac{91893853320467}{100000000000000}} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\\ t_1 := t\_0 + \frac{\left(y \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{if}\;y + \frac{7936500793651}{10000000000000000} \leq \frac{-1152921504606847}{288230376151711744}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y + \frac{7936500793651}{10000000000000000} \leq \frac{7320129949063641}{9223372036854775808}:\\ \;\;\;\;t\_0 + \frac{\left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0
        (+
         (- (* (- x 1/2) (log x)) x)
         91893853320467/100000000000000))
       (t_1
        (+
         t_0
         (/
          (+
           (* (- (* y z) 13888888888889/5000000000000000) z)
           83333333333333/1000000000000000)
          x))))
  (if (<=
       (+ y 7936500793651/10000000000000000)
       -1152921504606847/288230376151711744)
    t_1
    (if (<=
         (+ y 7936500793651/10000000000000000)
         7320129949063641/9223372036854775808)
      (+
       t_0
       (/
        (+
         (*
          (-
           (* 7936500793651/10000000000000000 z)
           13888888888889/5000000000000000)
          z)
         83333333333333/1000000000000000)
        x))
      t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double t_1 = t_0 + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if ((y + 0.0007936500793651) <= -0.004) {
		tmp = t_1;
	} else if ((y + 0.0007936500793651) <= 0.0007936500793651004) {
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0
    t_1 = t_0 + (((((y * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    if ((y + 0.0007936500793651d0) <= (-0.004d0)) then
        tmp = t_1
    else if ((y + 0.0007936500793651d0) <= 0.0007936500793651004d0) then
        tmp = t_0 + (((((0.0007936500793651d0 * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * Math.log(x)) - x) + 0.91893853320467;
	double t_1 = t_0 + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if ((y + 0.0007936500793651) <= -0.004) {
		tmp = t_1;
	} else if ((y + 0.0007936500793651) <= 0.0007936500793651004) {
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	t_1 = t_0 + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	tmp = 0
	if (y + 0.0007936500793651) <= -0.004:
		tmp = t_1
	elif (y + 0.0007936500793651) <= 0.0007936500793651004:
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	t_1 = Float64(t_0 + Float64(Float64(Float64(Float64(Float64(y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -0.004)
		tmp = t_1;
	elseif (Float64(y + 0.0007936500793651) <= 0.0007936500793651004)
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	t_1 = t_0 + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -0.004)
		tmp = t_1;
	elseif ((y + 0.0007936500793651) <= 0.0007936500793651004)
		tmp = t_0 + (((((0.0007936500793651 * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 1/2), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(N[(N[(N[(y * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y + 7936500793651/10000000000000000), $MachinePrecision], -1152921504606847/288230376151711744], t$95$1, If[LessEqual[N[(y + 7936500793651/10000000000000000), $MachinePrecision], 7320129949063641/9223372036854775808], N[(t$95$0 + N[(N[(N[(N[(N[(7936500793651/10000000000000000 * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -0.0040000000000000001 or 7.9365007936510045e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 83.0%

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

   4. Applied rewrites 83.0%

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

   if -0.0040000000000000001 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.9365007936510045e-4

   1. Initial program 93.7%

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

   2. 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 78.1%

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\frac{7936500793651}{10000000000000000}} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 230000000000000:\\ \;\;\;\;\left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 230000000000000)
  (+
   (+ 91893853320467/100000000000000 (* -1/2 (log x)))
   (/
    (+
     (*
      (-
       (* (+ y 7936500793651/10000000000000000) z)
       13888888888889/5000000000000000)
      z)
     83333333333333/1000000000000000)
    x))
  (+
   (+ (- (* (- x 1/2) (log x)) x) 91893853320467/100000000000000)
   (/
    (+
     (* (- (* y z) 13888888888889/5000000000000000) z)
     83333333333333/1000000000000000)
    x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.3e+14) {
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.3d+14) then
        tmp = (0.91893853320467d0 + ((-0.5d0) * log(x))) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (((((y * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.3e+14) {
		tmp = (0.91893853320467 + (-0.5 * Math.log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.3e+14:
		tmp = (0.91893853320467 + (-0.5 * math.log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.3e+14)
		tmp = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.3e+14)
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((((y * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 230000000000000], N[(N[(91893853320467/100000000000000 + N[(-1/2 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 1/2), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision] + N[(N[(N[(N[(N[(y * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3e14

   1. Initial program 93.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   if 2.3e14 < x

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 83.0%

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

   4. Applied rewrites 83.0%

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{y \cdot z} - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 4199999999999999909983484408708288647004160:\\ \;\;\;\;\left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 4199999999999999909983484408708288647004160)
  (+
   (+ 91893853320467/100000000000000 (* -1/2 (log x)))
   (/
    (+
     (*
      (-
       (* (+ y 7936500793651/10000000000000000) z)
       13888888888889/5000000000000000)
      z)
     83333333333333/1000000000000000)
    x))
  (+
   (+ (* x (- (* -1 (log (/ 1 x))) 1)) 91893853320467/100000000000000)
   (/ 83333333333333/1000000000000000 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((x * ((-1.0 * log((1.0 / x))) - 1.0)) + 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.2d+42) then
        tmp = (0.91893853320467d0 + ((-0.5d0) * log(x))) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = ((x * (((-1.0d0) * log((1.0d0 / x))) - 1.0d0)) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = (0.91893853320467 + (-0.5 * Math.log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((x * ((-1.0 * Math.log((1.0 / x))) - 1.0)) + 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.2e+42:
		tmp = (0.91893853320467 + (-0.5 * math.log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = ((x * ((-1.0 * math.log((1.0 / x))) - 1.0)) + 0.91893853320467) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.2e+42)
		tmp = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(-1.0 * log(Float64(1.0 / x))) - 1.0)) + 0.91893853320467) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.2e+42)
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	else
		tmp = ((x * ((-1.0 * log((1.0 / x))) - 1.0)) + 0.91893853320467) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4199999999999999909983484408708288647004160], N[(N[(91893853320467/100000000000000 + N[(-1/2 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(-1 * N[Log[N[(1 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision] + N[(83333333333333/1000000000000000 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1999999999999999e42

   1. Initial program 93.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   if 4.1999999999999999e42 < x

   1. Initial program 93.7%

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

   2. 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 57.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 56.6%

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

   7. Applied rewrites 56.6%

      \[\leadsto \left(\color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 4199999999999999909983484408708288647004160:\\ \;\;\;\;\left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 4199999999999999909983484408708288647004160)
  (+
   (+ 91893853320467/100000000000000 (* -1/2 (log x)))
   (/
    (+
     (*
      (-
       (* (+ y 7936500793651/10000000000000000) z)
       13888888888889/5000000000000000)
      z)
     83333333333333/1000000000000000)
    x))
  (-
   (+
    91893853320467/100000000000000
    (+
     (* 83333333333333/1000000000000000 (/ 1 x))
     (* (log x) (- x 1/2))))
   x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (log(x) * (x - 0.5)))) - x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.2d+42) then
        tmp = (0.91893853320467d0 + ((-0.5d0) * log(x))) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 * (1.0d0 / x)) + (log(x) * (x - 0.5d0)))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = (0.91893853320467 + (-0.5 * Math.log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (Math.log(x) * (x - 0.5)))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.2e+42:
		tmp = (0.91893853320467 + (-0.5 * math.log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (math.log(x) * (x - 0.5)))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.2e+42)
		tmp = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 * Float64(1.0 / x)) + Float64(log(x) * Float64(x - 0.5)))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.2e+42)
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	else
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (log(x) * (x - 0.5)))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4199999999999999909983484408708288647004160], N[(N[(91893853320467/100000000000000 + N[(-1/2 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 7936500793651/10000000000000000), $MachinePrecision] * z), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] + 83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(91893853320467/100000000000000 + N[(N[(83333333333333/1000000000000000 * N[(1 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 1/2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1999999999999999e42

   1. Initial program 93.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   if 4.1999999999999999e42 < x

   1. Initial program 93.7%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 57.4%

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

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 4199999999999999909983484408708288647004160:\\ \;\;\;\;\frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 4199999999999999909983484408708288647004160)
  (/
   (-
    (*
     (-
      (* z (- y -7936500793651/10000000000000000))
      13888888888889/5000000000000000)
     z)
    -83333333333333/1000000000000000)
   x)
  (+
   (+ (- (* (- x 1/2) (log x)) x) 91893853320467/100000000000000)
   (/ 83333333333333/1000000000000000 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.2d+42) then
        tmp = ((((z * (y - (-0.0007936500793651d0))) - 0.0027777777777778d0) * z) - (-0.083333333333333d0)) / x
    else
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.2e+42:
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x
	else:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.2e+42)
		tmp = Float64(Float64(Float64(Float64(Float64(z * Float64(y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.2e+42)
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x;
	else
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4199999999999999909983484408708288647004160], N[(N[(N[(N[(N[(z * N[(y - -7936500793651/10000000000000000), $MachinePrecision]), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] - -83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 1/2), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 91893853320467/100000000000000), $MachinePrecision] + N[(83333333333333/1000000000000000 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1999999999999999e42

   1. Initial program 93.7%

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

   2. 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 57.4%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)} \cdot \frac{1}{x} \]
   8. 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 63.3%

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

   9. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)} \cdot \frac{1}{x} \]
   10. 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. lower-/.f64 63.3%

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

   11. Applied rewrites 63.3%

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

   if 4.1999999999999999e42 < x

   1. Initial program 93.7%

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

   2. 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 57.4%

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 4199999999999999909983484408708288647004160:\\ \;\;\;\;\frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{-83333333333333}{1000000000000000}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 4199999999999999909983484408708288647004160)
  (/
   (-
    (*
     (-
      (* z (- y -7936500793651/10000000000000000))
      13888888888889/5000000000000000)
     z)
    -83333333333333/1000000000000000)
   x)
  (-
   (+
    91893853320467/100000000000000
    (+
     (* 83333333333333/1000000000000000 (/ 1 x))
     (* (log x) (- x 1/2))))
   x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (log(x) * (x - 0.5)))) - x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.2d+42) then
        tmp = ((((z * (y - (-0.0007936500793651d0))) - 0.0027777777777778d0) * z) - (-0.083333333333333d0)) / x
    else
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 * (1.0d0 / x)) + (log(x) * (x - 0.5d0)))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.2e+42) {
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (Math.log(x) * (x - 0.5)))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.2e+42:
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x
	else:
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (math.log(x) * (x - 0.5)))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.2e+42)
		tmp = Float64(Float64(Float64(Float64(Float64(z * Float64(y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 * Float64(1.0 / x)) + Float64(log(x) * Float64(x - 0.5)))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.2e+42)
		tmp = ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x;
	else
		tmp = (0.91893853320467 + ((0.083333333333333 * (1.0 / x)) + (log(x) * (x - 0.5)))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4199999999999999909983484408708288647004160], N[(N[(N[(N[(N[(z * N[(y - -7936500793651/10000000000000000), $MachinePrecision]), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] - -83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision], N[(N[(91893853320467/100000000000000 + N[(N[(83333333333333/1000000000000000 * N[(1 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 1/2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1999999999999999e42

   1. Initial program 93.7%

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

   2. 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 57.4%

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

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

      3. add-to-fraction N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)} \cdot \frac{1}{x} \]
   8. 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 63.3%

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

   9. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)} \cdot \frac{1}{x} \]
   10. 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. lower-/.f64 63.3%

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

   11. Applied rewrites 63.3%

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

   if 4.1999999999999999e42 < x

   1. Initial program 93.7%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 57.4%

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

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.3% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (/
 (-
  (*
   (-
    (* z (- y -7936500793651/10000000000000000))
    13888888888889/5000000000000000)
   z)
  -83333333333333/1000000000000000)
 x))

C

double code(double x, double y, double z) {
	return ((((z * (y - -0.0007936500793651)) - 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 = ((((z * (y - (-0.0007936500793651d0))) - 0.0027777777777778d0) * z) - (-0.083333333333333d0)) / x
end function

Java

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

Python

def code(x, y, z):
	return ((((z * (y - -0.0007936500793651)) - 0.0027777777777778) * z) - -0.083333333333333) / x

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(z * N[(y - -7936500793651/10000000000000000), $MachinePrecision]), $MachinePrecision] - 13888888888889/5000000000000000), $MachinePrecision] * z), $MachinePrecision] - -83333333333333/1000000000000000), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

2. 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 57.4%

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

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

   3. add-to-fraction N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 N/A

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

   6. lower-*.f64 N/A

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

6. Applied rewrites 41.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)} \cdot \frac{1}{x} \]
8. 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 63.3%

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

9. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)} \cdot \frac{1}{x} \]
10. 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. lower-/.f64 63.3%

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

11. Applied rewrites 63.3%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 1/2) (log x)) x) 91893853320467/100000000000000) (/ (+ (* (- (* (+ y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))