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

Percentage Accurate: 93.4% → 99.5%

Time: 8.5s

Alternatives: 20

Speedup: 0.9×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;x \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;t\_0 + \frac{\mathsf{fma}\left(z, 0.0007936500793651, z \cdot y - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(z, \frac{y + 0.0007936500793651}{x} \cdot z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (<= x 2.15e-26)
     (+
      t_0
      (/
       (+
        (* (fma z 0.0007936500793651 (- (* z y) 0.0027777777777778)) z)
        0.083333333333333)
       x))
     (+
      t_0
      (fma z (* (/ (+ y 0.0007936500793651) x) z) (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 2.15e-26) {
		tmp = t_0 + (((fma(z, 0.0007936500793651, ((z * y) - 0.0027777777777778)) * z) + 0.083333333333333) / x);
	} else {
		tmp = t_0 + fma(z, (((y + 0.0007936500793651) / x) * z), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (x <= 2.15e-26)
		tmp = Float64(t_0 + Float64(Float64(Float64(fma(z, 0.0007936500793651, Float64(Float64(z * y) - 0.0027777777777778)) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(t_0 + fma(z, Float64(Float64(Float64(y + 0.0007936500793651) / x) * z), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 2.15e-26], N[(t$95$0 + N[(N[(N[(N[(z * 0.0007936500793651 + N[(N[(z * y), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999994e-26

   1. Initial program 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.4

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

   3. Applied rewrites 93.4%

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

   if 2.14999999999999994e-26 < x

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 94.0

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

   6. Applied rewrites 94.0%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{y + 0.0007936500793651}{x} \cdot z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.15e-26)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (fma z (* (/ (+ y 0.0007936500793651) x) z) (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.15e-26) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(z, (((y + 0.0007936500793651) / x) * z), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.15e-26)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(z, Float64(Float64(Float64(y + 0.0007936500793651) / x) * z), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.15e-26], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999994e-26

   1. Initial program 93.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 63.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 63.6

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

   4. Applied rewrites 63.6%

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

   if 2.14999999999999994e-26 < x

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 94.0

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

   6. Applied rewrites 94.0%

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

Alternative 3: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-fma.f64 N/A

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

3. Applied rewrites 97.7%

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

Alternative 4: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{fma}\left(y \cdot \frac{z}{x}, z, \frac{0.083333333333333}{x}\right) + \log x \cdot \left(x - 0.5\right)\right) + 0.91893853320467\right) - x\\ \mathbf{if}\;y \leq -0.00078:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-42}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{z \cdot 0.0007936500793651 - 0.0027777777777778}{x}, z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (+
           (+
            (fma (* y (/ z x)) z (/ 0.083333333333333 x))
            (* (log x) (- x 0.5)))
           0.91893853320467)
          x)))
   (if (<= y -0.00078)
     t_0
     (if (<= y 6.6e-42)
       (-
        (+
         (fma
          (/ (- (* z 0.0007936500793651) 0.0027777777777778) x)
          z
          (fma (log x) (- x 0.5) (/ 0.083333333333333 x)))
         0.91893853320467)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((fma((y * (z / x)), z, (0.083333333333333 / x)) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	double tmp;
	if (y <= -0.00078) {
		tmp = t_0;
	} else if (y <= 6.6e-42) {
		tmp = (fma((((z * 0.0007936500793651) - 0.0027777777777778) / x), z, fma(log(x), (x - 0.5), (0.083333333333333 / x))) + 0.91893853320467) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(fma(Float64(y * Float64(z / x)), z, Float64(0.083333333333333 / x)) + Float64(log(x) * Float64(x - 0.5))) + 0.91893853320467) - x)
	tmp = 0.0
	if (y <= -0.00078)
		tmp = t_0;
	elseif (y <= 6.6e-42)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(z * 0.0007936500793651) - 0.0027777777777778) / x), z, fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x))) + 0.91893853320467) - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[y, -0.00078], t$95$0, If[LessEqual[y, 6.6e-42], N[(N[(N[(N[(N[(N[(z * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.79999999999999986e-4 or 6.6000000000000005e-42 < y

   1. Initial program 93.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 94.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 94.5

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

   6. Applied rewrites 94.5%

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

   7. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.9

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

   9. Applied rewrites 83.9%

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

   if -7.79999999999999986e-4 < y < 6.6000000000000005e-42

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. associate-/l* N/A

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

      16. div-sub N/A

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.5%

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

Alternative 5: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 2e+264)
     (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ t_0 x))
     (+
      (fma -0.5 (log x) 0.91893853320467)
      (fma
       z
       (/ (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) x)
       (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= 2e+264) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (t_0 / x);
	} else {
		tmp = fma(-0.5, log(x), 0.91893853320467) + fma(z, ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= 2e+264)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(t_0 / x));
	else
		tmp = Float64(fma(-0.5, log(x), 0.91893853320467) + fma(z, Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+264], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.00000000000000009e264

   1. Initial program 93.4%

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

   if 2.00000000000000009e264 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 64.6

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

   6. Applied rewrites 64.6%

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

Alternative 6: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq 0.08333333333334:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot \frac{z}{x}, z, \frac{0.083333333333333}{x}\right) + \log x \cdot \left(x - 0.5\right)\right) + 0.91893853320467\right) - x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 0.08333333333334)
     (-
      (+
       (+ (fma (* y (/ z x)) z (/ 0.083333333333333 x)) (* (log x) (- x 0.5)))
       0.91893853320467)
      x)
     (if (<= t_0 2e+264)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ (* (* z z) (+ 0.0007936500793651 y)) x))
       (+
        (fma -0.5 (log x) 0.91893853320467)
        (fma
         z
         (/ (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) x)
         (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= 0.08333333333334) {
		tmp = ((fma((y * (z / x)), z, (0.083333333333333 / x)) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	} else if (t_0 <= 2e+264) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((z * z) * (0.0007936500793651 + y)) / x);
	} else {
		tmp = fma(-0.5, log(x), 0.91893853320467) + fma(z, ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= 0.08333333333334)
		tmp = Float64(Float64(Float64(fma(Float64(y * Float64(z / x)), z, Float64(0.083333333333333 / x)) + Float64(log(x) * Float64(x - 0.5))) + 0.91893853320467) - x);
	elseif (t_0 <= 2e+264)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x));
	else
		tmp = Float64(fma(-0.5, log(x), 0.91893853320467) + fma(z, Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, 0.08333333333334], N[(N[(N[(N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[t$95$0, 2e+264], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.083333333333340004

   1. Initial program 93.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 94.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 94.5

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

   6. Applied rewrites 94.5%

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

   7. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.9

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

   9. Applied rewrites 83.9%

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

   if 0.083333333333340004 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.00000000000000009e264

   1. Initial program 93.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

   if 2.00000000000000009e264 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 64.6

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

   6. Applied rewrites 64.6%

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

Alternative 7: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq 0.08333333333334:\\ \;\;\;\;t\_0 + \frac{\left(z \cdot y\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;t\_0 + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 0.08333333333334)
     (+ t_0 (/ (+ (* (* z y) z) 0.083333333333333) x))
     (if (<= t_1 2e+264)
       (+ t_0 (/ (* (* z z) (+ 0.0007936500793651 y)) x))
       (+
        (fma -0.5 (log x) 0.91893853320467)
        (fma
         z
         (/ (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) x)
         (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= 0.08333333333334) {
		tmp = t_0 + ((((z * y) * z) + 0.083333333333333) / x);
	} else if (t_1 <= 2e+264) {
		tmp = t_0 + (((z * z) * (0.0007936500793651 + y)) / x);
	} else {
		tmp = fma(-0.5, log(x), 0.91893853320467) + fma(z, ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= 0.08333333333334)
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(z * y) * z) + 0.083333333333333) / x));
	elseif (t_1 <= 2e+264)
		tmp = Float64(t_0 + Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x));
	else
		tmp = Float64(fma(-0.5, log(x), 0.91893853320467) + fma(z, Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, 0.08333333333334], N[(t$95$0 + N[(N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+264], N[(t$95$0 + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.083333333333340004

   1. Initial program 93.4%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if 0.083333333333340004 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.00000000000000009e264

   1. Initial program 93.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

   if 2.00000000000000009e264 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 64.6

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

   6. Applied rewrites 64.6%

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

Alternative 8: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq 0.08333333333334:\\ \;\;\;\;t\_0 + \frac{\left(z \cdot y\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;t\_0 + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\frac{0.0027777777777778 - \frac{0.083333333333333}{z}}{z \cdot x}\right) + \frac{0.0007936500793651 + y}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 0.08333333333334)
     (+ t_0 (/ (+ (* (* z y) z) 0.083333333333333) x))
     (if (<= t_1 2e+264)
       (+ t_0 (/ (* (* z z) (+ 0.0007936500793651 y)) x))
       (*
        (*
         (+
          (- (/ (- 0.0027777777777778 (/ 0.083333333333333 z)) (* z x)))
          (/ (+ 0.0007936500793651 y) x))
         z)
        z)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= 0.08333333333334) {
		tmp = t_0 + ((((z * y) * z) + 0.083333333333333) / x);
	} else if (t_1 <= 2e+264) {
		tmp = t_0 + (((z * z) * (0.0007936500793651 + y)) / x);
	} else {
		tmp = ((-((0.0027777777777778 - (0.083333333333333 / z)) / (z * x)) + ((0.0007936500793651 + y) / x)) * z) * z;
	}
	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 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_1 <= 0.08333333333334d0) then
        tmp = t_0 + ((((z * y) * z) + 0.083333333333333d0) / x)
    else if (t_1 <= 2d+264) then
        tmp = t_0 + (((z * z) * (0.0007936500793651d0 + y)) / x)
    else
        tmp = ((-((0.0027777777777778d0 - (0.083333333333333d0 / z)) / (z * x)) + ((0.0007936500793651d0 + y) / x)) * z) * z
    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 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= 0.08333333333334) {
		tmp = t_0 + ((((z * y) * z) + 0.083333333333333) / x);
	} else if (t_1 <= 2e+264) {
		tmp = t_0 + (((z * z) * (0.0007936500793651 + y)) / x);
	} else {
		tmp = ((-((0.0027777777777778 - (0.083333333333333 / z)) / (z * x)) + ((0.0007936500793651 + y) / x)) * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_1 <= 0.08333333333334:
		tmp = t_0 + ((((z * y) * z) + 0.083333333333333) / x)
	elif t_1 <= 2e+264:
		tmp = t_0 + (((z * z) * (0.0007936500793651 + y)) / x)
	else:
		tmp = ((-((0.0027777777777778 - (0.083333333333333 / z)) / (z * x)) + ((0.0007936500793651 + y) / x)) * z) * z
	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(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= 0.08333333333334)
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(z * y) * z) + 0.083333333333333) / x));
	elseif (t_1 <= 2e+264)
		tmp = Float64(t_0 + Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x));
	else
		tmp = Float64(Float64(Float64(Float64(-Float64(Float64(0.0027777777777778 - Float64(0.083333333333333 / z)) / Float64(z * x))) + Float64(Float64(0.0007936500793651 + y) / x)) * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_1 <= 0.08333333333334)
		tmp = t_0 + ((((z * y) * z) + 0.083333333333333) / x);
	elseif (t_1 <= 2e+264)
		tmp = t_0 + (((z * z) * (0.0007936500793651 + y)) / x);
	else
		tmp = ((-((0.0027777777777778 - (0.083333333333333 / z)) / (z * x)) + ((0.0007936500793651 + y) / x)) * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, 0.08333333333334], N[(t$95$0 + N[(N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+264], N[(t$95$0 + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[(N[(0.0027777777777778 - N[(0.083333333333333 / z), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]) + N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.083333333333340004

   1. Initial program 93.4%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if 0.083333333333340004 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.00000000000000009e264

   1. Initial program 93.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

   if 2.00000000000000009e264 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.7

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

   7. Applied rewrites 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 48.4%

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

Alternative 9: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.05e-12)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/ (* (* z z) (+ 0.0007936500793651 y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.05e-12) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((z * z) * (0.0007936500793651 + y)) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.05e-12)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * 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(z * z) * Float64(0.0007936500793651 + y)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.05e-12], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.04999999999999995e-12

   1. Initial program 93.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 63.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 63.6

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

   4. Applied rewrites 63.6%

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

   if 2.04999999999999995e-12 < x

   1. Initial program 93.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

Alternative 10: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 12:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot z\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 12.0)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (* (* z z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 12.0) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((z * z) * y) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 12.0)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * 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(z * z) * y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 12.0], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 12

   1. Initial program 93.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 63.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 63.6

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

   4. Applied rewrites 63.6%

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

   if 12 < x

   1. Initial program 93.4%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 59.9

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

   4. Applied rewrites 59.9%

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

Alternative 11: 83.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 29500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 29500000000000.0)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 29500000000000.0) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 29500000000000.0)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 29500000000000.0], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.95e13

   1. Initial program 93.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 63.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 63.6

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

   4. Applied rewrites 63.6%

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

   if 2.95e13 < x

   1. Initial program 93.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 34.5

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

   4. Applied rewrites 34.5%

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

Alternative 12: 67.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ t_1 := \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)))
        (t_1 (* (/ (+ y 0.0007936500793651) x) (* z z))))
   (if (<= t_0 -4e+104)
     t_1
     (if (<= t_0 INFINITY)
       (-
        (fma
         (log x)
         (- x 0.5)
         (fma (/ 1.0 x) 0.083333333333333 0.91893853320467))
        x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double t_1 = ((y + 0.0007936500793651) / x) * (z * z);
	double tmp;
	if (t_0 <= -4e+104) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(log(x), (x - 0.5), fma((1.0 / x), 0.083333333333333, 0.91893853320467)) - x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	t_1 = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -4e+104)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(fma(log(x), Float64(x - 0.5), fma(Float64(1.0 / x), 0.083333333333333, 0.91893853320467)) - x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+104], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4e104 or +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 42.3

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

   7. Applied rewrites 42.3%

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

   if -4e104 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. +-commutative 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. *-commutative N/A

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

      4. div-add N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 57.0

         \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\right) - x \]

   8. Applied rewrites 57.0%

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

Alternative 13: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+104}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\frac{0.0027777777777778 - \frac{0.083333333333333}{z}}{z \cdot x}\right) + \frac{0.0007936500793651 + y}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -4e+104)
     (* (/ (+ y 0.0007936500793651) x) (* z z))
     (if (<= t_0 INFINITY)
       (-
        (fma
         (log x)
         (- x 0.5)
         (fma (/ 1.0 x) 0.083333333333333 0.91893853320467))
        x)
       (*
        (*
         (+
          (- (/ (- 0.0027777777777778 (/ 0.083333333333333 z)) (* z x)))
          (/ (+ 0.0007936500793651 y) x))
         z)
        z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -4e+104) {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(log(x), (x - 0.5), fma((1.0 / x), 0.083333333333333, 0.91893853320467)) - x;
	} else {
		tmp = ((-((0.0027777777777778 - (0.083333333333333 / z)) / (z * x)) + ((0.0007936500793651 + y) / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -4e+104)
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(log(x), Float64(x - 0.5), fma(Float64(1.0 / x), 0.083333333333333, 0.91893853320467)) - x);
	else
		tmp = Float64(Float64(Float64(Float64(-Float64(Float64(0.0027777777777778 - Float64(0.083333333333333 / z)) / Float64(z * x))) + Float64(Float64(0.0007936500793651 + y) / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+104], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[((-N[(N[(0.0027777777777778 - N[(0.083333333333333 / z), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]) + N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4e104

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 42.3

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

   7. Applied rewrites 42.3%

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

   if -4e104 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. +-commutative 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. *-commutative N/A

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

      4. div-add N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 57.0

         \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\right) - x \]

   8. Applied rewrites 57.0%

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\right) - x \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.7

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

   7. Applied rewrites 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 48.4%

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

Alternative 14: 67.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ t_1 := \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)))
        (t_1 (* (/ (+ y 0.0007936500793651) x) (* z z))))
   (if (<= t_0 -4e+104)
     t_1
     (if (<= t_0 INFINITY)
       (-
        (+ (fma (log x) (- x 0.5) (/ 0.083333333333333 x)) 0.91893853320467)
        x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double t_1 = ((y + 0.0007936500793651) / x) * (z * z);
	double tmp;
	if (t_0 <= -4e+104) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fma(log(x), (x - 0.5), (0.083333333333333 / x)) + 0.91893853320467) - x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	t_1 = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -4e+104)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)) + 0.91893853320467) - x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+104], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4e104 or +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 42.3

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

   7. Applied rewrites 42.3%

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

   if -4e104 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 93.4%

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

   2. Taylor expanded in z around 0

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

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 57.0

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

   4. Applied rewrites 57.0%

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

Alternative 15: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 3800000000000:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e-43)
   (/ (fma -0.0027777777777778 z 0.083333333333333) x)
   (if (<= x 3800000000000.0) (/ (* (* z z) y) x) (* (- (log x) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e-43) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else if (x <= 3800000000000.0) {
		tmp = ((z * z) * y) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e-43)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	elseif (x <= 3800000000000.0)
		tmp = Float64(Float64(Float64(z * z) * y) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e-43], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3800000000000.0], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.89999999999999985e-43

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.5

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

   10. Applied rewrites 29.5%

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

   if 1.89999999999999985e-43 < x < 3.8e12

   1. Initial program 93.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot {z}^{2}}{\color{blue}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 29.9

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

   4. Applied rewrites 29.9%

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

   if 3.8e12 < x

   1. Initial program 93.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 34.5

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

   4. Applied rewrites 34.5%

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

Alternative 16: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 8000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* y (* z (/ z x)))))
   (if (<= t_0 -5e+58)
     t_1
     (if (<= t_0 8000000000.0)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = y * (z * (z / x));
	double tmp;
	if (t_0 <= -5e+58) {
		tmp = t_1;
	} else if (t_0 <= 8000000000.0) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	t_1 = Float64(y * Float64(z * Float64(z / x)))
	tmp = 0.0
	if (t_0 <= -5e+58)
		tmp = t_1;
	elseif (t_0 <= 8000000000.0)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+58], t$95$1, If[LessEqual[t$95$0, 8000000000.0], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -4.99999999999999986e58 or 8e9 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 93.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot {z}^{2}}{\color{blue}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 29.9

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

   4. Applied rewrites 29.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot {z}^{2}}{x} \]

      6. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{{z}^{2}}{x}} \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{{z}^{2}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto y \cdot \frac{{z}^{2}}{\color{blue}{x}} \]

      9. pow2 N/A

         \[\leadsto y \cdot \frac{z \cdot z}{x} \]

      10. lift-*.f64 31.8

          \[\leadsto y \cdot \frac{z \cdot z}{x} \]

   6. Applied rewrites 31.8%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \frac{z \cdot z}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto y \cdot \frac{z \cdot z}{\color{blue}{x}} \]

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 32.1

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

   8. Applied rewrites 32.1%

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

   if -4.99999999999999986e58 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 8e9

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.5

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

   10. Applied rewrites 29.5%

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

Alternative 17: 46.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ t_1 := \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+79}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)))
        (t_1 (* (/ (+ y 0.0007936500793651) x) (* z z))))
   (if (<= t_0 -4e+104)
     t_1
     (if (<= t_0 1e+79)
       (/ 0.083333333333333 x)
       (if (<= t_0 INFINITY) (* (- (log x) 1.0) x) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double t_1 = ((y + 0.0007936500793651) / x) * (z * z);
	double tmp;
	if (t_0 <= -4e+104) {
		tmp = t_1;
	} else if (t_0 <= 1e+79) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (log(x) - 1.0) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double t_1 = ((y + 0.0007936500793651) / x) * (z * z);
	double tmp;
	if (t_0 <= -4e+104) {
		tmp = t_1;
	} else if (t_0 <= 1e+79) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.log(x) - 1.0) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	t_1 = ((y + 0.0007936500793651) / x) * (z * z)
	tmp = 0
	if t_0 <= -4e+104:
		tmp = t_1
	elif t_0 <= 1e+79:
		tmp = 0.083333333333333 / x
	elif t_0 <= math.inf:
		tmp = (math.log(x) - 1.0) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	t_1 = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -4e+104)
		tmp = t_1;
	elseif (t_0 <= 1e+79)
		tmp = Float64(0.083333333333333 / x);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(log(x) - 1.0) * 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) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	t_1 = ((y + 0.0007936500793651) / x) * (z * z);
	tmp = 0.0;
	if (t_0 <= -4e+104)
		tmp = t_1;
	elseif (t_0 <= 1e+79)
		tmp = 0.083333333333333 / x;
	elseif (t_0 <= Inf)
		tmp = (log(x) - 1.0) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+104], t$95$1, If[LessEqual[t$95$0, 1e+79], N[(0.083333333333333 / x), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4e104 or +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 42.3

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

   7. Applied rewrites 42.3%

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

   if -4e104 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 9.99999999999999967e78

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 23.8%

       \[\leadsto \frac{0.083333333333333}{x} \]

   if 9.99999999999999967e78 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 93.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 34.5

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

   4. Applied rewrites 34.5%

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

Alternative 18: 45.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ t_1 := \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)))
        (t_1 (* (/ (+ y 0.0007936500793651) x) (* z z))))
   (if (<= t_0 -4e+104)
     t_1
     (if (<= t_0 INFINITY)
       (- (fma (log x) (- x 0.5) 0.91893853320467) x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double t_1 = ((y + 0.0007936500793651) / x) * (z * z);
	double tmp;
	if (t_0 <= -4e+104) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(log(x), (x - 0.5), 0.91893853320467) - x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	t_1 = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -4e+104)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(fma(log(x), Float64(x - 0.5), 0.91893853320467) - x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+104], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4e104 or +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 93.4%

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

   2. Taylor expanded in z around -inf

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 42.3

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

   7. Applied rewrites 42.3%

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

   if -4e104 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. +-commutative 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. *-commutative N/A

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

      4. div-add N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 35.9%

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

Alternative 19: 29.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

2. Taylor expanded in z around -inf

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

4. Applied rewrites 58.7%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 50.7%

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

8. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   2. lower-fma.f64 29.5

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

10. Applied rewrites 29.5%

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

Alternative 20: 23.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

2. Taylor expanded in z around -inf

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

4. Applied rewrites 58.7%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 50.7%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 23.8%

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

Reproduce

herbie shell --seed 2025128 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))