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

Percentage Accurate: 93.9% → 98.6%

Time: 5.4s

Alternatives: 18

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right)\\ \mathbf{if}\;x \leq 7.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right), x, \mathsf{fma}\left(t\_0, z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{z}{x}, \mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (- y -0.0007936500793651) -0.0027777777777778)))
   (if (<= x 7.6e+53)
     (/
      (fma
       (fma (log x) (- x 0.5) (- 0.91893853320467 x))
       x
       (fma t_0 z 0.083333333333333))
      x)
     (fma t_0 (/ z x) (fma (- x 0.5) (log x) (* -1.0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (y - -0.0007936500793651), -0.0027777777777778);
	double tmp;
	if (x <= 7.6e+53) {
		tmp = fma(fma(log(x), (x - 0.5), (0.91893853320467 - x)), x, fma(t_0, z, 0.083333333333333)) / x;
	} else {
		tmp = fma(t_0, (z / x), fma((x - 0.5), log(x), (-1.0 * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778)
	tmp = 0.0
	if (x <= 7.6e+53)
		tmp = Float64(fma(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)), x, fma(t_0, z, 0.083333333333333)) / x);
	else
		tmp = fma(t_0, Float64(z / x), fma(Float64(x - 0.5), log(x), Float64(-1.0 * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision]}, If[LessEqual[x, 7.6e+53], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] * x + N[(t$95$0 * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(t$95$0 * N[(z / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.59999999999999995e53

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 78.2%

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

   if 7.59999999999999995e53 < x

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. div-add-rev N/A

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

   3. Applied rewrites 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.6e+53)
   (-
    (fma (log x) (- x 0.5) (- 0.91893853320467 x))
    (/
     (fma
      (fma (- -0.0007936500793651 y) z 0.0027777777777778)
      z
      -0.083333333333333)
     x))
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (fma (- x 0.5) (log x) (* -1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.6e+53) {
		tmp = fma(log(x), (x - 0.5), (0.91893853320467 - x)) - (fma(fma((-0.0007936500793651 - y), z, 0.0027777777777778), z, -0.083333333333333) / x);
	} else {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), fma((x - 0.5), log(x), (-1.0 * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.6e+53)
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)) - Float64(fma(fma(Float64(-0.0007936500793651 - y), z, 0.0027777777777778), z, -0.083333333333333) / x));
	else
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), fma(Float64(x - 0.5), log(x), Float64(-1.0 * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.6e+53], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-0.0007936500793651 - y), $MachinePrecision] * z + 0.0027777777777778), $MachinePrecision] * z + -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.59999999999999995e53

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+l- N/A

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

      7. sub-negate-rev N/A

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

      8. add-flip-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 94.0%

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

   if 7.59999999999999995e53 < x

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. div-add-rev N/A

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

   3. Applied rewrites 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.6e+53)
   (fma
    (- x 0.5)
    (log x)
    (-
     (- 0.91893853320467 x)
     (/
      (fma
       (fma (- -0.0007936500793651 y) z 0.0027777777777778)
       z
       -0.083333333333333)
      x)))
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (fma (- x 0.5) (log x) (* -1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.6e+53) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (fma(fma((-0.0007936500793651 - y), z, 0.0027777777777778), z, -0.083333333333333) / x)));
	} else {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), fma((x - 0.5), log(x), (-1.0 * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.6e+53)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) - Float64(fma(fma(Float64(-0.0007936500793651 - y), z, 0.0027777777777778), z, -0.083333333333333) / x)));
	else
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), fma(Float64(x - 0.5), log(x), Float64(-1.0 * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.6e+53], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] - N[(N[(N[(N[(-0.0007936500793651 - y), $MachinePrecision] * z + 0.0027777777777778), $MachinePrecision] * z + -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.59999999999999995e53

   1. Initial program 93.9%

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

   3. Applied rewrites 94.0%

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

   if 7.59999999999999995e53 < x

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. div-add-rev N/A

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

   3. Applied rewrites 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. div-add N/A

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

   9. div-add-rev N/A

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

3. Applied rewrites 98.6%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. div-add N/A

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

   9. div-add-rev N/A

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

3. Applied rewrites 98.6%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. div-add N/A

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

   8. associate-+r+ N/A

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

   9. associate-+l+ N/A

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

   10. lower-+.f64 N/A

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

3. Applied rewrites 98.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-+r- N/A

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

   6. associate-+l- N/A

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

   7. lower--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 98.6

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

   13. lift-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 7: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.45e+19)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (fma (- x 0.5) (log x) (* -1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.45e+19) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), fma((x - 0.5), log(x), (-1.0 * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.45e+19)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), fma(Float64(x - 0.5), log(x), Float64(-1.0 * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.45e+19], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e19

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 63.2

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

   4. Applied rewrites 63.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. add-flip N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 63.2

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

   6. Applied rewrites 63.2%

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

   if 1.45e19 < x

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. div-add-rev N/A

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

   3. Applied rewrites 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

Alternative 8: 86.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -4e+33)
     (* (- y -0.0007936500793651) (* z (/ z x)))
     (if (<= t_0 1e+51)
       (fma
        -0.0027777777777778
        (/ z x)
        (fma
         (- x 0.5)
         (log x)
         (- (- 0.91893853320467 x) (/ -0.083333333333333 x))))
       (* z (* (- y -0.0007936500793651) (/ z 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 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 1e+51) {
		tmp = fma(-0.0027777777777778, (z / x), fma((x - 0.5), log(x), ((0.91893853320467 - x) - (-0.083333333333333 / x))));
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / 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 <= -4e+33)
		tmp = Float64(Float64(y - -0.0007936500793651) * Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e+51)
		tmp = fma(-0.0027777777777778, Float64(z / x), fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) - Float64(-0.083333333333333 / x))));
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / 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, -4e+33], N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+51], N[(-0.0027777777777778 * N[(z / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / 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)) < -3.9999999999999998e33

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

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

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. div-add-rev N/A

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

   3. Applied rewrites 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.7%

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

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

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 44.7

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 44.7

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

   8. Applied rewrites 44.7%

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

Alternative 9: 86.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -4e+33)
     (* (- y -0.0007936500793651) (* z (/ z x)))
     (if (<= t_0 1e+51)
       (-
        (fma (- x 0.5) (log x) (- 0.91893853320467 x))
        (/ (- -0.083333333333333 (* -0.0027777777777778 z)) x))
       (* z (* (- y -0.0007936500793651) (/ z 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 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 1e+51) {
		tmp = fma((x - 0.5), log(x), (0.91893853320467 - x)) - ((-0.083333333333333 - (-0.0027777777777778 * z)) / x);
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / 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 <= -4e+33)
		tmp = Float64(Float64(y - -0.0007936500793651) * Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e+51)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)) - Float64(Float64(-0.083333333333333 - Float64(-0.0027777777777778 * z)) / x));
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / 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, -4e+33], N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+51], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.083333333333333 - N[(-0.0027777777777778 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / 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)) < -3.9999999999999998e33

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

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

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. div-add-rev N/A

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

   3. Applied rewrites 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. add-flip-rev N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. associate-+l- N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. associate-+l- N/A

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

   10. Applied rewrites 62.6%

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

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

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 44.7

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 44.7

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

   8. Applied rewrites 44.7%

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

Alternative 10: 86.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -4e+33)
     (* (- y -0.0007936500793651) (* z (/ z x)))
     (if (<= t_0 1e+51)
       (-
        (fma (- x 0.5) (log x) (- 0.91893853320467 x))
        (/ -0.083333333333333 x))
       (* z (* (- y -0.0007936500793651) (/ z 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 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 1e+51) {
		tmp = fma((x - 0.5), log(x), (0.91893853320467 - x)) - (-0.083333333333333 / x);
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / 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 <= -4e+33)
		tmp = Float64(Float64(y - -0.0007936500793651) * Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e+51)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)) - Float64(-0.083333333333333 / x));
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / 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, -4e+33], N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+51], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / 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)) < -3.9999999999999998e33

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

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

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 56.8

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

   4. Applied rewrites 56.8%

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

   6. Applied rewrites 56.9%

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

   8. Applied rewrites 56.9%

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

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

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 44.7

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 44.7

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

   8. Applied rewrites 44.7%

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

Alternative 11: 86.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -4e+33)
     (* (- y -0.0007936500793651) (* z (/ z x)))
     (if (<= t_0 1e+51)
       (-
        (- (/ 0.083333333333333 x) (fma (- 0.5 x) (log x) -0.91893853320467))
        x)
       (* z (* (- y -0.0007936500793651) (/ z 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 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 1e+51) {
		tmp = ((0.083333333333333 / x) - fma((0.5 - x), log(x), -0.91893853320467)) - x;
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / 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 <= -4e+33)
		tmp = Float64(Float64(y - -0.0007936500793651) * Float64(z * Float64(z / x)));
	elseif (t_0 <= 1e+51)
		tmp = Float64(Float64(Float64(0.083333333333333 / x) - fma(Float64(0.5 - x), log(x), -0.91893853320467)) - x);
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / 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, -4e+33], N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+51], N[(N[(N[(0.083333333333333 / x), $MachinePrecision] - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + -0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / 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)) < -3.9999999999999998e33

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

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

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 56.8

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

   4. Applied rewrites 56.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. sub-flip N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 56.8

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

   6. Applied rewrites 56.8%

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

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

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 44.7

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 44.7

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

   8. Applied rewrites 44.7%

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

Alternative 12: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.1e+78)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (fma (log x) (- x 0.5) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.1e+78) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma(log(x), (x - 0.5), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.1e+78)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = fma(log(x), Float64(x - 0.5), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.1e+78], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.10000000000000011e78

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 63.2

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

   4. Applied rewrites 63.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. add-flip N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 63.2

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

   6. Applied rewrites 63.2%

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

   if 6.10000000000000011e78 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 56.8

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

   4. Applied rewrites 56.8%

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 36.2

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

   9. Applied rewrites 36.2%

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

Alternative 13: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+50)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* z (* (- y -0.0007936500793651) (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+50) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+50)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+50], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6999999999999999e50

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 63.2

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

   4. Applied rewrites 63.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. add-flip N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 63.2

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

   6. Applied rewrites 63.2%

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

   if 1.6999999999999999e50 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.0

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.0

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

   6. Applied rewrites 31.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 44.7

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 44.7

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

   8. Applied rewrites 44.7%

      \[\leadsto z \cdot \color{blue}{\left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\left(y - -0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0007936500793651 \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -4e+33)
     (* (- y -0.0007936500793651) (* z (/ z x)))
     (if (<= t_0 5e+15)
       (/
        (fma
         (- (* 0.0007936500793651 z) 0.0027777777777778)
         z
         0.083333333333333)
        x)
       (* z (* (- y -0.0007936500793651) (/ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 5e+15) {
		tmp = fma(((0.0007936500793651 * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -4e+33)
		tmp = Float64(Float64(y - -0.0007936500793651) * Float64(z * Float64(z / x)));
	elseif (t_0 <= 5e+15)
		tmp = Float64(fma(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / x)));
	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]}, If[LessEqual[t$95$0, -4e+33], N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+15], N[(N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -3.9999999999999998e33

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \color{blue}{\frac{y}{x}}\right) \]

      3. +-commutative N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{y}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{y}{x} \cdot {z}^{2} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]

      5. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      6. lift-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) \cdot {z}^{2} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{\color{blue}{x}}\right) \]

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      18. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot 0.0007936500793651\right)}{x} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      20. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      21. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{x} \]

   6. Applied rewrites 31.0%

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{\color{blue}{x}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot z\right)}{x} \]

      7. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      10. add-flip N/A

          \[\leadsto \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      11. metadata-eval N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot \color{blue}{z}}{x} \]

      12. lift--.f64 N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot z}{x} \]

      14. associate-/l* N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \frac{z}{\color{blue}{x}}\right) \]

      16. lower-*.f64 45.0

          \[\leadsto \left(y - -0.0007936500793651\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

   8. Applied rewrites 45.0%

      \[\leadsto \left(y - -0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   if -3.9999999999999998e33 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5e15

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.2

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 46.7

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x} \]

   7. Applied rewrites 46.7%

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lower-fma.f64 46.7

         \[\leadsto \frac{\mathsf{fma}\left(0.0007936500793651 \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} \]

   9. Applied rewrites 46.7%

      \[\leadsto \frac{\mathsf{fma}\left(0.0007936500793651 \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{\color{blue}{x}} \]

   if 5e15 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \color{blue}{\frac{y}{x}}\right) \]

      3. +-commutative N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{y}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{y}{x} \cdot {z}^{2} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]

      5. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      6. lift-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) \cdot {z}^{2} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{\color{blue}{x}}\right) \]

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      18. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot 0.0007936500793651\right)}{x} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      20. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      21. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{x} \]

   6. Applied rewrites 31.0%

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{\color{blue}{x}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot z\right)}{x} \]

      7. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot z}{x} \]

      9. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \frac{z}{\color{blue}{x}}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      12. *-commutative N/A

          \[\leadsto \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right) \cdot \frac{\color{blue}{z}}{x} \]

      13. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right)} \]

      15. lower-*.f64 44.7

          \[\leadsto z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \color{blue}{\frac{z}{x}}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

      17. add-flip N/A

          \[\leadsto z \cdot \left(\left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

      18. metadata-eval N/A

          \[\leadsto z \cdot \left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right) \]

      19. lift--.f64 44.7

          \[\leadsto z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

   8. Applied rewrites 44.7%

      \[\leadsto z \cdot \color{blue}{\left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\left(y - -0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -4e+33)
     (* (- y -0.0007936500793651) (* z (/ z x)))
     (if (<= t_0 4e-6)
       (/ 0.083333333333333 x)
       (* z (* (- y -0.0007936500793651) (/ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 4e-6) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / x));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    if (t_0 <= (-4d+33)) then
        tmp = (y - (-0.0007936500793651d0)) * (z * (z / x))
    else if (t_0 <= 4d-6) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = z * ((y - (-0.0007936500793651d0)) * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -4e+33) {
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	} else if (t_0 <= 4e-6) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = z * ((y - -0.0007936500793651) * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	tmp = 0
	if t_0 <= -4e+33:
		tmp = (y - -0.0007936500793651) * (z * (z / x))
	elif t_0 <= 4e-6:
		tmp = 0.083333333333333 / x
	else:
		tmp = z * ((y - -0.0007936500793651) * (z / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -4e+33)
		tmp = Float64(Float64(y - -0.0007936500793651) * Float64(z * Float64(z / x)));
	elseif (t_0 <= 4e-6)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_0 <= -4e+33)
		tmp = (y - -0.0007936500793651) * (z * (z / x));
	elseif (t_0 <= 4e-6)
		tmp = 0.083333333333333 / x;
	else
		tmp = z * ((y - -0.0007936500793651) * (z / x));
	end
	tmp_2 = 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]}, If[LessEqual[t$95$0, -4e+33], N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-6], N[(0.083333333333333 / x), $MachinePrecision], N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -3.9999999999999998e33

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \color{blue}{\frac{y}{x}}\right) \]

      3. +-commutative N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{y}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{y}{x} \cdot {z}^{2} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]

      5. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      6. lift-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) \cdot {z}^{2} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{\color{blue}{x}}\right) \]

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      18. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot 0.0007936500793651\right)}{x} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      20. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      21. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{x} \]

   6. Applied rewrites 31.0%

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{\color{blue}{x}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot z\right)}{x} \]

      7. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      10. add-flip N/A

          \[\leadsto \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      11. metadata-eval N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot \color{blue}{z}}{x} \]

      12. lift--.f64 N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot z}{x} \]

      14. associate-/l* N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \frac{z}{\color{blue}{x}}\right) \]

      16. lower-*.f64 45.0

          \[\leadsto \left(y - -0.0007936500793651\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

   8. Applied rewrites 45.0%

      \[\leadsto \left(y - -0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   if -3.9999999999999998e33 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 3.99999999999999982e-6

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.2

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.2%

      \[\leadsto \frac{0.083333333333333}{x} \]

   if 3.99999999999999982e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \color{blue}{\frac{y}{x}}\right) \]

      3. +-commutative N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{y}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{y}{x} \cdot {z}^{2} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]

      5. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      6. lift-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) \cdot {z}^{2} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{\color{blue}{x}}\right) \]

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      18. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot 0.0007936500793651\right)}{x} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      20. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      21. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{x} \]

   6. Applied rewrites 31.0%

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{\color{blue}{x}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot z\right)}{x} \]

      7. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot z}{x} \]

      9. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \frac{z}{\color{blue}{x}}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      12. *-commutative N/A

          \[\leadsto \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right) \cdot \frac{\color{blue}{z}}{x} \]

      13. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right)} \]

      15. lower-*.f64 44.7

          \[\leadsto z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \color{blue}{\frac{z}{x}}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

      17. add-flip N/A

          \[\leadsto z \cdot \left(\left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

      18. metadata-eval N/A

          \[\leadsto z \cdot \left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right) \]

      19. lift--.f64 44.7

          \[\leadsto z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

   8. Applied rewrites 44.7%

      \[\leadsto z \cdot \color{blue}{\left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* z (* (- y -0.0007936500793651) (/ z x)))))
   (if (<= t_0 -4e+33) t_1 (if (<= t_0 4e-6) (/ 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 = z * ((y - -0.0007936500793651) * (z / x));
	double tmp;
	if (t_0 <= -4e+33) {
		tmp = t_1;
	} else if (t_0 <= 4e-6) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    t_1 = z * ((y - (-0.0007936500793651d0)) * (z / x))
    if (t_0 <= (-4d+33)) then
        tmp = t_1
    else if (t_0 <= 4d-6) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = z * ((y - -0.0007936500793651) * (z / x));
	double tmp;
	if (t_0 <= -4e+33) {
		tmp = t_1;
	} else if (t_0 <= 4e-6) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	t_1 = z * ((y - -0.0007936500793651) * (z / x))
	tmp = 0
	if t_0 <= -4e+33:
		tmp = t_1
	elif t_0 <= 4e-6:
		tmp = 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(z * Float64(Float64(y - -0.0007936500793651) * Float64(z / x)))
	tmp = 0.0
	if (t_0 <= -4e+33)
		tmp = t_1;
	elseif (t_0 <= 4e-6)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	t_1 = z * ((y - -0.0007936500793651) * (z / x));
	tmp = 0.0;
	if (t_0 <= -4e+33)
		tmp = t_1;
	elseif (t_0 <= 4e-6)
		tmp = 0.083333333333333 / x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(N[(y - -0.0007936500793651), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+33], t$95$1, If[LessEqual[t$95$0, 4e-6], N[(0.083333333333333 / 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) < -3.9999999999999998e33 or 3.99999999999999982e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.3

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.3%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \color{blue}{\frac{y}{x}}\right) \]

      3. +-commutative N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{y}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{y}{x} \cdot {z}^{2} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]

      5. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      6. lift-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \frac{y}{x} + \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) \cdot {z}^{2} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot {z}^{2} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{\color{blue}{x}}\right) \]

      10. mult-flip-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{{z}^{2} \cdot y}{x} + \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      12. div-add-rev N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{z}^{2} \cdot y + {z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      18. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot 0.0007936500793651\right)}{x} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, {z}^{2} \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      20. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{x} \]

      21. lower-*.f64 31.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{x} \]

   6. Applied rewrites 31.0%

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot 0.0007936500793651\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}\right)}{\color{blue}{x}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y + \left(z \cdot z\right) \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot z\right)}{x} \]

      7. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z \cdot z}{x} \]

      9. associate-/l* N/A

         \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \left(z \cdot \frac{z}{\color{blue}{x}}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      12. *-commutative N/A

          \[\leadsto \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right) \cdot \frac{\color{blue}{z}}{x} \]

      13. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right)} \]

      15. lower-*.f64 44.7

          \[\leadsto z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \color{blue}{\frac{z}{x}}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

      17. add-flip N/A

          \[\leadsto z \cdot \left(\left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

      18. metadata-eval N/A

          \[\leadsto z \cdot \left(\left(y - \frac{-7936500793651}{10000000000000000}\right) \cdot \frac{z}{x}\right) \]

      19. lift--.f64 44.7

          \[\leadsto z \cdot \left(\left(y - -0.0007936500793651\right) \cdot \frac{\color{blue}{z}}{x}\right) \]

   8. Applied rewrites 44.7%

      \[\leadsto z \cdot \color{blue}{\left(\left(y - -0.0007936500793651\right) \cdot \frac{z}{x}\right)} \]

   if -3.9999999999999998e33 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 3.99999999999999982e-6

   1. Initial program 93.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.2

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.2%

      \[\leadsto \frac{0.083333333333333}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (+ 0.083333333333333 (* -0.0027777777777778 z)) x))

C

double code(double x, double y, double z) {
	return (0.083333333333333 + (-0.0027777777777778 * z)) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (0.083333333333333d0 + ((-0.0027777777777778d0) * z)) / x
end function

Java

public static double code(double x, double y, double z) {
	return (0.083333333333333 + (-0.0027777777777778 * z)) / x;
}

Python

def code(x, y, z):
	return (0.083333333333333 + (-0.0027777777777778 * z)) / x

Julia

function code(x, y, z)
	return Float64(Float64(0.083333333333333 + Float64(-0.0027777777777778 * z)) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(0.083333333333333 + N[(-0.0027777777777778 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.2

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.2%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{x} \]
6. Step-by-step derivation

   1. lower-*.f64 29.0

      \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

7. Applied rewrites 29.0%

   \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]
8. Add Preprocessing

Alternative 18: 23.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.2

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.2%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 23.2%

   \[\leadsto \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))