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

Percentage Accurate: 94.0% → 99.6%

Time: 14.2s

Alternatives: 23

Speedup: 0.9×

• 31.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log x - 1, x, -0.5 \cdot \log x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{{t\_0}^{2} - 0.8444480278083504}{t\_0 - 0.91893853320467} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z - \frac{0.0027777777777778}{x}, z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (log x) 1.0) x (* -0.5 (log x)))))
   (if (<= x 5.5e+25)
     (+
      (/ (- (pow t_0 2.0) 0.8444480278083504) (- t_0 0.91893853320467))
      (/
       (+
        (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
        0.083333333333333)
       x))
     (+
      (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
      (fma
       (- (* (+ (/ y x) (/ 0.0007936500793651 x)) z) (/ 0.0027777777777778 x))
       z
       (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((log(x) - 1.0), x, (-0.5 * log(x)));
	double tmp;
	if (x <= 5.5e+25) {
		tmp = ((pow(t_0, 2.0) - 0.8444480278083504) / (t_0 - 0.91893853320467)) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(((((y / x) + (0.0007936500793651 / x)) * z) - (0.0027777777777778 / x)), z, (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(log(x) - 1.0), x, Float64(-0.5 * log(x)))
	tmp = 0.0
	if (x <= 5.5e+25)
		tmp = Float64(Float64(Float64((t_0 ^ 2.0) - 0.8444480278083504) / Float64(t_0 - 0.91893853320467)) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) - Float64(0.0027777777777778 / x)), z, Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.5e+25], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 0.8444480278083504), $MachinePrecision] / N[(t$95$0 - 0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000018e25

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(\log x - 1, x, -0.5 \cdot \log x\right)\right)}^{2} - 0.8444480278083504}{\mathsf{fma}\left(\log x - 1, x, -0.5 \cdot \log x\right) - 0.91893853320467}} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 5.50000000000000018e25 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log x - 1, x, -0.5 \cdot \log x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z - \frac{0.0027777777777778}{x}, z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.5e+25)
   (+
    (+ (fma (- (log x) 1.0) x (* -0.5 (log x))) 0.91893853320467)
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (fma
     (- (* (+ (/ y x) (/ 0.0007936500793651 x)) z) (/ 0.0027777777777778 x))
     z
     (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.5e+25) {
		tmp = (fma((log(x) - 1.0), x, (-0.5 * log(x))) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(((((y / x) + (0.0007936500793651 / x)) * z) - (0.0027777777777778 / x)), z, (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.5e+25)
		tmp = Float64(Float64(fma(Float64(log(x) - 1.0), x, Float64(-0.5 * log(x))) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) - Float64(0.0027777777777778 / x)), z, Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.5e+25], N[(N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $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], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000018e25

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 5.50000000000000018e25 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 99.5%

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

Alternative 3: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log x - 1\right) \cdot x + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq 0.0833333332:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+ (* (- (log x) 1.0) x) (/ (* (* z z) (+ 0.0007936500793651 y)) x)))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 0.0833333332)
     t_0
     (if (<= t_1 0.1)
       (+
        (fma (log x) (- x 0.5) (/ 0.083333333333333 x))
        (- 0.91893853320467 x))
       (if (<= t_1 2e+299)
         t_0
         (*
          (- (* (/ (+ 0.0007936500793651 y) x) z) (/ 0.0027777777777778 x))
          z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((log(x) - 1.0) * x) + (((z * z) * (0.0007936500793651 + y)) / x);
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= 0.0833333332) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = fma(log(x), (x - 0.5), (0.083333333333333 / x)) + (0.91893853320467 - x);
	} else if (t_1 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = ((((0.0007936500793651 + y) / x) * z) - (0.0027777777777778 / x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x))
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= 0.0833333332)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)) + Float64(0.91893853320467 - x));
	elseif (t_1 <= 2e+299)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) - Float64(0.0027777777777778 / x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0833333332], t$95$0, If[LessEqual[t$95$1, 0.1], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], t$95$0, N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.083333333199999998 or 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.0000000000000001e299

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 93.7

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

   8. Applied rewrites 93.7%

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

   if 0.083333333199999998 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower--.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 2.0000000000000001e299 < (+.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 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 91.8

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

   10. Applied rewrites 91.8%

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

Alternative 4: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z \cdot y, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 0.1)
     (+
      (- (* (log x) (- x 0.5)) x)
      (+ 0.91893853320467 (/ (fma (* z y) z 0.083333333333333) x)))
     (if (<= t_0 2e+299)
       (+ (* (- (log x) 1.0) x) (/ (* (* z z) (+ 0.0007936500793651 y)) x))
       (*
        (- (* (/ (+ 0.0007936500793651 y) x) z) (/ 0.0027777777777778 x))
        z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= 0.1) {
		tmp = ((log(x) * (x - 0.5)) - x) + (0.91893853320467 + (fma((z * y), z, 0.083333333333333) / x));
	} else if (t_0 <= 2e+299) {
		tmp = ((log(x) - 1.0) * x) + (((z * z) * (0.0007936500793651 + y)) / x);
	} else {
		tmp = ((((0.0007936500793651 + y) / x) * z) - (0.0027777777777778 / x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) + Float64(0.91893853320467 + Float64(fma(Float64(z * y), z, 0.083333333333333) / x)));
	elseif (t_0 <= 2e+299)
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) - Float64(0.0027777777777778 / x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(z * y), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.8%

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.8

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

   9. Applied rewrites 97.8%

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

   if 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.0000000000000001e299

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 98.1

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

   8. Applied rewrites 98.1%

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

   if 2.0000000000000001e299 < (+.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 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 91.8

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

   10. Applied rewrites 91.8%

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

Alternative 5: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z - \frac{0.0027777777777778}{x}, z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.5e-7)
   (+
    (- (* (log x) (- x 0.5)) x)
    (+
     0.91893853320467
     (/
      (fma
       (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (fma
     (- (* (+ (/ y x) (/ 0.0007936500793651 x)) z) (/ 0.0027777777777778 x))
     z
     (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e-7) {
		tmp = ((log(x) * (x - 0.5)) - x) + (0.91893853320467 + (fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(((((y / x) + (0.0007936500793651 / x)) * z) - (0.0027777777777778 / x)), z, (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.5e-7)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) + Float64(0.91893853320467 + Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) - Float64(0.0027777777777778 / x)), z, Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.5e-7], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999998e-7

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   if 4.4999999999999998e-7 < x

   1. Initial program 89.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 99.5%

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

Alternative 6: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      2e+299)
   (+
    (- (* (log x) (- x 0.5)) x)
    (+
     0.91893853320467
     (/
      (fma
       (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (* (- (* (/ (+ 0.0007936500793651 y) x) z) (/ 0.0027777777777778 x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 2e+299) {
		tmp = ((log(x) * (x - 0.5)) - x) + (0.91893853320467 + (fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((((0.0007936500793651 + y) / x) * z) - (0.0027777777777778 / x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 2e+299)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) + Float64(0.91893853320467 + Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) - Float64(0.0027777777777778 / x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 2e+299], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2.0000000000000001e299

   1. Initial program 98.1%

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

   4. Applied rewrites 98.1%

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

   if 2.0000000000000001e299 < (+.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 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 91.8

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

   10. Applied rewrites 91.8%

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

Alternative 7: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{+189}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(z, y \cdot \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.8e+189)
   (+
    (- (* (log x) (- x 0.5)) x)
    (+
     0.91893853320467
     (/
      (fma
       (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (fma z (* y (/ z x)) (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.8e+189) {
		tmp = ((log(x) * (x - 0.5)) - x) + (0.91893853320467 + (fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(z, (y * (z / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.8e+189)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) + Float64(0.91893853320467 + Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(z, Float64(y * Float64(z / x)), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.8e+189], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7999999999999998e189

   1. Initial program 97.3%

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

   4. Applied rewrites 97.3%

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

   if 3.7999999999999998e189 < x

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.0

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

   7. Applied rewrites 96.0%

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

Alternative 8: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 97.8%

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

Alternative 9: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00033:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00033)
   (+
    (fma -0.5 (log x) 0.91893853320467)
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (+ (* (- (log x) 1.0) x) (/ (* (* z z) (+ 0.0007936500793651 y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00033) {
		tmp = fma(-0.5, log(x), 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((log(x) - 1.0) * x) + (((z * z) * (0.0007936500793651 + y)) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00033)
		tmp = Float64(fma(-0.5, log(x), 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00033], N[(N[(-0.5 * N[Log[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], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 3.3e-4 < x

   1. Initial program 89.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 10: 85.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+41} \lor \neg \left(z \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.32e+41) (not (<= z 1.05e-5)))
   (* (- (* (/ (+ 0.0007936500793651 y) x) z) (/ 0.0027777777777778 x)) z)
   (+ (fma (log x) (- x 0.5) (/ 0.083333333333333 x)) (- 0.91893853320467 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.32e+41) || !(z <= 1.05e-5)) {
		tmp = ((((0.0007936500793651 + y) / x) * z) - (0.0027777777777778 / x)) * z;
	} else {
		tmp = fma(log(x), (x - 0.5), (0.083333333333333 / x)) + (0.91893853320467 - x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.32e+41) || !(z <= 1.05e-5))
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) - Float64(0.0027777777777778 / x)) * z);
	else
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)) + Float64(0.91893853320467 - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.32e+41], N[Not[LessEqual[z, 1.05e-5]], $MachinePrecision]], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3199999999999999e41 or 1.04999999999999994e-5 < z

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 83.8

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

   10. Applied rewrites 83.8%

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

   if -1.3199999999999999e41 < z < 1.04999999999999994e-5

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower--.f64 96.2

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

   7. Applied rewrites 96.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+41} \lor \neg \left(z \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+15)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+15) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+15)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+15], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 96.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1e15 < x

   1. Initial program 89.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 77.4

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

   5. Applied rewrites 77.4%

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

Alternative 12: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{z \cdot \left(0.0007936500793651 \cdot z\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -0.2)
     (* y (/ (* z z) x))
     (if (<= t_0 0.1)
       (/ (+ (* z (* 0.0007936500793651 z)) 0.083333333333333) x)
       (/ (* z (* (+ 0.0007936500793651 y) 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 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = ((z * (0.0007936500793651 * z)) + 0.083333333333333) / x;
	} else {
		tmp = (z * ((0.0007936500793651 + y) * 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) + 0.083333333333333d0
    if (t_0 <= (-0.2d0)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 0.1d0) then
        tmp = ((z * (0.0007936500793651d0 * z)) + 0.083333333333333d0) / x
    else
        tmp = (z * ((0.0007936500793651d0 + y) * 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) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = ((z * (0.0007936500793651 * z)) + 0.083333333333333) / x;
	} else {
		tmp = (z * ((0.0007936500793651 + y) * z)) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -0.2:
		tmp = y * ((z * z) / x)
	elif t_0 <= 0.1:
		tmp = ((z * (0.0007936500793651 * z)) + 0.083333333333333) / x
	else:
		tmp = (z * ((0.0007936500793651 + y) * 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 <= -0.2)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(Float64(Float64(z * Float64(0.0007936500793651 * z)) + 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * z)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 0.1)
		tmp = ((z * (0.0007936500793651 * z)) + 0.083333333333333) / x;
	else
		tmp = (z * ((0.0007936500793651 + y) * z)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(N[(z * N[(0.0007936500793651 * z), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 41.7

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

   10. Applied rewrites 41.7%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 41.7%

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

   if 0.10000000000000001 < (+.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 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.8%

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

   6. Applied rewrites 88.2%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 69.2

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

   9. Applied rewrites 69.2%

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

Alternative 13: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -0.2)
     (* y (/ (* z z) x))
     (if (<= t_0 0.1)
       (/ 0.083333333333333 x)
       (/ (* z (* (+ 0.0007936500793651 y) 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 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = (z * ((0.0007936500793651 + y) * 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) + 0.083333333333333d0
    if (t_0 <= (-0.2d0)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 0.1d0) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = (z * ((0.0007936500793651d0 + y) * 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) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = (z * ((0.0007936500793651 + y) * z)) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -0.2:
		tmp = y * ((z * z) / x)
	elif t_0 <= 0.1:
		tmp = 0.083333333333333 / x
	else:
		tmp = (z * ((0.0007936500793651 + y) * 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 <= -0.2)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * z)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 0.1)
		tmp = 0.083333333333333 / x;
	else
		tmp = (z * ((0.0007936500793651 + y) * z)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(0.083333333333333 / x), $MachinePrecision], N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 41.7

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

   8. Applied rewrites 41.7%

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

   if 0.10000000000000001 < (+.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 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.8%

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

   6. Applied rewrites 88.2%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 69.2

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

   9. Applied rewrites 69.2%

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

Alternative 14: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -0.2)
     (* y (/ (* z z) x))
     (if (<= t_0 0.1)
       (/ 0.083333333333333 x)
       (/ (* (* z z) (+ 0.0007936500793651 y)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((z * z) * (0.0007936500793651 + y)) / 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) + 0.083333333333333d0
    if (t_0 <= (-0.2d0)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 0.1d0) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = ((z * z) * (0.0007936500793651d0 + y)) / 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) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((z * z) * (0.0007936500793651 + y)) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -0.2:
		tmp = y * ((z * z) / x)
	elif t_0 <= 0.1:
		tmp = 0.083333333333333 / x
	else:
		tmp = ((z * z) * (0.0007936500793651 + y)) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 0.1)
		tmp = 0.083333333333333 / x;
	else
		tmp = ((z * z) * (0.0007936500793651 + y)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(0.083333333333333 / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 41.7

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

   8. Applied rewrites 41.7%

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

   if 0.10000000000000001 < (+.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 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in z around inf

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 69.2

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

   10. Applied rewrites 69.2%

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

   11. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-+.f64 69.1

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

   13. Applied rewrites 69.1%

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

Alternative 15: 62.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -0.2)
     (* y (/ (* z z) x))
     (if (<= t_0 0.1)
       (/ 0.083333333333333 x)
       (* (/ (+ y 0.0007936500793651) x) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_0 <= (-0.2d0)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 0.1d0) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = ((y + 0.0007936500793651d0) / x) * (z * z)
    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) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -0.2:
		tmp = y * ((z * z) / x)
	elif t_0 <= 0.1:
		tmp = 0.083333333333333 / x
	else:
		tmp = ((y + 0.0007936500793651) / x) * (z * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 0.1)
		tmp = 0.083333333333333 / x;
	else
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(0.083333333333333 / x), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 41.7

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

   8. Applied rewrites 41.7%

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

   if 0.10000000000000001 < (+.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 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. div-add N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 68.2%

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

Alternative 16: 52.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (or (<= t_0 -0.2) (not (<= t_0 2e+20)))
     (* y (/ (* z z) x))
     (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e+20)) {
		tmp = y * ((z * z) / x);
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 2d+20))) then
        tmp = y * ((z * z) / x)
    else
        tmp = 0.083333333333333d0 / 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) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e+20)) {
		tmp = y * ((z * z) / x);
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 2e+20):
		tmp = y * ((z * z) / x)
	else:
		tmp = 0.083333333333333 / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 2e+20))
		tmp = Float64(y * Float64(Float64(z * z) / x));
	else
		tmp = Float64(0.083333333333333 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 2e+20)))
		tmp = y * ((z * z) / x);
	else
		tmp = 0.083333333333333 / x;
	end
	tmp_2 = 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[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 2e+20]], $MachinePrecision]], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.083333333333333 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001 or 2e20 < (+.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 88.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2e20

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 40.3

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

   8. Applied rewrites 40.3%

      \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq -0.2 \lor \neg \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -0.2)
     (* y (/ (* z z) x))
     (if (<= t_0 2e+20) (/ 0.083333333333333 x) (/ (* (* z z) y) x)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 2e+20) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((z * z) * y) / 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) + 0.083333333333333d0
    if (t_0 <= (-0.2d0)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 2d+20) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = ((z * z) * y) / 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) + 0.083333333333333;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 2e+20) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((z * z) * y) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -0.2:
		tmp = y * ((z * z) / x)
	elif t_0 <= 2e+20:
		tmp = 0.083333333333333 / x
	else:
		tmp = ((z * z) * y) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 2e+20)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(Float64(Float64(z * z) * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 2e+20)
		tmp = 0.083333333333333 / x;
	else
		tmp = ((z * z) * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+20], N[(0.083333333333333 / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2e20

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 40.3

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

   8. Applied rewrites 40.3%

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

   if 2e20 < (+.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 87.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.2

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

   5. Applied rewrites 44.2%

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

Alternative 18: 29.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{z}{x} \cdot -0.0027777777777778\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (or (<= t_0 -0.2) (not (<= t_0 2e+129)))
     (* (/ z x) -0.0027777777777778)
     (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e+129)) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 2d+129))) then
        tmp = (z / x) * (-0.0027777777777778d0)
    else
        tmp = 0.083333333333333d0 / 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) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e+129)) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 2e+129):
		tmp = (z / x) * -0.0027777777777778
	else:
		tmp = 0.083333333333333 / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 2e+129))
		tmp = Float64(Float64(z / x) * -0.0027777777777778);
	else
		tmp = Float64(0.083333333333333 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 2e+129)))
		tmp = (z / x) * -0.0027777777777778;
	else
		tmp = 0.083333333333333 / x;
	end
	tmp_2 = 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[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 2e+129]], $MachinePrecision]], N[(N[(z / x), $MachinePrecision] * -0.0027777777777778), $MachinePrecision], N[(0.083333333333333 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -0.20000000000000001 or 2e129 < (+.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 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z} - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. div-add N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} \cdot 1}}{x}\right) \]

      13. associate-*r/ N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   4. Applied rewrites 98.2%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]

   7. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\left(\frac{y + 0.0007936500793651}{x} - \frac{\frac{0.0027777777777778}{x}}{z}\right) \cdot \left(z \cdot z\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{z}{x} \cdot \frac{-13888888888889}{5000000000000000} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z}{x} \cdot \frac{-13888888888889}{5000000000000000} \]

      3. lower-/.f64 15.9

         \[\leadsto \frac{z}{x} \cdot -0.0027777777777778 \]

   10. Applied rewrites 15.9%

       \[\leadsto \frac{z}{x} \cdot \color{blue}{-0.0027777777777778} \]

   if -0.20000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 2e129

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      7. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      8. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      10. lower-/.f64 93.2

          \[\leadsto \left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x \]

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 36.7

         \[\leadsto \frac{0.083333333333333}{x} \]

   8. Applied rewrites 36.7%

      \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq -0.2 \lor \neg \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 2 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{z}{x} \cdot -0.0027777777777778\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 4:\\ \;\;\;\;\frac{z \cdot \left(y \cdot z\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      4.0)
   (/ (+ (* z (* y z)) 0.083333333333333) x)
   (/ (* z (* (+ 0.0007936500793651 y) z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4.0) {
		tmp = ((z * (y * z)) + 0.083333333333333) / x;
	} else {
		tmp = (z * ((0.0007936500793651 + y) * 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) :: tmp
    if ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) <= 4.0d0) then
        tmp = ((z * (y * z)) + 0.083333333333333d0) / x
    else
        tmp = (z * ((0.0007936500793651d0 + y) * z)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4.0) {
		tmp = ((z * (y * z)) + 0.083333333333333) / x;
	} else {
		tmp = (z * ((0.0007936500793651 + y) * z)) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4.0:
		tmp = ((z * (y * z)) + 0.083333333333333) / x
	else:
		tmp = (z * ((0.0007936500793651 + y) * z)) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4.0)
		tmp = Float64(Float64(Float64(z * Float64(y * z)) + 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * z)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4.0)
		tmp = ((z * (y * z)) + 0.083333333333333) / x;
	else
		tmp = (z * ((0.0007936500793651 + y) * z)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 4.0], N[(N[(N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4

   1. Initial program 97.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z} - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. div-add N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} \cdot 1}}{x}\right) \]

      13. associate-*r/ N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   4. Applied rewrites 97.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   7. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, \left(\mathsf{fma}\left(\log x - 1, x, -0.5 \cdot \log x\right) + 0.91893853320467\right) \cdot x\right) + 0.083333333333333}{x}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{83333333333333}{1000000000000000}}{x} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      2. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-+.f64 49.1

         \[\leadsto \frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right) + 0.083333333333333}{x} \]

   10. Applied rewrites 49.1%

       \[\leadsto \frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right) + 0.083333333333333}{x} \]

   11. Taylor expanded in y around inf

       \[\leadsto \frac{z \cdot \left(y \cdot z\right) + \frac{83333333333333}{1000000000000000}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 49.1%

       \[\leadsto \frac{z \cdot \left(y \cdot z\right) + 0.083333333333333}{x} \]

   if 4 < (+.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 88.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z} - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. div-add N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} \cdot 1}}{x}\right) \]

      13. associate-*r/ N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   4. Applied rewrites 97.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)} \]

   6. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\left(\log x \cdot \left(x - 0.5\right) - x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{{z}^{2}} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \]

      3. associate-*r/ N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} + \frac{\color{blue}{y}}{x}\right) \]

      4. metadata-eval N/A

         \[\leadsto {z}^{2} \cdot \left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}\right) \]

      5. div-add N/A

         \[\leadsto {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{\color{blue}{x}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)}{x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z\right)}{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z\right)}{x} \]

      13. lift-+.f64 69.9

          \[\leadsto \frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right)}{x} \]

   9. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 64.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.65e+102)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (* (/ (+ 0.0007936500793651 y) x) z) (/ 0.0027777777777778 x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.65e+102) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((0.0007936500793651 + y) / x) * z) - (0.0027777777777778 / x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.65e+102)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) - Float64(0.0027777777777778 / x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.65e+102], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.64999999999999999e102

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lift--.f64 79.0

         \[\leadsto \frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      12. lower-+.f64 79.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}} \]

   if 1.64999999999999999e102 < x

   1. Initial program 84.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z} - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. div-add N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} \cdot 1}}{x}\right) \]

      13. associate-*r/ N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   4. Applied rewrites 97.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]

   7. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\left(\frac{y + 0.0007936500793651}{x} - \frac{\frac{0.0027777777777778}{x}}{z}\right) \cdot \left(z \cdot z\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      2. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      3. lower--.f64 N/A

         \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      6. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} + \frac{y}{x}\right) \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}\right) \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      8. div-add N/A

         \[\leadsto \left(\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot z - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot z - \frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot z - \frac{\frac{13888888888889}{5000000000000000}}{x}\right) \cdot z \]

      13. lift-/.f64 24.9

          \[\leadsto \left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot z \]

   10. Applied rewrites 24.9%

       \[\leadsto \left(\frac{0.0007936500793651 + y}{x} \cdot z - \frac{0.0027777777777778}{x}\right) \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 62.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (fma
   (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
   z
   0.083333333333333)
  x))

C

double code(double x, double y, double z) {
	return fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} \]

   4. +-commutative N/A

      \[\leadsto \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   9. lift--.f64 57.0

      \[\leadsto \frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   12. lower-+.f64 57.0

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} \]

5. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}} \]
6. Add Preprocessing

Alternative 22: 62.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right) + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (+ (* z (* (+ 0.0007936500793651 y) z)) 0.083333333333333) x))

C

double code(double x, double y, double z) {
	return ((z * ((0.0007936500793651 + y) * z)) + 0.083333333333333) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((z * ((0.0007936500793651d0 + y) * z)) + 0.083333333333333d0) / x
end function

Java

public static double code(double x, double y, double z) {
	return ((z * ((0.0007936500793651 + y) * z)) + 0.083333333333333) / x;
}

Python

def code(x, y, z):
	return ((z * ((0.0007936500793651 + y) * z)) + 0.083333333333333) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * z)) + 0.083333333333333) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z * ((0.0007936500793651 + y) * z)) + 0.083333333333333) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

   4. lift--.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   5. lift-+.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z} - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   7. div-add N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right)} \]

   8. *-commutative N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   9. +-commutative N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\frac{z \cdot \left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)} - \frac{13888888888889}{5000000000000000}\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   11. associate-/l* N/A

       \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}} + \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} \cdot 1}}{x}\right) \]

   13. associate-*r/ N/A

       \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

4. Applied rewrites 97.8%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

7. Applied rewrites 75.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, \left(\mathsf{fma}\left(\log x - 1, x, -0.5 \cdot \log x\right) + 0.91893853320467\right) \cdot x\right) + 0.083333333333333}{x}} \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{83333333333333}{1000000000000000}}{x} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   2. associate-*l* N/A

      \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   4. *-commutative N/A

      \[\leadsto \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{z \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   6. lift-+.f64 56.8

      \[\leadsto \frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right) + 0.083333333333333}{x} \]

10. Applied rewrites 56.8%

    \[\leadsto \frac{z \cdot \left(\left(0.0007936500793651 + y\right) \cdot z\right) + 0.083333333333333}{x} \]
11. Add Preprocessing

Alternative 23: 23.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - \color{blue}{x} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   6. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   7. lift--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   8. associate-*r/ N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   10. lower-/.f64 62.3

       \[\leadsto \left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x \]

5. Applied rewrites 62.3%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]
7. Step-by-step derivation

   1. lift-/.f64 22.4

      \[\leadsto \frac{0.083333333333333}{x} \]

8. Applied rewrites 22.4%

   \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
9. Add Preprocessing

Developer Target 1: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x))
  (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

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)) + (0.91893853320467d0 - x)) + (0.083333333333333d0 / x)) + ((z / x) * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) + Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x)) + Float64(Float64(z / x) * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025046 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))