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

Percentage Accurate: 93.7% → 99.6%

Time: 6.2s

Alternatives: 18

Speedup: 1.2×

• 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: 93.7% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) \cdot x\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z - \frac{0.0027777777777778}{x}, z, \frac{0.083333333333333}{x}\right) + \log x \cdot \left(x - 0.5\right)\right) + 0.91893853320467\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7e-16)
   (/
    (+
     (fma
      (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
      z
      (* (fma -0.5 (log x) 0.91893853320467) x))
     0.083333333333333)
    x)
   (-
    (+
     (+
      (fma
       (- (* (+ (/ y x) (/ 0.0007936500793651 x)) z) (/ 0.0027777777777778 x))
       z
       (/ 0.083333333333333 x))
      (* (log x) (- x 0.5)))
     0.91893853320467)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e-16) {
		tmp = (fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, (fma(-0.5, log(x), 0.91893853320467) * x)) + 0.083333333333333) / x;
	} else {
		tmp = ((fma(((((y / x) + (0.0007936500793651 / x)) * z) - (0.0027777777777778 / x)), z, (0.083333333333333 / x)) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7e-16)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, Float64(fma(-0.5, log(x), 0.91893853320467) * x)) + 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * z) - Float64(0.0027777777777778 / x)), z, Float64(0.083333333333333 / x)) + Float64(log(x) * Float64(x - 0.5))) + 0.91893853320467) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7e-16], N[(N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(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] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000035e-16

   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. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 7.00000000000000035e-16 < x

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.5%

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

Alternative 2: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\left(-\log x\right)\right) - 1\right) \cdot x + \mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e+15)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (+
    (* (- (- (- (log x))) 1.0) x)
    (fma
     z
     (/ (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) x)
     (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+15) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((-(-log(x)) - 1.0) * x) + fma(z, ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e+15)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(Float64(-Float64(-log(x))) - 1.0) * x) + fma(z, Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e+15], 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], N[(N[(N[((-(-N[Log[x], $MachinePrecision])) - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e15

   1. Initial program 99.7%

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

   if 3e15 < x

   1. Initial program 87.2%

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

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

   3. Applied rewrites 97.4%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 97.5

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

   6. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\left(\left(-\left(-\log x\right)\right) - 1\right) \cdot x} + \mathsf{fma}\left(z, \frac{\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

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

3. Applied rewrites 97.4%

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

Alternative 4: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3300000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) \cdot x\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3300000.0)
   (/
    (+
     (fma
      (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
      z
      (* (fma -0.5 (log x) 0.91893853320467) x))
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* (* z z) (/ (+ 0.0007936500793651 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3300000.0) {
		tmp = (fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, (fma(-0.5, log(x), 0.91893853320467) * x)) + 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((z * z) * ((0.0007936500793651 + y) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3300000.0)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, Float64(fma(-0.5, log(x), 0.91893853320467) * x)) + 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(z * z) * Float64(Float64(0.0007936500793651 + y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3300000.0], N[(N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e6

   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. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

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

   if 3.3e6 < x

   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. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 91.4

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

   4. Applied rewrites 91.4%

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

Alternative 5: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((-(-log(x)) - 1.0) * x) + fma(z, ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(-Float64(-log(x))) - 1.0) * x) + 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[Log[x], $MachinePrecision])) - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lift-log.f64 96.5

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

6. Applied rewrites 96.5%

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

Alternative 6: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3300000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3300000.0)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* (* z z) (/ (+ 0.0007936500793651 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3300000.0) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((z * z) * ((0.0007936500793651 + y) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3300000.0)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(z * z) * Float64(Float64(0.0007936500793651 + y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3300000.0], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e6

   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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      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 97.7

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   if 3.3e6 < x

   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. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 91.4

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

   4. Applied rewrites 91.4%

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

Alternative 7: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3300000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\left(-\log x\right)\right) - 1\right) \cdot x + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3300000.0)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (* (- (- (- (log x))) 1.0) x)
    (* (/ (+ y 0.0007936500793651) x) (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3300000.0) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((-(-log(x)) - 1.0) * x) + (((y + 0.0007936500793651) / x) * (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3300000.0)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(-Float64(-log(x))) - 1.0) * x) + Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3300000.0], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[((-(-N[Log[x], $MachinePrecision])) - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e6

   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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      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 97.7

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   if 3.3e6 < x

   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. 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

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 97.3

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

   6. Applied rewrites 97.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 91.2

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

   9. Applied rewrites 91.2%

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

Alternative 8: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+151)
     (* (* y (/ z x)) z)
     (if (<= t_0 4e+307)
       (-
        (+ (fma (log x) (- x 0.5) (/ 0.083333333333333 x)) 0.91893853320467)
        x)
       (* (* (/ (+ 0.0007936500793651 y) x) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+151) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 4e+307) {
		tmp = (fma(log(x), (x - 0.5), (0.083333333333333 / x)) + 0.91893853320467) - x;
	} else {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -1e+151)
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 4e+307)
		tmp = Float64(Float64(fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)) + 0.91893853320467) - x);
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+151], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 4e+307], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000002e151

   1. Initial program 88.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 89.9%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 90.8

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

   10. Applied rewrites 90.8%

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

   if -1.00000000000000002e151 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 3.99999999999999994e307

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

   if 3.99999999999999994e307 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 82.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 88.7

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

   10. Applied rewrites 88.7%

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

Alternative 9: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.85e+107)
   (/
    (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 <= 2.85e+107) {
		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 <= 2.85e+107)
		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, 2.85e+107], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.84999999999999986e107

   1. Initial program 98.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      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 84.4

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

      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 84.4

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

   4. Applied rewrites 84.4%

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

   if 2.84999999999999986e107 < x

   1. Initial program 83.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 77.1

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

   4. Applied rewrites 77.1%

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

Alternative 10: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-42}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq 10000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* (- (log x) 1.0) x)))
   (if (<= t_0 (- INFINITY))
     (* (* y (/ z x)) z)
     (if (<= t_0 5e-270)
       t_1
       (if (<= t_0 1e-42)
         (/ 0.083333333333333 x)
         (if (<= t_0 10000000.0)
           t_1
           (* (* (/ (+ 0.0007936500793651 y) x) z) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 10000000.0) {
		tmp = t_1;
	} else {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (Math.log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 10000000.0) {
		tmp = t_1;
	} else {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	t_1 = (math.log(x) - 1.0) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (y * (z / x)) * z
	elif t_0 <= 5e-270:
		tmp = t_1
	elif t_0 <= 1e-42:
		tmp = 0.083333333333333 / x
	elif t_0 <= 10000000.0:
		tmp = t_1
	else:
		tmp = (((0.0007936500793651 + y) / x) * z) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = Float64(0.083333333333333 / x);
	elseif (t_0 <= 10000000.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	t_1 = (log(x) - 1.0) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = 0.083333333333333 / x;
	elseif (t_0 <= 10000000.0)
		tmp = t_1;
	else
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e-270], t$95$1, If[LessEqual[t$95$0, 1e-42], N[(0.083333333333333 / x), $MachinePrecision], If[LessEqual[t$95$0, 10000000.0], t$95$1, N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -inf.0

   1. Initial program 83.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 91.7

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

   10. Applied rewrites 91.7%

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

   if -inf.0 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.9999999999999998e-270 or 1.00000000000000004e-42 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1e7

   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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 46.0

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

   4. Applied rewrites 46.0%

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

   if 4.9999999999999998e-270 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000004e-42

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 49.7

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

   7. Applied rewrites 49.7%

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

   if 1e7 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 11.9

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 13.1

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

   10. Applied rewrites 13.1%

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

Alternative 11: 62.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot y + 0.083333333333333}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0037)
   (/ (+ (* (* z z) y) 0.083333333333333) x)
   (if (<= x 3.3e+109)
     (* (* (/ (+ 0.0007936500793651 y) x) z) z)
     (* (- (log x) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0037) {
		tmp = (((z * z) * y) + 0.083333333333333) / x;
	} else if (x <= 3.3e+109) {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.0037d0) then
        tmp = (((z * z) * y) + 0.083333333333333d0) / x
    else if (x <= 3.3d+109) then
        tmp = (((0.0007936500793651d0 + y) / x) * z) * z
    else
        tmp = (log(x) - 1.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0037) {
		tmp = (((z * z) * y) + 0.083333333333333) / x;
	} else if (x <= 3.3e+109) {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	} else {
		tmp = (Math.log(x) - 1.0) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0037:
		tmp = (((z * z) * y) + 0.083333333333333) / x
	elif x <= 3.3e+109:
		tmp = (((0.0007936500793651 + y) / x) * z) * z
	else:
		tmp = (math.log(x) - 1.0) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0037)
		tmp = Float64(Float64(Float64(Float64(z * z) * y) + 0.083333333333333) / x);
	elseif (x <= 3.3e+109)
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0037)
		tmp = (((z * z) * y) + 0.083333333333333) / x;
	elseif (x <= 3.3e+109)
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	else
		tmp = (log(x) - 1.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0037], N[(N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.3e+109], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0037000000000000002

   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. 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

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot {z}^{2} + \frac{83333333333333}{1000000000000000}}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot y + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot y + \frac{83333333333333}{1000000000000000}}{x} \]

      3. pow2 N/A

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

      4. lower-*.f64 82.5

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

   9. Applied rewrites 82.5%

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

   if 0.0037000000000000002 < x < 3.2999999999999999e109

   1. Initial program 95.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 45.1

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 47.9

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

   10. Applied rewrites 47.9%

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

   if 3.2999999999999999e109 < x

   1. Initial program 83.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 77.3

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

   4. Applied rewrites 77.3%

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

Alternative 12: 57.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-42}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq 8 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* (- (log x) 1.0) x)))
   (if (<= t_0 (- INFINITY))
     (* (* y (/ z x)) z)
     (if (<= t_0 5e-270)
       t_1
       (if (<= t_0 1e-42)
         (/ 0.083333333333333 x)
         (if (<= t_0 8e+59)
           t_1
           (* (/ (- (* 0.0007936500793651 z) 0.0027777777777778) x) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 8e+59) {
		tmp = t_1;
	} else {
		tmp = (((0.0007936500793651 * z) - 0.0027777777777778) / x) * z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (Math.log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 8e+59) {
		tmp = t_1;
	} else {
		tmp = (((0.0007936500793651 * z) - 0.0027777777777778) / x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	t_1 = (math.log(x) - 1.0) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (y * (z / x)) * z
	elif t_0 <= 5e-270:
		tmp = t_1
	elif t_0 <= 1e-42:
		tmp = 0.083333333333333 / x
	elif t_0 <= 8e+59:
		tmp = t_1
	else:
		tmp = (((0.0007936500793651 * z) - 0.0027777777777778) / x) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = Float64(0.083333333333333 / x);
	elseif (t_0 <= 8e+59)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) / x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	t_1 = (log(x) - 1.0) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = 0.083333333333333 / x;
	elseif (t_0 <= 8e+59)
		tmp = t_1;
	else
		tmp = (((0.0007936500793651 * z) - 0.0027777777777778) / x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e-270], t$95$1, If[LessEqual[t$95$0, 1e-42], N[(0.083333333333333 / x), $MachinePrecision], If[LessEqual[t$95$0, 8e+59], t$95$1, N[(N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -inf.0

   1. Initial program 83.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 91.7

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

   10. Applied rewrites 91.7%

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

   if -inf.0 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.9999999999999998e-270 or 1.00000000000000004e-42 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 7.99999999999999977e59

   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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 46.1

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

   4. Applied rewrites 46.1%

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

   if 4.9999999999999998e-270 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000004e-42

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 49.7

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

   7. Applied rewrites 49.7%

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

   if 7.99999999999999977e59 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 14.6

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

   4. Applied rewrites 14.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}}{x} \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 4.9%

       \[\leadsto \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x} \cdot z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-42}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot 0.0007936500793651 - 0.0027777777777778\right) \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* (- (log x) 1.0) x)))
   (if (<= t_0 (- INFINITY))
     (* (* y (/ z x)) z)
     (if (<= t_0 5e-270)
       t_1
       (if (<= t_0 1e-42)
         (/ 0.083333333333333 x)
         (if (<= t_0 2e+43)
           t_1
           (/ (* (- (* z 0.0007936500793651) 0.0027777777777778) z) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 2e+43) {
		tmp = t_1;
	} else {
		tmp = (((z * 0.0007936500793651) - 0.0027777777777778) * z) / x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (Math.log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 2e+43) {
		tmp = t_1;
	} else {
		tmp = (((z * 0.0007936500793651) - 0.0027777777777778) * z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	t_1 = (math.log(x) - 1.0) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (y * (z / x)) * z
	elif t_0 <= 5e-270:
		tmp = t_1
	elif t_0 <= 1e-42:
		tmp = 0.083333333333333 / x
	elif t_0 <= 2e+43:
		tmp = t_1
	else:
		tmp = (((z * 0.0007936500793651) - 0.0027777777777778) * z) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = Float64(0.083333333333333 / x);
	elseif (t_0 <= 2e+43)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(z * 0.0007936500793651) - 0.0027777777777778) * z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	t_1 = (log(x) - 1.0) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = 0.083333333333333 / x;
	elseif (t_0 <= 2e+43)
		tmp = t_1;
	else
		tmp = (((z * 0.0007936500793651) - 0.0027777777777778) * z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e-270], t$95$1, If[LessEqual[t$95$0, 1e-42], N[(0.083333333333333 / x), $MachinePrecision], If[LessEqual[t$95$0, 2e+43], t$95$1, N[(N[(N[(N[(z * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -inf.0

   1. Initial program 83.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 91.7

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

   10. Applied rewrites 91.7%

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

   if -inf.0 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.9999999999999998e-270 or 1.00000000000000004e-42 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.00000000000000003e43

   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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 46.1

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

   4. Applied rewrites 46.1%

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

   if 4.9999999999999998e-270 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000004e-42

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 49.7

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

   7. Applied rewrites 49.7%

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

   if 2.00000000000000003e43 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 13.8

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

   4. Applied rewrites 13.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 15.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.1

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

   10. Applied rewrites 4.1%

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

Alternative 14: 53.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-42}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* (- (log x) 1.0) x)))
   (if (<= t_0 (- INFINITY))
     (* (* y (/ z x)) z)
     (if (<= t_0 5e-270)
       t_1
       (if (<= t_0 1e-42)
         (/ 0.083333333333333 x)
         (if (<= t_0 2e+190) t_1 (* y (/ (* z z) x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 2e+190) {
		tmp = t_1;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (Math.log(x) - 1.0) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 5e-270) {
		tmp = t_1;
	} else if (t_0 <= 1e-42) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 2e+190) {
		tmp = t_1;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	t_1 = (math.log(x) - 1.0) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (y * (z / x)) * z
	elif t_0 <= 5e-270:
		tmp = t_1
	elif t_0 <= 1e-42:
		tmp = 0.083333333333333 / x
	elif t_0 <= 2e+190:
		tmp = t_1
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = Float64(0.083333333333333 / x);
	elseif (t_0 <= 2e+190)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	t_1 = (log(x) - 1.0) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 5e-270)
		tmp = t_1;
	elseif (t_0 <= 1e-42)
		tmp = 0.083333333333333 / x;
	elseif (t_0 <= 2e+190)
		tmp = t_1;
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e-270], t$95$1, If[LessEqual[t$95$0, 1e-42], N[(0.083333333333333 / x), $MachinePrecision], If[LessEqual[t$95$0, 2e+190], t$95$1, N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -inf.0

   1. Initial program 83.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 91.7

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

   10. Applied rewrites 91.7%

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

   if -inf.0 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.9999999999999998e-270 or 1.00000000000000004e-42 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.0000000000000001e190

   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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 45.1

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

   4. Applied rewrites 45.1%

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

   if 4.9999999999999998e-270 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000004e-42

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 49.7

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

   7. Applied rewrites 49.7%

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

   if 2.0000000000000001e190 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 16.2

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

   4. Applied rewrites 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 16.9

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

   6. Applied rewrites 16.9%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+64}:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -5e+64)
     (* (* y (/ z x)) z)
     (if (<= t_0 4e-5) (/ 0.083333333333333 x) (* y (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5e+64) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 4e-5) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    if (t_0 <= (-5d+64)) then
        tmp = (y * (z / x)) * z
    else if (t_0 <= 4d-5) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = y * ((z * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5e+64) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 4e-5) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	tmp = 0
	if t_0 <= -5e+64:
		tmp = (y * (z / x)) * z
	elif t_0 <= 4e-5:
		tmp = 0.083333333333333 / x
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -5e+64)
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 4e-5)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_0 <= -5e+64)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 4e-5)
		tmp = 0.083333333333333 / x;
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+64], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 4e-5], N[(0.083333333333333 / x), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5e64

   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. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 80.4

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

   10. Applied rewrites 80.4%

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

   if -5e64 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.00000000000000033e-5

   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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 48.5

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

   7. Applied rewrites 48.5%

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

   if 4.00000000000000033e-5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 47.6

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

   6. Applied rewrites 47.6%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot \frac{z}{x}\right) \cdot z\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y (/ z x)) z))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -5e+64) t_0 (if (<= t_1 0.1) (/ 0.083333333333333 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * (z / x)) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -5e+64) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * (z / x)) * z
    t_1 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_1 <= (-5d+64)) then
        tmp = t_0
    else if (t_1 <= 0.1d0) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * (z / x)) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -5e+64) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * (z / x)) * z
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_1 <= -5e+64:
		tmp = t_0
	elif t_1 <= 0.1:
		tmp = 0.083333333333333 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(z / x)) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -5e+64)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * (z / x)) * z;
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_1 <= -5e+64)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = 0.083333333333333 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $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, -5e+64], t$95$0, If[LessEqual[t$95$1, 0.1], N[(0.083333333333333 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -5e64 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))

   1. Initial program 88.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)} \]
   6. 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. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 52.6

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

   10. Applied rewrites 52.6%

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

   if -5e64 < (+.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. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 48.4

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

   7. Applied rewrites 48.4%

      \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0027777777777778 \cdot \frac{z}{x}\\ \mathbf{if}\;z \leq -30:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.0027777777777778 (/ z x))))
   (if (<= z -30.0) t_0 (if (<= z 8.2e+119) (/ 0.083333333333333 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -0.0027777777777778 * (z / x);
	double tmp;
	if (z <= -30.0) {
		tmp = t_0;
	} else if (z <= 8.2e+119) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = t_0;
	}
	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 = (-0.0027777777777778d0) * (z / x)
    if (z <= (-30.0d0)) then
        tmp = t_0
    else if (z <= 8.2d+119) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.0027777777777778 * (z / x);
	double tmp;
	if (z <= -30.0) {
		tmp = t_0;
	} else if (z <= 8.2e+119) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.0027777777777778 * (z / x)
	tmp = 0
	if z <= -30.0:
		tmp = t_0
	elif z <= 8.2e+119:
		tmp = 0.083333333333333 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.0027777777777778 * Float64(z / x))
	tmp = 0.0
	if (z <= -30.0)
		tmp = t_0;
	elseif (z <= 8.2e+119)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.0027777777777778 * (z / x);
	tmp = 0.0;
	if (z <= -30.0)
		tmp = t_0;
	elseif (z <= 8.2e+119)
		tmp = 0.083333333333333 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.0027777777777778 * N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -30.0], t$95$0, If[LessEqual[z, 8.2e+119], N[(0.083333333333333 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -30 or 8.1999999999999994e119 < z

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in z around 0

      \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{\color{blue}{x}} \]

      2. lower-/.f64 16.8

         \[\leadsto -0.0027777777777778 \cdot \frac{z}{x} \]

   7. Applied rewrites 16.8%

      \[\leadsto -0.0027777777777778 \cdot \color{blue}{\frac{z}{x}} \]

   if -30 < z < 8.1999999999999994e119

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 37.2

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

   7. Applied rewrites 37.2%

      \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 23.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   7. lift--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   8. associate-*r/ N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

   10. lower-/.f64 57.4

       \[\leadsto \left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x \]

4. Applied rewrites 57.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]
6. Step-by-step derivation

   1. lift-/.f64 23.9

      \[\leadsto \frac{0.083333333333333}{x} \]

7. Applied rewrites 23.9%

   \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))