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

Percentage Accurate: 94.0% → 99.6%

Time: 12.4s

Alternatives: 15

Speedup: 1.0×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.85e+22)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (+ (* (- (log x) 1.0) x) (* (* (/ (+ 0.0007936500793651 y) x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.85e+22) {
		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) + ((((0.0007936500793651 + y) / x) * z) * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.85d+22) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = ((log(x) - 1.0d0) * x) + ((((0.0007936500793651d0 + y) / x) * z) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.85e+22) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = ((Math.log(x) - 1.0) * x) + ((((0.0007936500793651 + y) / x) * z) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.85e+22:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = ((math.log(x) - 1.0) * x) + ((((0.0007936500793651 + y) / x) * z) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.85e+22)
		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(log(x) - 1.0) * x) + Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.85e+22)
		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) + ((((0.0007936500793651 + y) / x) * z) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.85e+22], 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[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8499999999999999e22

   1. Initial program 99.6%

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

   if 2.8499999999999999e22 < x

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.4

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in z around inf

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

       1. associate-/l* N/A

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

       2. div-add N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. associate-*r/ N/A

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

       13. metadata-eval N/A

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

       14. div-add N/A

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

       15. lower-/.f64 N/A

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

       16. lower-+.f64 99.7

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

   11. Applied rewrites 99.7%

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

Alternative 2: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+110} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (or (<= t_0 -5e+110) (not (<= t_0 5e+158)))
     (* (* (/ (+ 0.0007936500793651 y) x) z) z)
     (fma
      (log x)
      (- x 0.5)
      (- (+ 0.91893853320467 (/ 0.083333333333333 x)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -5e+110) || !(t_0 <= 5e+158)) {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	} else {
		tmp = fma(log(x), (x - 0.5), ((0.91893853320467 + (0.083333333333333 / x)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if ((t_0 <= -5e+110) || !(t_0 <= 5e+158))
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z);
	else
		tmp = fma(log(x), Float64(x - 0.5), Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+110], N[Not[LessEqual[t$95$0, 5e+158]], $MachinePrecision]], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -4.99999999999999978e110 or 4.9999999999999996e158 < (+.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 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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -4.99999999999999978e110 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.9999999999999996e158

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 89.6

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

   7. Applied rewrites 89.6%

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

   9. Applied rewrites 89.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq -5 \cdot 10^{+110} \lor \neg \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+110} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x} + 0.91893853320467\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (or (<= t_0 -5e+110) (not (<= t_0 5e+158)))
     (* (* (/ (+ 0.0007936500793651 y) x) z) z)
     (-
      (fma (log x) (- x 0.5) (+ (/ 0.083333333333333 x) 0.91893853320467))
      x))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -5e+110) || !(t_0 <= 5e+158)) {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	} else {
		tmp = fma(log(x), (x - 0.5), ((0.083333333333333 / x) + 0.91893853320467)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if ((t_0 <= -5e+110) || !(t_0 <= 5e+158))
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z);
	else
		tmp = Float64(fma(log(x), Float64(x - 0.5), Float64(Float64(0.083333333333333 / x) + 0.91893853320467)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+110], N[Not[LessEqual[t$95$0, 5e+158]], $MachinePrecision]], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -4.99999999999999978e110 or 4.9999999999999996e158 < (+.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 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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -4.99999999999999978e110 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.9999999999999996e158

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 89.6

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

   7. Applied rewrites 89.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq -5 \cdot 10^{+110} \lor \neg \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x} + 0.91893853320467\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \frac{\left(z \cdot z\right) \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -5.0)
     (+ (* (- (log x) 1.0) x) (/ (* (* z z) y) x))
     (if (<= t_0 5e+158)
       (fma
        (log x)
        (- x 0.5)
        (- (+ 0.91893853320467 (/ 0.083333333333333 x)) x))
       (* (* (/ (+ 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) + 0.083333333333333;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = ((log(x) - 1.0) * x) + (((z * z) * y) / x);
	} else if (t_0 <= 5e+158) {
		tmp = fma(log(x), (x - 0.5), ((0.91893853320467 + (0.083333333333333 / x)) - x));
	} else {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(Float64(z * z) * y) / x));
	elseif (t_0 <= 5e+158)
		tmp = fma(log(x), Float64(x - 0.5), Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) - 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[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+158], N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $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 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -5

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if -5 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.9999999999999996e158

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 91.3

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 91.3%

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

   if 4.9999999999999996e158 < (+.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 84.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 90.7

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

   5. Applied rewrites 90.7%

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

Alternative 5: 97.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z / x)) - Float64(0.0027777777777778 / x)), z, Float64(Float64(0.083333333333333 / x) + Float64(Float64(log(x) - 1.0) * x)))
end

Wolfram

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

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-log.f64 91.8

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

5. Applied rewrites 91.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

7. Applied rewrites 97.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/l* N/A

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

   10. lift-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 98.6

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

9. Applied rewrites 98.6%

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

Alternative 6: 97.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. div-add N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 98.5%

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

Alternative 7: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.057:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.057)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* (* (/ (+ 0.0007936500793651 y) x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.057) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((0.0007936500793651 + y) / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.057)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.057], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0570000000000000021

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.0570000000000000021 < x

   1. Initial program 86.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. div-add-rev N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. div-add-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.15)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+ (* (- (log x) 1.0) x) (* (* (/ (+ 0.0007936500793651 y) x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.15) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((log(x) - 1.0) * x) + ((((0.0007936500793651 + y) / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.15)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.15], 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[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.149999999999999994

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.149999999999999994 < x

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in z around inf

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

       1. associate-/l* N/A

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

       2. div-add N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. associate-*r/ N/A

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

       13. metadata-eval N/A

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

       14. div-add N/A

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

       15. lower-/.f64 N/A

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

       16. lower-+.f64 98.7

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

   11. Applied rewrites 98.7%

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

Alternative 9: 83.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.55e+18)
   (/
    (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 <= 2.55e+18) {
		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 <= 2.55e+18)
		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, 2.55e+18], 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 < 2.55e18

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if 2.55e18 < x

   1. Initial program 85.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 69.6

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

   7. Applied rewrites 69.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -5.0)
     (* (* y (/ z x)) z)
     (if (<= t_0 2e-5)
       (/ 0.083333333333333 x)
       (* (* (/ (+ 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 tmp;
	if (t_0 <= -5.0) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 2e-5) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    if (t_0 <= (-5.0d0)) then
        tmp = (y * (z / x)) * z
    else if (t_0 <= 2d-5) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = (((0.0007936500793651d0 + y) / x) * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 2e-5) {
		tmp = 0.083333333333333 / x;
	} 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
	tmp = 0
	if t_0 <= -5.0:
		tmp = (y * (z / x)) * z
	elif t_0 <= 2e-5:
		tmp = 0.083333333333333 / x
	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)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 2e-5)
		tmp = Float64(0.083333333333333 / x);
	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;
	tmp = 0.0;
	if (t_0 <= -5.0)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 2e-5)
		tmp = 0.083333333333333 / x;
	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]}, If[LessEqual[t$95$0, -5.0], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(0.083333333333333 / 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 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 79.1%

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

   if -5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.00000000000000016e-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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 47.2%

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

   if 2.00000000000000016e-5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 75.7

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

   5. Applied rewrites 75.7%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.6% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -5.0)
     (* (* y (/ z x)) z)
     (if (<= t_0 2e-5)
       (/ 0.083333333333333 x)
       (* (/ (+ 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 tmp;
	if (t_0 <= -5.0) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 2e-5) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((0.0007936500793651 + y) / x) * (z * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    if (t_0 <= (-5.0d0)) then
        tmp = (y * (z / x)) * z
    else if (t_0 <= 2d-5) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = ((0.0007936500793651d0 + y) / x) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 2e-5) {
		tmp = 0.083333333333333 / x;
	} 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
	tmp = 0
	if t_0 <= -5.0:
		tmp = (y * (z / x)) * z
	elif t_0 <= 2e-5:
		tmp = 0.083333333333333 / x
	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)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 2e-5)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 + y) / x) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_0 <= -5.0)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 2e-5)
		tmp = 0.083333333333333 / x;
	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]}, If[LessEqual[t$95$0, -5.0], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(0.083333333333333 / x), $MachinePrecision], N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $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) < -5

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 79.1%

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

   if -5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.00000000000000016e-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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 47.2%

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

   if 2.00000000000000016e-5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 70.0

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

   7. Applied rewrites 70.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651 + y}{x} \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+114}\right):\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (or (<= t_0 -5.0) (not (<= t_0 2e+114)))
     (* (* y (/ z x)) z)
     (/ 0.083333333333333 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 <= -5.0) || !(t_0 <= 2e+114)) {
		tmp = (y * (z / x)) * z;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 2d+114))) then
        tmp = (y * (z / x)) * z
    else
        tmp = 0.083333333333333d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e+114)) {
		tmp = (y * (z / x)) * z;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 2e+114):
		tmp = (y * (z / x)) * z
	else:
		tmp = 0.083333333333333 / 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 <= -5.0) || !(t_0 <= 2e+114))
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	else
		tmp = Float64(0.083333333333333 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 2e+114)))
		tmp = (y * (z / x)) * z;
	else
		tmp = 0.083333333333333 / 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[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 2e+114]], $MachinePrecision]], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(0.083333333333333 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5 or 2e114 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 55.5%

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

   if -5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2e114

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 94.2

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.2%

       \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \lor \neg \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+114}\right):\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -5.0)
     (* (* y (/ z x)) z)
     (if (<= t_0 2e+114) (/ 0.083333333333333 x) (* (* (/ z x) z) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 2e+114) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((z / x) * z) * y;
	}
	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 <= (-5.0d0)) then
        tmp = (y * (z / x)) * z
    else if (t_0 <= 2d+114) then
        tmp = 0.083333333333333d0 / x
    else
        tmp = ((z / x) * z) * y
    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 <= -5.0) {
		tmp = (y * (z / x)) * z;
	} else if (t_0 <= 2e+114) {
		tmp = 0.083333333333333 / x;
	} else {
		tmp = ((z / x) * z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	tmp = 0
	if t_0 <= -5.0:
		tmp = (y * (z / x)) * z
	elif t_0 <= 2e+114:
		tmp = 0.083333333333333 / x
	else:
		tmp = ((z / x) * z) * y
	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 <= -5.0)
		tmp = Float64(Float64(y * Float64(z / x)) * z);
	elseif (t_0 <= 2e+114)
		tmp = Float64(0.083333333333333 / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * y);
	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 <= -5.0)
		tmp = (y * (z / x)) * z;
	elseif (t_0 <= 2e+114)
		tmp = 0.083333333333333 / x;
	else
		tmp = ((z / x) * z) * y;
	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, -5.0], N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e+114], N[(0.083333333333333 / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * y), $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) < -5

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 79.1%

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

   if -5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2e114

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 94.2

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.2%

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

   if 2e114 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.1

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

   5. Applied rewrites 40.1%

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

   7. Applied rewrites 44.3%

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

   9. Applied rewrites 50.5%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5:\\ \;\;\;\;\left(y \cdot \frac{z}{x}\right) \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651 + y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+25)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ (+ 0.0007936500793651 y) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+25) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((0.0007936500793651 + y) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+25)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.05e+25], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e25

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 1.05e25 < x

   1. Initial program 85.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 33.8

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

   5. Applied rewrites 33.8%

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

Alternative 15: 23.7% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. div-add N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 56.6

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

7. Applied rewrites 56.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 22.3%

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

11. Final simplification 22.3%

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

Developer Target 1: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025006 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))