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

Percentage Accurate: 93.9% → 99.7%

Time: 11.7s

Alternatives: 18

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: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) - \left(\left(x - 0.91893853320467\right) - \frac{\mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7e+15)
   (-
    (* (log x) (- x 0.5))
    (-
     (- x 0.91893853320467)
     (/
      (fma
       (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (+ (* (- (log x) 1.0) x) (* (* (/ (- y -0.0007936500793651) x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e+15) {
		tmp = (log(x) * (x - 0.5)) - ((x - 0.91893853320467) - (fma(((z * (0.0007936500793651 + y)) - 0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((log(x) - 1.0) * x) + ((((y - -0.0007936500793651) / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7e+15)
		tmp = Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(Float64(x - 0.91893853320467) - Float64(fma(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7e+15], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(x - 0.91893853320467), $MachinePrecision] - N[(N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e15

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 7e15 < x

   1. Initial program 88.1%

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

   3. 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 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\log x - 1\right) \cdot x + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. 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-rev 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-rev 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. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 99.7

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

   8. Applied rewrites 99.7%

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

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(\left(\frac{0.083333333333333}{z \cdot z} + y\right) - -0.0007936500793651\right) - \frac{0.0027777777777778}{z}}{x} \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - \frac{-0.083333333333333}{x}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -5e+206)
     (*
      (/
       (-
        (- (+ (/ 0.083333333333333 (* z z)) y) -0.0007936500793651)
        (/ 0.0027777777777778 z))
       x)
      (* z z))
     (if (<= t_0 1e+308)
       (-
        (fma (- x 0.5) (log x) (- 0.91893853320467 (/ -0.083333333333333 x)))
        x)
       (* (* (/ (- y -0.0007936500793651) x) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -5e+206) {
		tmp = (((((0.083333333333333 / (z * z)) + y) - -0.0007936500793651) - (0.0027777777777778 / z)) / x) * (z * z);
	} else if (t_0 <= 1e+308) {
		tmp = fma((x - 0.5), log(x), (0.91893853320467 - (-0.083333333333333 / x))) - x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -5e+206)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.083333333333333 / Float64(z * z)) + y) - -0.0007936500793651) - Float64(0.0027777777777778 / z)) / x) * Float64(z * z));
	elseif (t_0 <= 1e+308)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - Float64(-0.083333333333333 / x))) - x);
	else
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+206], N[(N[(N[(N[(N[(N[(0.083333333333333 / N[(z * z), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - -0.0007936500793651), $MachinePrecision] - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+308], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -5.0000000000000002e206

   1. Initial program 83.8%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

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

   if -5.0000000000000002e206 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 1e308

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

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

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval 89.0

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

   5. Applied rewrites 89.0%

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

   if 1e308 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 87.0%

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

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

      13. metadata-eval N/A

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

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

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e-6)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (* (* (/ (- y -0.0007936500793651) x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e-6) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((y - -0.0007936500793651) / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e-6)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e-6], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -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[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999982e-6

   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. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.2

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

   5. Applied rewrites 99.2%

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

   if 3.99999999999999982e-6 < x

   1. Initial program 88.6%

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

   3. Taylor expanded in z around inf

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

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

   5. Applied rewrites 99.5%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t\_0 + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (log x) 1.0) x)))
   (if (<= x 5e+15)
     (+
      t_0
      (/
       (+
        (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
        0.083333333333333)
       x))
     (+ t_0 (* (* (/ (- y -0.0007936500793651) x) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (log(x) - 1.0) * x;
	double tmp;
	if (x <= 5e+15) {
		tmp = t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = t_0 + ((((y - -0.0007936500793651) / x) * z) * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (log(x) - 1.0d0) * x
    if (x <= 5d+15) then
        tmp = t_0 + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    else
        tmp = t_0 + ((((y - (-0.0007936500793651d0)) / x) * z) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(x) - 1.0) * x;
	double tmp;
	if (x <= 5e+15) {
		tmp = t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = t_0 + ((((y - -0.0007936500793651) / x) * z) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(x) - 1.0) * x
	tmp = 0
	if x <= 5e+15:
		tmp = t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	else:
		tmp = t_0 + ((((y - -0.0007936500793651) / x) * z) * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (x <= 5e+15)
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(x) - 1.0) * x;
	tmp = 0.0;
	if (x <= 5e+15)
		tmp = t_0 + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	else
		tmp = t_0 + ((((y - -0.0007936500793651) / x) * z) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 5e+15], N[(t$95$0 + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e15

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

   if 5e15 < x

   1. Initial program 88.1%

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

   3. 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 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\log x - 1\right) \cdot x + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. 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-rev 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-rev 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. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 99.7

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

   8. Applied rewrites 99.7%

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

Alternative 5: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+196}:\\ \;\;\;\;t\_0 + \frac{\left(z \cdot z\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (log x) 1.0) x)))
   (if (<= x 1.66)
     (/
      (fma
       (fma (- y -0.0007936500793651) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (if (<= x 1.6e+196) (+ t_0 (/ (* (* z z) y) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (log(x) - 1.0) * x;
	double tmp;
	if (x <= 1.66) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else if (x <= 1.6e+196) {
		tmp = t_0 + (((z * z) * y) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (x <= 1.66)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	elseif (x <= 1.6e+196)
		tmp = Float64(t_0 + Float64(Float64(Float64(z * z) * y) / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1.66], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.6e+196], N[(t$95$0 + N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.65999999999999992

   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. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.2

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

   5. Applied rewrites 99.2%

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

   if 1.65999999999999992 < x < 1.59999999999999996e196

   1. Initial program 93.5%

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

   3. Taylor expanded in 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 92.9

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

   5. Applied rewrites 92.9%

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

   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 86.1

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

   8. Applied rewrites 86.1%

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

   if 1.59999999999999996e196 < x

   1. Initial program 80.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 93.7%

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

   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 92.2

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

   7. Applied rewrites 92.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+196}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \frac{\left(z \cdot z\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e-6)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+ (* (- (log x) 1.0) x) (* (* (/ (- y -0.0007936500793651) x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e-6) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((log(x) - 1.0) * x) + ((((y - -0.0007936500793651) / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e-6)
		tmp = Float64(fma(fma(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(y - -0.0007936500793651) / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e-6], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -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[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999982e-6

   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. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.2

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

   5. Applied rewrites 99.2%

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

   if 3.99999999999999982e-6 < x

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 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 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\log x - 1\right) \cdot x + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. 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-rev 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-rev 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. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 99.1

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

   8. Applied rewrites 99.1%

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

Alternative 7: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.5e+48)
   (/
    (fma
     (fma (- y -0.0007936500793651) 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 <= 8.5e+48) {
		tmp = fma(fma((y - -0.0007936500793651), 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 <= 8.5e+48)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), 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, 8.5e+48], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -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 < 8.5000000000000001e48

   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. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 94.2

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

   5. Applied rewrites 94.2%

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

   if 8.5000000000000001e48 < x

   1. Initial program 86.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. 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. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 95.1%

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

   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 75.6

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

   7. Applied rewrites 75.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(\frac{\left(0.0007936500793651 - \frac{0.0027777777777778}{z}\right) \cdot z}{y} + z\right), y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      5e+65)
   (/
    (fma
     (* z (+ (/ (* (- 0.0007936500793651 (/ 0.0027777777777778 z)) z) y) z))
     y
     0.083333333333333)
    x)
   (* (* (/ (- y -0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 5e+65) {
		tmp = fma((z * ((((0.0007936500793651 - (0.0027777777777778 / z)) * z) / y) + z)), y, 0.083333333333333) / x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 5e+65)
		tmp = Float64(fma(Float64(z * Float64(Float64(Float64(Float64(0.0007936500793651 - Float64(0.0027777777777778 / z)) * z) / y) + z)), y, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 5e+65], N[(N[(N[(z * N[(N[(N[(N[(0.0007936500793651 - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.99999999999999973e65

   1. Initial program 97.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 53.5%

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

   if 4.99999999999999973e65 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 88.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      13. metadata-eval N/A

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

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

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 87.4

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

   5. Applied rewrites 87.4%

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

Alternative 9: 54.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -10 \lor \neg \left(t\_0 \leq 0.1\right):\\ \;\;\;\;y \cdot \left(\frac{z}{x} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (or (<= t_0 -10.0) (not (<= t_0 0.1)))
     (* y (* (/ z x) z))
     (/ (fma -0.0027777777777778 z 0.083333333333333) x))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 0.1)) {
		tmp = y * ((z / x) * z);
	} else {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if ((t_0 <= -10.0) || !(t_0 <= 0.1))
		tmp = Float64(y * Float64(Float64(z / x) * z));
	else
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10.0], N[Not[LessEqual[t$95$0, 0.1]], $MachinePrecision]], N[(y * N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $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)) < -10 or 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 55.9%

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

   if -10 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 49.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq -10 \lor \neg \left(\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 0.1\right):\\ \;\;\;\;y \cdot \left(\frac{z}{x} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.9% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -10.0)
     (* z (* z (/ y x)))
     (if (<= t_0 0.1)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (* y (* (/ z x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = z * (z * (y / x));
	} else if (t_0 <= 0.1) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = y * ((z / x) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(z * Float64(z * Float64(y / x)));
	elseif (t_0 <= 0.1)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(y * Float64(Float64(z / x) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -10

   1. Initial program 87.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 59.0%

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

   if -10 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 49.8%

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

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

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 53.1%

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

   9. Applied rewrites 54.8%

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

Alternative 11: 65.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{y} + z\right), y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      5e+65)
   (/
    (fma
     (* z (+ (/ (fma 0.0007936500793651 z -0.0027777777777778) y) z))
     y
     0.083333333333333)
    x)
   (* (* (/ (- y -0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 5e+65) {
		tmp = fma((z * ((fma(0.0007936500793651, z, -0.0027777777777778) / y) + z)), y, 0.083333333333333) / x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 5e+65)
		tmp = Float64(fma(Float64(z * Float64(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / y) + z)), y, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 5e+65], N[(N[(N[(z * N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.99999999999999973e65

   1. Initial program 97.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.5%

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

   if 4.99999999999999973e65 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 88.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      13. metadata-eval N/A

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

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

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 87.4

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

   5. Applied rewrites 87.4%

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

Alternative 12: 65.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 4 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      4e+186)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ (- y -0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4e+186) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 4e+186)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 4e+186], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 3.99999999999999992e186

   1. Initial program 97.3%

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

   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. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 55.1

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

   5. Applied rewrites 55.1%

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

   if 3.99999999999999992e186 < (+.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 85.9%

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

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

      13. metadata-eval N/A

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

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

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 13: 64.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      500000.0)
   (/ (fma (* z z) y 0.083333333333333) x)
   (* (* (/ (- y -0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000.0) {
		tmp = fma((z * z), y, 0.083333333333333) / x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000.0)
		tmp = Float64(fma(Float64(z * z), y, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 500000.0], N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5e5

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.6%

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

   if 5e5 < (+.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 89.8%

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

   3. Taylor expanded in z around inf

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

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

      13. metadata-eval N/A

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

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

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 82.4

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

   5. Applied rewrites 82.4%

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

Alternative 14: 63.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - -0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      500000.0)
   (/ (fma (* z z) y 0.083333333333333) x)
   (* (/ (- y -0.0007936500793651) x) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000.0) {
		tmp = fma((z * z), y, 0.083333333333333) / x;
	} else {
		tmp = ((y - -0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000.0)
		tmp = Float64(fma(Float64(z * z), y, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(y - -0.0007936500793651) / x) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 500000.0], N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $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)) < 5e5

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.6%

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

   if 5e5 < (+.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 89.8%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.4%

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

Alternative 15: 53.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{z}{x} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      1e+152)
   (/ (fma (* z z) y 0.083333333333333) x)
   (* y (* (/ z x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+152) {
		tmp = fma((z * z), y, 0.083333333333333) / x;
	} else {
		tmp = y * ((z / x) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+152)
		tmp = Float64(fma(Float64(z * z), y, 0.083333333333333) / x);
	else
		tmp = Float64(y * Float64(Float64(z / x) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 1e+152], N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(z / x), $MachinePrecision] * z), $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)) < 1e152

   1. Initial program 97.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.2%

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

   if 1e152 < (+.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 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 y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 60.3%

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

   9. Applied rewrites 62.6%

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

Alternative 16: 63.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.034 \lor \neg \left(y \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.034) (not (<= y 4e-9)))
   (/ (fma (* z z) y 0.083333333333333) x)
   (/
    (fma (fma 0.0007936500793651 z -0.0027777777777778) z 0.083333333333333)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.034) || !(y <= 4e-9)) {
		tmp = fma((z * z), y, 0.083333333333333) / x;
	} else {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.034) || !(y <= 4e-9))
		tmp = Float64(fma(Float64(z * z), y, 0.083333333333333) / x);
	else
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.034], N[Not[LessEqual[y, 4e-9]], $MachinePrecision]], N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.034000000000000002 or 4.00000000000000025e-9 < y

   1. Initial program 91.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left({z}^{2}, y, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.6%

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

   if -0.034000000000000002 < y < 4.00000000000000025e-9

   1. Initial program 96.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 63.0%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.034 \lor \neg \left(y \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.2% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 61.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 62.3%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 30.4%

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

Alternative 18: 24.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 61.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 62.3%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 24.9%

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

Developer Target 1: 98.5% 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 2025008 
(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)))