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

Percentage Accurate: 94.2% → 97.8%

Time: 5.4s

Alternatives: 19

Speedup: 1.0×

• 31.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

   12. mult-flip-rev N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 97.8%

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

Alternative 2: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log x \cdot x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+209)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (+
    (+ (- (* (log x) x) x) 0.91893853320467)
    (fma z (/ (* y z) x) (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+209) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = (((log(x) * x) - x) + 0.91893853320467) + fma(z, ((y * z) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+209)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(Float64(log(x) * x) - x) + 0.91893853320467) + fma(z, Float64(Float64(y * z) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+209], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e209

   1. Initial program 94.2%

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

   if 2.0000000000000001e209 < x

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. neg-log N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 82.3

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

   9. Applied rewrites 82.3%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-log.f64 82.3

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

   12. Applied rewrites 82.3%

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

Alternative 3: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.3e-32)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (fma z (/ (* y z) x) (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.3e-32) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(z, ((y * z) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.3e-32)
		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) + fma(z, Float64(Float64(y * z) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.3e-32], 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[(z * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2999999999999999e-32

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   if 4.2999999999999999e-32 < x

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

Alternative 4: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log x \cdot x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.6e-6)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (log x) x) x) 0.91893853320467)
    (fma z (/ (* y z) x) (/ 0.083333333333333 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.6e-6) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((log(x) * x) - x) + 0.91893853320467) + fma(z, ((y * z) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.6e-6)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(log(x) * x) - x) + 0.91893853320467) + fma(z, Float64(Float64(y * z) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.6e-6], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(z * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.60000000000000034e-6

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   if 6.60000000000000034e-6 < x

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. neg-log N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 82.3

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

   9. Applied rewrites 82.3%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-log.f64 82.3

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

   12. Applied rewrites 82.3%

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

Alternative 5: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot z\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.3)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (* (* z z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.3) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((z * z) * y) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.3)
		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(z * z) * y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.3], 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[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.29999999999999982

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   if 6.29999999999999982 < x

   1. Initial program 94.2%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 61.8

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

   4. Applied rewrites 61.8%

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

Alternative 6: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+79}:\\ \;\;\;\;-\left(-z \cdot \frac{z}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x} + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+79)
     (- (* (- (* z (/ z x))) y))
     (if (<= t_0 2e+306)
       (-
        (fma (- x 0.5) (log x) (+ (/ 0.083333333333333 x) 0.91893853320467))
        x)
       (fma
        (/ (fma (- y -0.0007936500793651) z -0.0027777777777778) x)
        z
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+79) {
		tmp = -(-(z * (z / x)) * y);
	} else if (t_0 <= 2e+306) {
		tmp = fma((x - 0.5), log(x), ((0.083333333333333 / x) + 0.91893853320467)) - x;
	} else {
		tmp = fma((fma((y - -0.0007936500793651), z, -0.0027777777777778) / x), z, (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -1e+79)
		tmp = Float64(-Float64(Float64(-Float64(z * Float64(z / x))) * y));
	elseif (t_0 <= 2e+306)
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(0.083333333333333 / x) + 0.91893853320467)) - x);
	else
		tmp = fma(Float64(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778) / x), z, Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+79], (-N[((-N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]) * y), $MachinePrecision]), If[LessEqual[t$95$0, 2e+306], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $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)) < -9.99999999999999967e78

   1. Initial program 94.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   10. Applied rewrites 32.2%

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

   if -9.99999999999999967e78 < (+.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)) < 2.00000000000000003e306

   1. Initial program 94.2%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lift-/.f64 57.5

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

   6. Applied rewrites 57.5%

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

   if 2.00000000000000003e306 < (+.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 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift--.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lift-/.f64 63.8

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

   6. Applied rewrites 63.8%

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

Alternative 7: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+79}:\\ \;\;\;\;-\left(-z \cdot \frac{z}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) - -0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+79)
     (- (* (- (* z (/ z x))) y))
     (if (<= t_0 2e+306)
       (-
        (- (fma (log x) (- x 0.5) (/ 0.083333333333333 x)) -0.91893853320467)
        x)
       (fma
        (/ (fma (- y -0.0007936500793651) z -0.0027777777777778) x)
        z
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -1e+79) {
		tmp = -(-(z * (z / x)) * y);
	} else if (t_0 <= 2e+306) {
		tmp = (fma(log(x), (x - 0.5), (0.083333333333333 / x)) - -0.91893853320467) - x;
	} else {
		tmp = fma((fma((y - -0.0007936500793651), z, -0.0027777777777778) / x), z, (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -1e+79)
		tmp = Float64(-Float64(Float64(-Float64(z * Float64(z / x))) * y));
	elseif (t_0 <= 2e+306)
		tmp = Float64(Float64(fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)) - -0.91893853320467) - x);
	else
		tmp = fma(Float64(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778) / x), z, Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+79], (-N[((-N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]) * y), $MachinePrecision]), If[LessEqual[t$95$0, 2e+306], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - -0.91893853320467), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $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)) < -9.99999999999999967e78

   1. Initial program 94.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   10. Applied rewrites 32.2%

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

   if -9.99999999999999967e78 < (+.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)) < 2.00000000000000003e306

   1. Initial program 94.2%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 57.5

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

   4. Applied rewrites 57.5%

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

   6. Applied rewrites 57.5%

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

   if 2.00000000000000003e306 < (+.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 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift--.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lift-/.f64 63.8

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

   6. Applied rewrites 63.8%

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

Alternative 8: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2900000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2900000000.0)
   (/
    (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 <= 2900000000.0) {
		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 <= 2900000000.0)
		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, 2900000000.0], 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 < 2.9e9

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   if 2.9e9 < x

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

   9. Applied rewrites 35.8%

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

Alternative 9: 65.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+79}:\\ \;\;\;\;-\left(-z \cdot \frac{z}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+305}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

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 -1e+79)
     (- (* (- (* z (/ z x))) y))
     (if (<= t_0 2e+134)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       (if (<= t_0 1e+305)
         (* (- (log x) 1.0) 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 <= -1e+79) {
		tmp = -(-(z * (z / x)) * y);
	} else if (t_0 <= 2e+134) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else if (t_0 <= 1e+305) {
		tmp = (log(x) - 1.0) * 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 <= -1e+79)
		tmp = Float64(-Float64(Float64(-Float64(z * Float64(z / x))) * y));
	elseif (t_0 <= 2e+134)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	elseif (t_0 <= 1e+305)
		tmp = Float64(Float64(log(x) - 1.0) * 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, -1e+79], (-N[((-N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]) * y), $MachinePrecision]), If[LessEqual[t$95$0, 2e+134], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e+305], 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]]]]]

Derivation

1. Split input into 4 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)) < -9.99999999999999967e78

   1. Initial program 94.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   10. Applied rewrites 32.2%

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

   if -9.99999999999999967e78 < (+.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)) < 1.99999999999999984e134

   1. Initial program 94.2%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 46.7

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

   7. Applied rewrites 46.7%

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

   if 1.99999999999999984e134 < (+.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)) < 9.9999999999999994e304

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

   9. Applied rewrites 35.8%

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

   if 9.9999999999999994e304 < (+.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 94.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      10. unpow2 N/A

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

      11. lower-*.f64 42.0

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

   4. Applied rewrites 42.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. add-flip N/A

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

      8. +-commutative N/A

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

      9. div-add N/A

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

      10. mult-flip-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   6. Applied rewrites 43.7%

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

Alternative 10: 64.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+79}:\\ \;\;\;\;-\left(-z \cdot \frac{z}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+305}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

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 -1e+79)
     (- (* (- (* z (/ z x))) y))
     (if (<= t_0 2e+134)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (if (<= t_0 1e+305)
         (* (- (log x) 1.0) 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 <= -1e+79) {
		tmp = -(-(z * (z / x)) * y);
	} else if (t_0 <= 2e+134) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else if (t_0 <= 1e+305) {
		tmp = (log(x) - 1.0) * 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 <= -1e+79)
		tmp = Float64(-Float64(Float64(-Float64(z * Float64(z / x))) * y));
	elseif (t_0 <= 2e+134)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	elseif (t_0 <= 1e+305)
		tmp = Float64(Float64(log(x) - 1.0) * 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, -1e+79], (-N[((-N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]) * y), $MachinePrecision]), If[LessEqual[t$95$0, 2e+134], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e+305], 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]]]]]

Derivation

1. Split input into 4 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)) < -9.99999999999999967e78

   1. Initial program 94.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   10. Applied rewrites 32.2%

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

   if -9.99999999999999967e78 < (+.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)) < 1.99999999999999984e134

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   if 1.99999999999999984e134 < (+.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)) < 9.9999999999999994e304

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

   9. Applied rewrites 35.8%

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

   if 9.9999999999999994e304 < (+.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 94.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      10. unpow2 N/A

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

      11. lower-*.f64 42.0

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

   4. Applied rewrites 42.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. add-flip N/A

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

      8. +-commutative N/A

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

      9. div-add N/A

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

      10. mult-flip-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   6. Applied rewrites 43.7%

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

Alternative 11: 60.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+79}:\\ \;\;\;\;-\left(-z \cdot \frac{z}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+305}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -1e+79)
     (- (* (- (* z (/ z x))) y))
     (if (<= t_0 2e+134)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (if (<= t_0 1e+305)
         (* (- (log x) 1.0) x)
         (* (/ 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 <= -1e+79) {
		tmp = -(-(z * (z / x)) * y);
	} else if (t_0 <= 2e+134) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else if (t_0 <= 1e+305) {
		tmp = (log(x) - 1.0) * x;
	} else {
		tmp = (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 <= -1e+79)
		tmp = Float64(-Float64(Float64(-Float64(z * Float64(z / x))) * y));
	elseif (t_0 <= 2e+134)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	elseif (t_0 <= 1e+305)
		tmp = Float64(Float64(log(x) - 1.0) * x);
	else
		tmp = Float64(Float64(0.0007936500793651 / x) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+79], (-N[((-N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]) * y), $MachinePrecision]), If[LessEqual[t$95$0, 2e+134], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e+305], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(0.0007936500793651 / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)) < -9.99999999999999967e78

   1. Initial program 94.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   10. Applied rewrites 32.2%

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

   if -9.99999999999999967e78 < (+.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)) < 1.99999999999999984e134

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   if 1.99999999999999984e134 < (+.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)) < 9.9999999999999994e304

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. mult-flip-rev N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

   9. Applied rewrites 35.8%

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

   if 9.9999999999999994e304 < (+.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 94.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      10. unpow2 N/A

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

      11. lower-*.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 26.2%

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

Alternative 12: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+51}:\\ \;\;\;\;-\left(-z \cdot \frac{z}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -1e+51)
     (- (* (- (* z (/ z x))) y))
     (if (<= t_0 0.01)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (* (/ 0.0007936500793651 x) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -1e+51) {
		tmp = -(-(z * (z / x)) * y);
	} else if (t_0 <= 0.01) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = (0.0007936500793651 / x) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -1e+51)
		tmp = Float64(-Float64(Float64(-Float64(z * Float64(z / x))) * y));
	elseif (t_0 <= 0.01)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(0.0007936500793651 / x) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+51], (-N[((-N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]) * y), $MachinePrecision]), If[LessEqual[t$95$0, 0.01], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.0007936500793651 / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   10. Applied rewrites 32.2%

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

   if -1e51 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 0.0100000000000000002

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   if 0.0100000000000000002 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 94.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      10. unpow2 N/A

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

      11. lower-*.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 26.2%

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

Alternative 13: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+51)
     (* y (/ (* z z) x))
     (if (<= t_0 0.1)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (* (/ 0.0007936500793651 x) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+51) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = (0.0007936500793651 / x) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+51)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(0.0007936500793651 / x) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+51], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.0007936500793651 / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.1

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

   4. Applied rewrites 30.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 31.7

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

   6. Applied rewrites 31.7%

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

   if -1e51 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   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 94.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{y - \frac{-7936500793651}{10000000000000000}}{x} \cdot {z}^{2} \]

      10. unpow2 N/A

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

      11. lower-*.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 26.2%

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

Alternative 14: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot 0.0007936500793651}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+51)
     (* y (/ (* z z) x))
     (if (<= t_0 0.1)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       (/ (* (* z z) 0.0007936500793651) 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 <= -1e+51) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 0.1) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = ((z * z) * 0.0007936500793651) / 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 <= -1e+51)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 0.1)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z * z) * 0.0007936500793651) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+51], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.1

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

   4. Applied rewrites 30.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 31.7

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

   6. Applied rewrites 31.7%

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

   if -1e51 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   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 94.2%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. pow2 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      5. lift-*.f64 26.2

         \[\leadsto \frac{z \cdot z}{x} \cdot 0.0007936500793651 \]

   7. Applied rewrites 26.2%

      \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{0.0007936500793651} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      4. *-commutative N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{z \cdot z}{\color{blue}{x}} \]

      5. pow2 N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      10. pow2 N/A

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

      11. lift-*.f64 26.2

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

   9. Applied rewrites 26.2%

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

Alternative 15: 48.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot 0.0007936500793651}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.01)
   (/ (fma -0.0027777777777778 z 0.083333333333333) x)
   (/ (* (* z z) 0.0007936500793651) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.01) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = ((z * z) * 0.0007936500793651) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.01)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z * z) * 0.0007936500793651) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 0.01], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 0.0100000000000000002

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   if 0.0100000000000000002 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 94.2%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. pow2 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      5. lift-*.f64 26.2

         \[\leadsto \frac{z \cdot z}{x} \cdot 0.0007936500793651 \]

   7. Applied rewrites 26.2%

      \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{0.0007936500793651} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      4. *-commutative N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{z \cdot z}{\color{blue}{x}} \]

      5. pow2 N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{x} \]

      10. pow2 N/A

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

      11. lift-*.f64 26.2

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

   9. Applied rewrites 26.2%

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

Alternative 16: 48.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot z}{x} \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.01)
   (/ (fma -0.0027777777777778 z 0.083333333333333) x)
   (* (/ (* z z) x) 0.0007936500793651)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.01) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = ((z * z) / x) * 0.0007936500793651;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.01)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z * z) / x) * 0.0007936500793651);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 0.01], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 0.0100000000000000002

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   if 0.0100000000000000002 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 94.2%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. pow2 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      5. lift-*.f64 26.2

         \[\leadsto \frac{z \cdot z}{x} \cdot 0.0007936500793651 \]

   7. Applied rewrites 26.2%

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

Alternative 17: 48.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{x}\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.01)
   (/ (fma -0.0027777777777778 z 0.083333333333333) x)
   (* (* z (/ z x)) 0.0007936500793651)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.01) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = (z * (z / x)) * 0.0007936500793651;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.01)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * Float64(z / x)) * 0.0007936500793651);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 0.01], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 0.0100000000000000002

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

   if 0.0100000000000000002 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 94.2%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. pow2 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      5. lift-*.f64 26.2

         \[\leadsto \frac{z \cdot z}{x} \cdot 0.0007936500793651 \]

   7. Applied rewrites 26.2%

      \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{0.0007936500793651} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \frac{7936500793651}{10000000000000000} \]

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 26.3

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

   9. Applied rewrites 26.3%

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

Alternative 18: 29.0% accurate, 4.0× 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 94.2%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   2. lower-fma.f64 29.0

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

7. Applied rewrites 29.0%

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

Alternative 19: 23.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 23.1%

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

Reproduce

herbie shell --seed 2025136 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))