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

Percentage Accurate: 93.8% → 98.1%

Time: 19.6s

Alternatives: 20

Speedup: 1.0×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% 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: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;y \leq -7.2 \lor \neg \left(y \leq 3.6 \cdot 10^{+28}\right):\\ \;\;\;\;t\_0 + \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, z \cdot \left(y \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (or (<= y -7.2) (not (<= y 3.6e+28)))
     (+ t_0 (fma (/ 1.0 x) 0.083333333333333 (* z (* y (/ z x)))))
     (+
      t_0
      (fma
       z
       (/ (- (* 0.0007936500793651 z) 0.0027777777777778) x)
       (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if ((y <= -7.2) || !(y <= 3.6e+28)) {
		tmp = t_0 + fma((1.0 / x), 0.083333333333333, (z * (y * (z / x))));
	} else {
		tmp = t_0 + fma(z, (((0.0007936500793651 * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if ((y <= -7.2) || !(y <= 3.6e+28))
		tmp = Float64(t_0 + fma(Float64(1.0 / x), 0.083333333333333, Float64(z * Float64(y * Float64(z / x)))));
	else
		tmp = Float64(t_0 + fma(z, Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[Or[LessEqual[y, -7.2], N[Not[LessEqual[y, 3.6e+28]], $MachinePrecision]], N[(t$95$0 + N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + N[(z * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000018 or 3.5999999999999999e28 < y

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      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. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.1%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-/.f64 95.1

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 98.7

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

   9. Applied rewrites 98.7%

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

   if -7.20000000000000018 < y < 3.5999999999999999e28

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. div-sub N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 99.0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \lor \neg \left(y \leq 3.6 \cdot 10^{+28}\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, z \cdot \left(y \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+61}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -2e+127)
     (* y (/ (* z z) x))
     (if (<= t_0 1e+61)
       (+
        (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
        (/ 0.083333333333333 x))
       (*
        (-
         (+
          (+ (/ 0.0007936500793651 x) (/ 0.91893853320467 (* z z)))
          (+ (/ y x) (/ 0.083333333333333 (* (* z z) x))))
         (/ 0.0027777777777778 (* z 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 <= -2e+127) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 1e+61) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else {
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_0 <= (-2d+127)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 1d+61) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + (0.083333333333333d0 / x)
    else
        tmp = ((((0.0007936500793651d0 / x) + (0.91893853320467d0 / (z * z))) + ((y / x) + (0.083333333333333d0 / ((z * z) * x)))) - (0.0027777777777778d0 / (z * x))) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -2e+127) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 1e+61) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	} else {
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -2e+127:
		tmp = y * ((z * z) / x)
	elif t_0 <= 1e+61:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x)
	else:
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * 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 <= -2e+127)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 1e+61)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(0.91893853320467 / Float64(z * z))) + Float64(Float64(y / x) + Float64(0.083333333333333 / Float64(Float64(z * z) * x)))) - Float64(0.0027777777777778 / Float64(z * x))) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -2e+127)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 1e+61)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (0.083333333333333 / x);
	else
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+127], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+61], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(0.91893853320467 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] + N[(0.083333333333333 / N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0027777777777778 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $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)) < -1.99999999999999991e127

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -1.99999999999999991e127 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.99999999999999949e60

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 94.6%

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

   if 9.99999999999999949e60 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 86.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\mathsf{fma}\left(\log x, \frac{x - \frac{1}{2}}{z \cdot z}, \frac{y}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{{z}^{2}}\right) \cdot \left(z \cdot z\right) \]
   7. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{z \cdot z}\right) \cdot \left(z \cdot z\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 78.5

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

   11. Applied rewrites 78.5%

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

   12. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 79.0

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

   14. Applied rewrites 79.0%

       \[\leadsto \left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+61}:\\ \;\;\;\;\left(\left(\frac{0.083333333333333}{x} + \log x \cdot \left(x - 0.5\right)\right) + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -2e+127)
     (* y (/ (* z z) x))
     (if (<= t_0 1e+61)
       (-
        (+ (+ (/ 0.083333333333333 x) (* (log x) (- x 0.5))) 0.91893853320467)
        x)
       (*
        (-
         (+
          (+ (/ 0.0007936500793651 x) (/ 0.91893853320467 (* z z)))
          (+ (/ y x) (/ 0.083333333333333 (* (* z z) x))))
         (/ 0.0027777777777778 (* z 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 <= -2e+127) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 1e+61) {
		tmp = (((0.083333333333333 / x) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	} else {
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_0 <= (-2d+127)) then
        tmp = y * ((z * z) / x)
    else if (t_0 <= 1d+61) then
        tmp = (((0.083333333333333d0 / x) + (log(x) * (x - 0.5d0))) + 0.91893853320467d0) - x
    else
        tmp = ((((0.0007936500793651d0 / x) + (0.91893853320467d0 / (z * z))) + ((y / x) + (0.083333333333333d0 / ((z * z) * x)))) - (0.0027777777777778d0 / (z * x))) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -2e+127) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 1e+61) {
		tmp = (((0.083333333333333 / x) + (Math.log(x) * (x - 0.5))) + 0.91893853320467) - x;
	} else {
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -2e+127:
		tmp = y * ((z * z) / x)
	elif t_0 <= 1e+61:
		tmp = (((0.083333333333333 / x) + (math.log(x) * (x - 0.5))) + 0.91893853320467) - x
	else:
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * 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 <= -2e+127)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 1e+61)
		tmp = Float64(Float64(Float64(Float64(0.083333333333333 / x) + Float64(log(x) * Float64(x - 0.5))) + 0.91893853320467) - x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(0.91893853320467 / Float64(z * z))) + Float64(Float64(y / x) + Float64(0.083333333333333 / Float64(Float64(z * z) * x)))) - Float64(0.0027777777777778 / Float64(z * x))) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -2e+127)
		tmp = y * ((z * z) / x);
	elseif (t_0 <= 1e+61)
		tmp = (((0.083333333333333 / x) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	else
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+127], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+61], N[(N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(0.91893853320467 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] + N[(0.083333333333333 / N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0027777777777778 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $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)) < -1.99999999999999991e127

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -1.99999999999999991e127 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.99999999999999949e60

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      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. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 94.6%

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

   if 9.99999999999999949e60 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 86.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\mathsf{fma}\left(\log x, \frac{x - \frac{1}{2}}{z \cdot z}, \frac{y}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{{z}^{2}}\right) \cdot \left(z \cdot z\right) \]
   7. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{z \cdot z}\right) \cdot \left(z \cdot z\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 78.5

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

   11. Applied rewrites 78.5%

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

   12. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 79.0

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

   14. Applied rewrites 79.0%

       \[\leadsto \left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;t\_0 + \frac{t\_1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(z, \frac{0.0007936500793651 \cdot z - 0.0027777777777778}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 1e+300)
     (+ t_0 (/ t_1 x))
     (+
      t_0
      (fma
       z
       (/ (- (* 0.0007936500793651 z) 0.0027777777777778) x)
       (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= 1e+300) {
		tmp = t_0 + (t_1 / x);
	} else {
		tmp = t_0 + fma(z, (((0.0007936500793651 * z) - 0.0027777777777778) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= 1e+300)
		tmp = Float64(t_0 + Float64(t_1 / x));
	else
		tmp = Float64(t_0 + fma(z, Float64(Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778) / x), Float64(0.083333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+300], N[(t$95$0 + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1.0000000000000001e300

   1. Initial program 97.6%

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

   if 1.0000000000000001e300 < (+.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 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-/.f64 94.8

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

   6. Applied rewrites 94.8%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. div-sub N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 87.4

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

   9. Applied rewrites 87.4%

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

Alternative 5: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+61}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right) + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -2e+127)
     (* y (/ (* z z) x))
     (if (<= t_0 1e+61)
       (-
        (+ (fma (log x) (- x 0.5) (/ 0.083333333333333 x)) 0.91893853320467)
        x)
       (*
        (-
         (+
          (+ (/ 0.0007936500793651 x) (/ 0.91893853320467 (* z z)))
          (+ (/ y x) (/ 0.083333333333333 (* (* z z) x))))
         (/ 0.0027777777777778 (* z 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 <= -2e+127) {
		tmp = y * ((z * z) / x);
	} else if (t_0 <= 1e+61) {
		tmp = (fma(log(x), (x - 0.5), (0.083333333333333 / x)) + 0.91893853320467) - x;
	} else {
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * 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 <= -2e+127)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	elseif (t_0 <= 1e+61)
		tmp = Float64(Float64(fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)) + 0.91893853320467) - x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(0.91893853320467 / Float64(z * z))) + Float64(Float64(y / x) + Float64(0.083333333333333 / Float64(Float64(z * z) * x)))) - Float64(0.0027777777777778 / Float64(z * 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, -2e+127], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+61], 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[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(0.91893853320467 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] + N[(0.083333333333333 / N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0027777777777778 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $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)) < -1.99999999999999991e127

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -1.99999999999999991e127 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.99999999999999949e60

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if 9.99999999999999949e60 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 86.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\mathsf{fma}\left(\log x, \frac{x - \frac{1}{2}}{z \cdot z}, \frac{y}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{{z}^{2}}\right) \cdot \left(z \cdot z\right) \]
   7. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{z \cdot z}\right) \cdot \left(z \cdot z\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 78.5

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

   11. Applied rewrites 78.5%

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

   12. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 79.0

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

   14. Applied rewrites 79.0%

       \[\leadsto \left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 500000000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot y\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right)}{x} + \log x \cdot \left(x - 0.5\right)\right) + 0.91893853320467\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      500000000.0)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/ (+ (* (* z y) z) 0.083333333333333) x))
   (-
    (+
     (+ (/ (* (* z z) (+ y 0.0007936500793651)) x) (* (log x) (- x 0.5)))
     0.91893853320467)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000.0) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((z * y) * z) + 0.083333333333333) / x);
	} else {
		tmp = (((((z * z) * (y + 0.0007936500793651)) / x) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) <= 500000000.0d0) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((z * y) * z) + 0.083333333333333d0) / x)
    else
        tmp = (((((z * z) * (y + 0.0007936500793651d0)) / x) + (log(x) * (x - 0.5d0))) + 0.91893853320467d0) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000.0) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((z * y) * z) + 0.083333333333333) / x);
	} else {
		tmp = (((((z * z) * (y + 0.0007936500793651)) / x) + (Math.log(x) * (x - 0.5))) + 0.91893853320467) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000.0:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((z * y) * z) + 0.083333333333333) / x)
	else:
		tmp = (((((z * z) * (y + 0.0007936500793651)) / x) + (math.log(x) * (x - 0.5))) + 0.91893853320467) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(z * y) * z) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(z * z) * Float64(y + 0.0007936500793651)) / x) + Float64(log(x) * Float64(x - 0.5))) + 0.91893853320467) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000.0)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((z * y) * z) + 0.083333333333333) / x);
	else
		tmp = (((((z * z) * (y + 0.0007936500793651)) / x) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 500000000.0], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(z * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5e8

   1. Initial program 97.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if 5e8 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 87.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 86.9

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

   10. Applied rewrites 86.9%

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

Alternative 7: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(1.0 / x), 0.083333333333333, Float64(z * Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) / 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[(1.0 / x), $MachinePrecision] * 0.083333333333333 + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. div-add N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

4. Applied rewrites 97.8%

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

   1. lift-pow.f64 N/A

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

   2. inv-pow N/A

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

   3. lower-/.f64 97.8

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

6. Applied rewrites 97.8%

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

Alternative 8: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 / x) + (z * ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / 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) + ((0.083333333333333d0 / x) + (z * ((((0.0007936500793651d0 + y) * z) - 0.0027777777777778d0) / x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 / x) + (z * ((((0.0007936500793651 + y) * z) - 0.0027777777777778) / 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[(0.083333333333333 / x), $MachinePrecision] + N[(z * N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. associate-*r/ N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

4. Applied rewrites 97.8%

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

Alternative 9: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 97.8%

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

Alternative 10: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\left(z \cdot z\right) \cdot \left(y + 0.0007936500793651\right)}{x} + \log x \cdot \left(x - 0.5\right)\right) + 0.91893853320467\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.4e-5)
   (-
    (+
     (/
      (fma
       (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)
     0.91893853320467)
    x)
   (-
    (+
     (+ (/ (* (* z z) (+ y 0.0007936500793651)) x) (* (log x) (- x 0.5)))
     0.91893853320467)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.4e-5) {
		tmp = ((fma((((y + 0.0007936500793651) * z) - 0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x;
	} else {
		tmp = (((((z * z) * (y + 0.0007936500793651)) / x) + (log(x) * (x - 0.5))) + 0.91893853320467) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.4e-5)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(z * z) * Float64(y + 0.0007936500793651)) / x) + Float64(log(x) * Float64(x - 0.5))) + 0.91893853320467) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.4e-5], N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(N[(N[(z * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4000000000000001e-5

   1. Initial program 99.7%

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

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-fma.f64 98.5

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

   10. Applied rewrites 98.5%

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

   if 2.4000000000000001e-5 < x

   1. Initial program 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. div-add N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 85.9

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

   10. Applied rewrites 85.9%

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

Alternative 11: 93.8% 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]

Derivation

1. Initial program 93.1%

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

Alternative 12: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. div-add N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 93.1%

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

Alternative 13: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(z \cdot z\right) \cdot 0.0007936500793651}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.25e+34)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/ (* (* z z) 0.0007936500793651) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.25e+34) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (((z * z) * 0.0007936500793651) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.25e+34)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(z * z) * 0.0007936500793651) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.25e+34], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e34

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 95.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 1.25e34 < x

   1. Initial program 86.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.0%

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

Alternative 14: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6e+97)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6e+97) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6e+97)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6e+97], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.9999999999999997e97

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 89.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 5.9999999999999997e97 < x

   1. Initial program 84.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 78.3

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

   5. Applied rewrites 78.3%

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

Alternative 15: 64.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.0007936500793651}{x} + \frac{0.91893853320467}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{0.083333333333333}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{0.0027777777777778}{z \cdot x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.06e+21)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (*
    (-
     (+
      (+ (/ 0.0007936500793651 x) (/ 0.91893853320467 (* z z)))
      (+ (/ y x) (/ 0.083333333333333 (* (* z z) x))))
     (/ 0.0027777777777778 (* z x)))
    (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.06e+21) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((((0.0007936500793651 / x) + (0.91893853320467 / (z * z))) + ((y / x) + (0.083333333333333 / ((z * z) * x)))) - (0.0027777777777778 / (z * x))) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.06e+21)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(0.91893853320467 / Float64(z * z))) + Float64(Float64(y / x) + Float64(0.083333333333333 / Float64(Float64(z * z) * x)))) - Float64(0.0027777777777778 / Float64(z * x))) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.06e+21], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(0.91893853320467 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] + N[(0.083333333333333 / N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0027777777777778 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.06e21

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 96.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if 1.06e21 < x

   1. Initial program 86.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\mathsf{fma}\left(\log x, \frac{x - \frac{1}{2}}{z \cdot z}, \frac{y}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{{z}^{2}}\right) \cdot \left(z \cdot z\right) \]
   7. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 55.7

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

   8. Applied rewrites 55.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{\frac{91893853320467}{100000000000000}}{z \cdot z}\right) + \left(\frac{y}{x} + \frac{\frac{83333333333333}{1000000000000000}}{\left(z \cdot z\right) \cdot x}\right)\right) - \frac{x}{z \cdot z}\right) \cdot \left(z \cdot z\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 26.2

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

   11. Applied rewrites 26.2%

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

   12. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 28.2

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

   14. Applied rewrites 28.2%

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

Alternative 16: 62.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot y + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 2e-9)
   (- (/ (+ (* (* z z) y) 0.083333333333333) x) x)
   (* (/ (+ y 0.0007936500793651) x) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e-9) {
		tmp = ((((z * z) * y) + 0.083333333333333) / x) - x;
	} else {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) <= 2d-9) then
        tmp = ((((z * z) * y) + 0.083333333333333d0) / x) - x
    else
        tmp = ((y + 0.0007936500793651d0) / x) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e-9) {
		tmp = ((((z * z) * y) + 0.083333333333333) / x) - x;
	} else {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e-9:
		tmp = ((((z * z) * y) + 0.083333333333333) / x) - x
	else:
		tmp = ((y + 0.0007936500793651) / x) * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e-9)
		tmp = Float64(Float64(Float64(Float64(Float64(z * z) * y) + 0.083333333333333) / x) - x);
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e-9)
		tmp = ((((z * z) * y) + 0.083333333333333) / x) - x;
	else
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 2e-9], N[(N[(N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.00000000000000012e-9

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 72.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 52.5

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

   11. Applied rewrites 52.5%

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

   if 2.00000000000000012e-9 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.5%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.4

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

   8. Applied rewrites 74.4%

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

Alternative 17: 63.5% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 62.0

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 62.0

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

5. Applied rewrites 62.0%

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

Alternative 18: 43.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ (+ y 0.0007936500793651) x) (* z z)))

C

double code(double x, double y, double z) {
	return ((y + 0.0007936500793651) / x) * (z * z);
}

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 = ((y + 0.0007936500793651d0) / x) * (z * z)
end function

Java

public static double code(double x, double y, double z) {
	return ((y + 0.0007936500793651) / x) * (z * z);
}

Python

def code(x, y, z):
	return ((y + 0.0007936500793651) / x) * (z * z)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + 0.0007936500793651) / x) * (z * z);
end

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 61.0%

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

6. Taylor expanded in z around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. div-add N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 43.1

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

8. Applied rewrites 43.1%

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

Alternative 19: 32.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{z \cdot z}{x} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return y * ((z * z) / 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 = y * ((z * z) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 29.5

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

5. Applied rewrites 29.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 32.7

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

7. Applied rewrites 32.7%

   \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
8. Add Preprocessing

Alternative 20: 32.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(z \cdot \frac{z}{x}\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return y * (z * (z / 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 = y * (z * (z / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 29.5

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

5. Applied rewrites 29.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 32.7

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

7. Applied rewrites 32.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 32.6

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

9. Applied rewrites 32.6%

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

Developer Target 1: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025045 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

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

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