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

Percentage Accurate: 93.5% → 98.5%

Time: 6.8s

Alternatives: 16

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.5% 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: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, \mathsf{fma}\left(\log x, -0.5 + x, 0.91893853320467\right) \cdot x\right) + 0.083333333333333}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + z \cdot \frac{\left(0.0007936500793651 + y\right) \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e+45)
   (-
    (/
     (+
      (fma
       (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)
       z
       (* (fma (log x) (+ -0.5 x) 0.91893853320467) x))
      0.083333333333333)
     x)
    x)
   (+ (* (- (log x) 1.0) x) (* z (/ (* (+ 0.0007936500793651 y) z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e+45) {
		tmp = ((fma((((y + 0.0007936500793651) * z) - 0.0027777777777778), z, (fma(log(x), (-0.5 + x), 0.91893853320467) * x)) + 0.083333333333333) / x) - x;
	} else {
		tmp = ((log(x) - 1.0) * x) + (z * (((0.0007936500793651 + y) * z) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e+45)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778), z, Float64(fma(log(x), Float64(-0.5 + x), 0.91893853320467) * x)) + 0.083333333333333) / x) - x);
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(z * Float64(Float64(Float64(0.0007936500793651 + y) * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e+45], N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + N[(N[(N[Log[x], $MachinePrecision] * N[(-0.5 + x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(z * N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.9999999999999994e45

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 99.4%

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

   if 7.9999999999999994e45 < x

   1. Initial program 85.8%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 90.7

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 97.1%

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

Alternative 2: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.0007936500793651 + y\right) \cdot z\\ \mathbf{if}\;x \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(t\_0 - 0.0027777777777778, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ 0.0007936500793651 y) z)))
   (if (<= x 2.5e+46)
     (+
      (- (* (log x) (- x 0.5)) x)
      (+
       0.91893853320467
       (/ (fma (- t_0 0.0027777777777778) z 0.083333333333333) x)))
     (+ (* (- (log x) 1.0) x) (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) * z;
	double tmp;
	if (x <= 2.5e+46) {
		tmp = ((log(x) * (x - 0.5)) - x) + (0.91893853320467 + (fma((t_0 - 0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((log(x) - 1.0) * x) + (z * (t_0 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) * z)
	tmp = 0.0
	if (x <= 2.5e+46)
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - x) + Float64(0.91893853320467 + Float64(fma(Float64(t_0 - 0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(z * Float64(t_0 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 2.5e+46], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(z * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e46

   1. Initial program 99.4%

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

   3. Applied rewrites 99.4%

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

   if 2.5000000000000001e46 < x

   1. Initial program 85.8%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 90.6

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

   7. Applied rewrites 90.6%

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

   9. Applied rewrites 97.1%

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

Alternative 3: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + z \cdot \frac{\left(0.0007936500793651 + y\right) \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+34)
   (-
    (+
     (/
      (fma
       (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)
     (fma (log x) (- x 0.5) 0.91893853320467))
    x)
   (+ (* (- (log x) 1.0) x) (* z (/ (* (+ 0.0007936500793651 y) z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+34) {
		tmp = ((fma((((y + 0.0007936500793651) * z) - 0.0027777777777778), z, 0.083333333333333) / x) + fma(log(x), (x - 0.5), 0.91893853320467)) - x;
	} else {
		tmp = ((log(x) - 1.0) * x) + (z * (((0.0007936500793651 + y) * z) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+34)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778), z, 0.083333333333333) / x) + fma(log(x), Float64(x - 0.5), 0.91893853320467)) - x);
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(z * Float64(Float64(Float64(0.0007936500793651 + y) * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+34], 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] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(z * N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999989e34

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 99.6%

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

   if 1.99999999999999989e34 < 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. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 90.9

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

   7. Applied rewrites 90.9%

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

   9. Applied rewrites 97.3%

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

Alternative 4: 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 93.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-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. lower-/.f64 N/A

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

   17. associate-/l* N/A

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

3. Applied rewrites 97.6%

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

Alternative 5: 97.6% 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 93.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-fma.f64 N/A

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

3. Applied rewrites 97.6%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.0007936500793651\right) \cdot z\\ \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(t\_0 - 0.0027777777777778, z, 0.083333333333333\right)}{x} + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot \left(x - 0.5\right) - \left(x - 0.91893853320467\right)\right) + z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.0007936500793651) z)))
   (if (<= x 1.1)
     (-
      (+
       (/ (fma (- t_0 0.0027777777777778) z 0.083333333333333) x)
       0.91893853320467)
      x)
     (+ (- (* (log x) (- x 0.5)) (- x 0.91893853320467)) (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double tmp;
	if (x <= 1.1) {
		tmp = ((fma((t_0 - 0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x;
	} else {
		tmp = ((log(x) * (x - 0.5)) - (x - 0.91893853320467)) + (z * (t_0 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.0007936500793651) * z)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64(Float64(Float64(fma(Float64(t_0 - 0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x);
	else
		tmp = Float64(Float64(Float64(log(x) * Float64(x - 0.5)) - Float64(x - 0.91893853320467)) + Float64(z * Float64(t_0 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, 1.1], N[(N[(N[(N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision] + N[(z * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 98.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.1000000000000001 < x

   1. Initial program 87.5%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate-+l- N/A

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

      7. *-commutative N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower--.f64 91.3

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

   6. Applied rewrites 97.0%

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

Alternative 7: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x} + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + z \cdot \frac{\left(0.0007936500793651 + y\right) \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.8)
   (-
    (+
     (/
      (fma
       (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)
     0.91893853320467)
    x)
   (+ (* (- (log x) 1.0) x) (* z (/ (* (+ 0.0007936500793651 y) z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.8) {
		tmp = ((fma((((y + 0.0007936500793651) * z) - 0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x;
	} else {
		tmp = ((log(x) - 1.0) * x) + (z * (((0.0007936500793651 + y) * z) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.8)
		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(log(x) - 1.0) * x) + Float64(z * Float64(Float64(Float64(0.0007936500793651 + y) * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.8], 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[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(z * N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 98.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 1.80000000000000004 < x

   1. Initial program 87.5%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 90.9

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

   7. Applied rewrites 90.9%

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

   9. Applied rewrites 96.6%

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

Alternative 8: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log x - 1\right) \cdot x\\ \mathbf{if}\;x \leq 3100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+228}:\\ \;\;\;\;t\_0 + \frac{\left(z \cdot z\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (log x) 1.0) x)))
   (if (<= x 3100000.0)
     (/
      (fma
       (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
       z
       0.083333333333333)
      x)
     (if (<= x 3e+228) (+ t_0 (/ (* (* z z) y) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (log(x) - 1.0) * x;
	double tmp;
	if (x <= 3100000.0) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else if (x <= 3e+228) {
		tmp = t_0 + (((z * z) * y) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(x) - 1.0) * x)
	tmp = 0.0
	if (x <= 3100000.0)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	elseif (x <= 3e+228)
		tmp = Float64(t_0 + Float64(Float64(Float64(z * z) * y) / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 3100000.0], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3e+228], N[(t$95$0 + N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1e6

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 97.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if 3.1e6 < x < 3.0000000000000001e228

   1. Initial program 90.4%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

       1. lift-neg.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. remove-double-neg N/A

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

       5. lift-log.f64 82.3

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

   12. Applied rewrites 82.3%

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

   if 3.0000000000000001e228 < x

   1. Initial program 78.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 87.0

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

   4. Applied rewrites 87.0%

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

Alternative 9: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.15e+45)
   (/
    (fma
     (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.15e+45) {
		tmp = fma((((0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.15e+45)
		tmp = Float64(fma(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.15e+45], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000002e45

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 92.5

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

      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 92.5

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

   4. Applied rewrites 92.5%

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

   if 2.1500000000000002e45 < x

   1. Initial program 85.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 73.3

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

   4. Applied rewrites 73.3%

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

Alternative 10: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.0007936500793651\right) \cdot z\\ t_1 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(t\_0 - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\left(\log x \cdot x + 0.91893853320467\right) - x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.0007936500793651) z))
        (t_1
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/ (+ (* (- t_0 0.0027777777777778) z) 0.083333333333333) x))))
   (if (<= t_1 -5e+38)
     (* y (* z (/ z x)))
     (if (<= t_1 4e+303)
       (- (+ (* (log x) x) 0.91893853320467) x)
       (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double t_1 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((t_0 - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_1 <= -5e+38) {
		tmp = y * (z * (z / x));
	} else if (t_1 <= 4e+303) {
		tmp = ((log(x) * x) + 0.91893853320467) - x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + 0.0007936500793651d0) * z
    t_1 = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((t_0 - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    if (t_1 <= (-5d+38)) then
        tmp = y * (z * (z / x))
    else if (t_1 <= 4d+303) then
        tmp = ((log(x) * x) + 0.91893853320467d0) - x
    else
        tmp = z * (t_0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double t_1 = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((t_0 - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_1 <= -5e+38) {
		tmp = y * (z * (z / x));
	} else if (t_1 <= 4e+303) {
		tmp = ((Math.log(x) * x) + 0.91893853320467) - x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + 0.0007936500793651) * z
	t_1 = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((t_0 - 0.0027777777777778) * z) + 0.083333333333333) / x)
	tmp = 0
	if t_1 <= -5e+38:
		tmp = y * (z * (z / x))
	elif t_1 <= 4e+303:
		tmp = ((math.log(x) * x) + 0.91893853320467) - x
	else:
		tmp = z * (t_0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.0007936500793651) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(t_0 - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_1 <= -5e+38)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_1 <= 4e+303)
		tmp = Float64(Float64(Float64(log(x) * x) + 0.91893853320467) - x);
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + 0.0007936500793651) * z;
	t_1 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((t_0 - 0.0027777777777778) * z) + 0.083333333333333) / x);
	tmp = 0.0;
	if (t_1 <= -5e+38)
		tmp = y * (z * (z / x));
	elseif (t_1 <= 4e+303)
		tmp = ((log(x) * x) + 0.91893853320467) - x;
	else
		tmp = z * (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+38], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+303], N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.9999999999999997e38

   1. Initial program 87.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 89.8

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

   6. Applied rewrites 89.8%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
   7. 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 91.2

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

   8. Applied rewrites 91.2%

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

   if -4.9999999999999997e38 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4e303

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\frac{1}{x}\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      4. lower-neg.f64 N/A

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

      5. log-rec N/A

         \[\leadsto \left(\left(-x\right) \cdot \left(\mathsf{neg}\left(\log x\right)\right) + \frac{91893853320467}{100000000000000}\right) - x \]

      6. lower-neg.f64 N/A

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

      7. lift-log.f64 54.5

         \[\leadsto \left(\left(-x\right) \cdot \left(-\log x\right) + 0.91893853320467\right) - x \]

   7. Applied rewrites 54.5%

      \[\leadsto \left(\left(-x\right) \cdot \left(-\log x\right) + 0.91893853320467\right) - x \]

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 54.5

         \[\leadsto \left(\log x \cdot x + 0.91893853320467\right) - x \]

   10. Applied rewrites 54.5%

       \[\leadsto \left(\log x \cdot x + 0.91893853320467\right) - x \]

   if 4e303 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 84.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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. lower-/.f64 N/A

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

      17. associate-/l* N/A

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-add N/A

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

      9. pow2 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-add N/A

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

   6. Applied rewrites 86.6%

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

Alternative 11: 64.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq 50:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 50.0)
     (* y (* z (/ z x)))
     (if (<= t_0 4e+303)
       (* (- (log x) 1.0) x)
       (* (* (/ z x) z) 0.0007936500793651)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= 50.0) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 4e+303) {
		tmp = (log(x) - 1.0) * x;
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	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 = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
    if (t_0 <= 50.0d0) then
        tmp = y * (z * (z / x))
    else if (t_0 <= 4d+303) then
        tmp = (log(x) - 1.0d0) * x
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= 50.0) {
		tmp = y * (z * (z / x));
	} else if (t_0 <= 4e+303) {
		tmp = (Math.log(x) - 1.0) * x;
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)
	tmp = 0
	if t_0 <= 50.0:
		tmp = y * (z * (z / x))
	elif t_0 <= 4e+303:
		tmp = (math.log(x) - 1.0) * x
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= 50.0)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (t_0 <= 4e+303)
		tmp = Float64(Float64(log(x) - 1.0) * x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	tmp = 0.0;
	if (t_0 <= 50.0)
		tmp = y * (z * (z / x));
	elseif (t_0 <= 4e+303)
		tmp = (log(x) - 1.0) * x;
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 50.0], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+303], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 50

   1. Initial program 87.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 87.7

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

   6. Applied rewrites 87.7%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
   7. 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 89.1

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

   8. Applied rewrites 89.1%

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

   if 50 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4e303

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 53.3

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

   4. Applied rewrites 53.3%

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

   if 4e303 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 84.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 74.9

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

   7. Applied rewrites 74.9%

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

Alternative 12: 64.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot \left(0.0007936500793651 + y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.15e+45)
   (/ (* (* z z) (+ 0.0007936500793651 y)) x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.15e+45) {
		tmp = ((z * z) * (0.0007936500793651 + y)) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.15d+45) then
        tmp = ((z * z) * (0.0007936500793651d0 + y)) / x
    else
        tmp = (log(x) - 1.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.15e+45) {
		tmp = ((z * z) * (0.0007936500793651 + y)) / x;
	} else {
		tmp = (Math.log(x) - 1.0) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.15e+45:
		tmp = ((z * z) * (0.0007936500793651 + y)) / x
	else:
		tmp = (math.log(x) - 1.0) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.15e+45)
		tmp = Float64(Float64(Float64(z * z) * Float64(0.0007936500793651 + y)) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.15e+45)
		tmp = ((z * z) * (0.0007936500793651 + y)) / x;
	else
		tmp = (log(x) - 1.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.15e+45], N[(N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000002e45

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 56.2

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

   4. Applied rewrites 56.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. div-add-rev N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-+.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 2.1500000000000002e45 < x

   1. Initial program 85.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 73.3

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

   4. Applied rewrites 73.3%

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

Alternative 13: 64.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \frac{\left(y + 0.0007936500793651\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.15e+45)
   (* z (/ (* (+ y 0.0007936500793651) z) x))
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.15e+45) {
		tmp = z * (((y + 0.0007936500793651) * z) / x);
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.15d+45) then
        tmp = z * (((y + 0.0007936500793651d0) * z) / x)
    else
        tmp = (log(x) - 1.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.15e+45) {
		tmp = z * (((y + 0.0007936500793651) * z) / x);
	} else {
		tmp = (Math.log(x) - 1.0) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.15e+45:
		tmp = z * (((y + 0.0007936500793651) * z) / x)
	else:
		tmp = (math.log(x) - 1.0) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.15e+45)
		tmp = Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) / x));
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.15e+45)
		tmp = z * (((y + 0.0007936500793651) * z) / x);
	else
		tmp = (log(x) - 1.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.15e+45], N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000002e45

   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. 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. lower-/.f64 N/A

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

      17. associate-/l* N/A

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

   3. Applied rewrites 97.9%

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

   4. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-add N/A

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

      9. pow2 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-add N/A

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

   6. Applied rewrites 57.2%

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

   if 2.1500000000000002e45 < x

   1. Initial program 85.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 73.3

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

   4. Applied rewrites 73.3%

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

Alternative 14: 44.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;y \leq 0.0011:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.6e+16)
   (* y (* z (/ z x)))
   (if (<= y 0.0011)
     (* (* (/ z x) z) 0.0007936500793651)
     (* y (/ (* z z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.6e+16) {
		tmp = y * (z * (z / x));
	} else if (y <= 0.0011) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8.6d+16)) then
        tmp = y * (z * (z / x))
    else if (y <= 0.0011d0) then
        tmp = ((z / x) * z) * 0.0007936500793651d0
    else
        tmp = y * ((z * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.6e+16) {
		tmp = y * (z * (z / x));
	} else if (y <= 0.0011) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.6e+16:
		tmp = y * (z * (z / x))
	elif y <= 0.0011:
		tmp = ((z / x) * z) * 0.0007936500793651
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.6e+16)
		tmp = Float64(y * Float64(z * Float64(z / x)));
	elseif (y <= 0.0011)
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.6e+16)
		tmp = y * (z * (z / x));
	elseif (y <= 0.0011)
		tmp = ((z / x) * z) * 0.0007936500793651;
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.6e+16], N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0011], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.6e16

   1. Initial program 92.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.7

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

   4. Applied rewrites 47.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 49.9

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

   6. Applied rewrites 49.9%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
   7. 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 50.5

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

   8. Applied rewrites 50.5%

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

   if -8.6e16 < y < 0.00110000000000000007

   1. Initial program 93.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 36.8

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 37.9

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

   7. Applied rewrites 37.9%

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

   if 0.00110000000000000007 < y

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.0

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

   4. Applied rewrites 48.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 49.5

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

   6. Applied rewrites 49.5%

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

Alternative 15: 44.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* z (/ z x)))))
   (if (<= y -8.6e+16)
     t_0
     (if (<= y 0.0008) (* (* (/ z x) z) 0.0007936500793651) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z * (z / x));
	double tmp;
	if (y <= -8.6e+16) {
		tmp = t_0;
	} else if (y <= 0.0008) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (z * (z / x))
    if (y <= (-8.6d+16)) then
        tmp = t_0
    else if (y <= 0.0008d0) then
        tmp = ((z / x) * z) * 0.0007936500793651d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (z * (z / x));
	double tmp;
	if (y <= -8.6e+16) {
		tmp = t_0;
	} else if (y <= 0.0008) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (z * (z / x))
	tmp = 0
	if y <= -8.6e+16:
		tmp = t_0
	elif y <= 0.0008:
		tmp = ((z / x) * z) * 0.0007936500793651
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z * Float64(z / x)))
	tmp = 0.0
	if (y <= -8.6e+16)
		tmp = t_0;
	elseif (y <= 0.0008)
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (z * (z / x));
	tmp = 0.0;
	if (y <= -8.6e+16)
		tmp = t_0;
	elseif (y <= 0.0008)
		tmp = ((z / x) * z) * 0.0007936500793651;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.6e+16], t$95$0, If[LessEqual[y, 0.0008], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.6e16 or 8.00000000000000038e-4 < y

   1. Initial program 93.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 49.7

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

   6. Applied rewrites 49.7%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
   7. 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 50.3

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

   8. Applied rewrites 50.3%

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

   if -8.6e16 < y < 8.00000000000000038e-4

   1. Initial program 93.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 36.8

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 37.9

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

   7. Applied rewrites 37.9%

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

Alternative 16: 26.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

2. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 42.5

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

4. Applied rewrites 42.5%

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

5. Taylor expanded in y around 0

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

   1. pow2 N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. div-add-rev N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 26.6

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

7. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))