Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

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

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

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

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, -0.5 - y, y + \left(x - z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (log y) (- -0.5 y) (+ y (- x z))))

C

double code(double x, double y, double z) {
	return fma(log(y), (-0.5 - y), (y + (x - z)));
}

Julia

function code(x, y, z)
	return fma(log(y), Float64(-0.5 - y), Float64(y + Float64(x - z)))
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+l+ 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. neg-sub0 99.9%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

   9. +-commutative 99.9%

      \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

   10. associate--r+ 99.9%

       \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

   11. metadata-eval 99.9%

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

   12. associate-+r- 99.9%

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

   13. +-commutative 99.9%

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

   14. associate-+r- 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+70}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-100}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y (- 1.0 (log y))) z)))
   (if (<= x -4.4e+70)
     (- x z)
     (if (<= x -1.2e-67)
       t_0
       (if (<= x -5.4e-100)
         (- (* (log y) -0.5) z)
         (if (<= x 1.1e+51) t_0 (- x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * (1.0 - log(y))) - z;
	double tmp;
	if (x <= -4.4e+70) {
		tmp = x - z;
	} else if (x <= -1.2e-67) {
		tmp = t_0;
	} else if (x <= -5.4e-100) {
		tmp = (log(y) * -0.5) - z;
	} else if (x <= 1.1e+51) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    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 * (1.0d0 - log(y))) - z
    if (x <= (-4.4d+70)) then
        tmp = x - z
    else if (x <= (-1.2d-67)) then
        tmp = t_0
    else if (x <= (-5.4d-100)) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (x <= 1.1d+51) then
        tmp = t_0
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * (1.0 - Math.log(y))) - z;
	double tmp;
	if (x <= -4.4e+70) {
		tmp = x - z;
	} else if (x <= -1.2e-67) {
		tmp = t_0;
	} else if (x <= -5.4e-100) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (x <= 1.1e+51) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * (1.0 - math.log(y))) - z
	tmp = 0
	if x <= -4.4e+70:
		tmp = x - z
	elif x <= -1.2e-67:
		tmp = t_0
	elif x <= -5.4e-100:
		tmp = (math.log(y) * -0.5) - z
	elif x <= 1.1e+51:
		tmp = t_0
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(1.0 - log(y))) - z)
	tmp = 0.0
	if (x <= -4.4e+70)
		tmp = Float64(x - z);
	elseif (x <= -1.2e-67)
		tmp = t_0;
	elseif (x <= -5.4e-100)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (x <= 1.1e+51)
		tmp = t_0;
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * (1.0 - log(y))) - z;
	tmp = 0.0;
	if (x <= -4.4e+70)
		tmp = x - z;
	elseif (x <= -1.2e-67)
		tmp = t_0;
	elseif (x <= -5.4e-100)
		tmp = (log(y) * -0.5) - z;
	elseif (x <= 1.1e+51)
		tmp = t_0;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -4.4e+70], N[(x - z), $MachinePrecision], If[LessEqual[x, -1.2e-67], t$95$0, If[LessEqual[x, -5.4e-100], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.1e+51], t$95$0, N[(x - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000001e70 or 1.09999999999999996e51 < x

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in x around inf 88.1%

      \[\leadsto \color{blue}{x} - z \]

   if -4.40000000000000001e70 < x < -1.2e-67 or -5.40000000000000031e-100 < x < 1.09999999999999996e51

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. log-rec 74.8%

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

      3. cancel-sign-sub 74.8%

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

      4. *-commutative 74.8%

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

      5. neg-mul-1 74.8%

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

      6. log-rec 74.8%

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

      7. log-rec 74.8%

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

      8. sub-neg 74.8%

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

   4. Simplified 74.8%

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

   if -1.2e-67 < x < -5.40000000000000031e-100

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+70}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-100}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.36 \cdot 10^{+83} \lor \neg \left(y \leq 9.5 \cdot 10^{+105}\right) \land y \leq 6.2 \cdot 10^{+159}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 2.36e+83) (and (not (<= y 9.5e+105)) (<= y 6.2e+159)))
   (- (- x (* (log y) 0.5)) z)
   (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 2.36e+83) || (!(y <= 9.5e+105) && (y <= 6.2e+159))) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= 2.36d+83) .or. (.not. (y <= 9.5d+105)) .and. (y <= 6.2d+159)) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= 2.36e+83) || (!(y <= 9.5e+105) && (y <= 6.2e+159))) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 2.36e+83) or (not (y <= 9.5e+105) and (y <= 6.2e+159)):
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 2.36e+83) || (!(y <= 9.5e+105) && (y <= 6.2e+159)))
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 2.36e+83) || (~((y <= 9.5e+105)) && (y <= 6.2e+159)))
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 2.36e+83], And[N[Not[LessEqual[y, 9.5e+105]], $MachinePrecision], LessEqual[y, 6.2e+159]]], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3599999999999999e83 or 9.4999999999999995e105 < y < 6.1999999999999996e159

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 91.4%

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

   if 2.3599999999999999e83 < y < 9.4999999999999995e105 or 6.1999999999999996e159 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 88.0%

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

      1. *-commutative 88.0%

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

      2. log-rec 88.0%

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

      3. cancel-sign-sub 88.0%

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

      4. *-commutative 88.0%

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

      5. neg-mul-1 88.0%

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

      6. log-rec 88.0%

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

      7. log-rec 88.0%

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

      8. sub-neg 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.36 \cdot 10^{+83} \lor \neg \left(y \leq 9.5 \cdot 10^{+105}\right) \land y \leq 6.2 \cdot 10^{+159}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 4: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \log y \cdot 0.5\right) - z\\ \mathbf{if}\;y \leq 6 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- x (* (log y) 0.5)) z)))
   (if (<= y 6e+83)
     t_0
     (if (<= y 7.5e+105)
       (- (- y (* y (log y))) z)
       (if (<= y 6.2e+159) t_0 (- (* y (- 1.0 (log y))) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - (log(y) * 0.5)) - z;
	double tmp;
	if (y <= 6e+83) {
		tmp = t_0;
	} else if (y <= 7.5e+105) {
		tmp = (y - (y * log(y))) - z;
	} else if (y <= 6.2e+159) {
		tmp = t_0;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    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 - (log(y) * 0.5d0)) - z
    if (y <= 6d+83) then
        tmp = t_0
    else if (y <= 7.5d+105) then
        tmp = (y - (y * log(y))) - z
    else if (y <= 6.2d+159) then
        tmp = t_0
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - (Math.log(y) * 0.5)) - z;
	double tmp;
	if (y <= 6e+83) {
		tmp = t_0;
	} else if (y <= 7.5e+105) {
		tmp = (y - (y * Math.log(y))) - z;
	} else if (y <= 6.2e+159) {
		tmp = t_0;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - (math.log(y) * 0.5)) - z
	tmp = 0
	if y <= 6e+83:
		tmp = t_0
	elif y <= 7.5e+105:
		tmp = (y - (y * math.log(y))) - z
	elif y <= 6.2e+159:
		tmp = t_0
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(log(y) * 0.5)) - z)
	tmp = 0.0
	if (y <= 6e+83)
		tmp = t_0;
	elseif (y <= 7.5e+105)
		tmp = Float64(Float64(y - Float64(y * log(y))) - z);
	elseif (y <= 6.2e+159)
		tmp = t_0;
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - (log(y) * 0.5)) - z;
	tmp = 0.0;
	if (y <= 6e+83)
		tmp = t_0;
	elseif (y <= 7.5e+105)
		tmp = (y - (y * log(y))) - z;
	elseif (y <= 6.2e+159)
		tmp = t_0;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 6e+83], t$95$0, If[LessEqual[y, 7.5e+105], N[(N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 6.2e+159], t$95$0, N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.9999999999999999e83 or 7.5000000000000002e105 < y < 6.1999999999999996e159

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 91.4%

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

   if 5.9999999999999999e83 < y < 7.5000000000000002e105

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 78.1%

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

      1. cancel-sign-sub-inv 78.1%

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

      2. metadata-eval 78.1%

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

      3. *-lft-identity 78.1%

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

      4. distribute-rgt-in 78.1%

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

      5. *-lft-identity 78.1%

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

      6. cancel-sign-sub 78.1%

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

      7. log-rec 78.1%

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

      8. remove-double-neg 78.1%

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

      9. *-commutative 78.1%

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

   4. Simplified 78.1%

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

   if 6.1999999999999996e159 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 90.7%

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

      1. *-commutative 90.7%

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

      2. log-rec 90.7%

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

      3. cancel-sign-sub 90.7%

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

      4. *-commutative 90.7%

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

      5. neg-mul-1 90.7%

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

      6. log-rec 90.7%

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

      7. log-rec 90.7%

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

      8. sub-neg 90.7%

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

   4. Simplified 90.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+83}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+159}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e-17)
   (- (- x (* (log y) 0.5)) z)
   (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-17) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y + (x - (y * log(y)))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.2d-17) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y + (x - (y * log(y)))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-17) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y + (x - (y * Math.log(y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.2e-17:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y + (x - (y * math.log(y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e-17)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y + Float64(x - Float64(y * log(y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e-17)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y + (x - (y * log(y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e-17], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000001e-17

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 100.0%

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

   if 8.2000000000000001e-17 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

      3. log-rec 98.8%

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

      4. remove-double-neg 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) - y \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e-17)
   (- (- x (* (log y) 0.5)) z)
   (- (- (+ y x) (* y (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-17) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = ((y + x) - (y * log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.2d-17) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = ((y + x) - (y * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-17) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = ((y + x) - (y * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.2e-17:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = ((y + x) - (y * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e-17)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(Float64(y + x) - Float64(y * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e-17)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = ((y + x) - (y * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e-17], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000001e-17

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 100.0%

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

   if 8.2000000000000001e-17 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. associate-+r- 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

      3. log-rec 98.8%

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

      4. remove-double-neg 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) - y \cdot \log y\right) - z\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (- x (* (log y) (+ y 0.5)))) z))

C

double code(double x, double y, double z) {
	return (y + (x - (log(y) * (y + 0.5)))) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y + (x - (log(y) * (y + 0.5d0)))) - z
end function

Java

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

Python

def code(x, y, z):
	return (y + (x - (math.log(y) * (y + 0.5)))) - z

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5)))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (x - (log(y) * (y + 0.5)))) - z;
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

2. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (+ y x) (* (log y) (+ y 0.5))) z))

C

double code(double x, double y, double z) {
	return ((y + x) - (log(y) * (y + 0.5))) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y + x) - (log(y) * (y + 0.5d0))) - z
end function

Java

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

Python

def code(x, y, z):
	return ((y + x) - (math.log(y) * (y + 0.5))) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) - Float64(log(y) * Float64(y + 0.5))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + x) - (log(y) * (y + 0.5))) - z;
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. associate-+r- 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 9: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -175:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -175.0) (- x z) (if (<= x 2.1e+23) (- (* (log y) -0.5) z) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -175.0) {
		tmp = x - z;
	} else if (x <= 2.1e+23) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-175.0d0)) then
        tmp = x - z
    else if (x <= 2.1d+23) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -175.0) {
		tmp = x - z;
	} else if (x <= 2.1e+23) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -175.0:
		tmp = x - z
	elif x <= 2.1e+23:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -175.0)
		tmp = Float64(x - z);
	elseif (x <= 2.1e+23)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -175.0)
		tmp = x - z;
	elseif (x <= 2.1e+23)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -175.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 2.1e+23], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -175 or 2.1000000000000001e23 < x

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in x around inf 85.4%

      \[\leadsto \color{blue}{x} - z \]

   if -175 < x < 2.1000000000000001e23

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 62.2%

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

   3. Taylor expanded in x around 0 59.9%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   4. Step-by-step derivation

      1. *-commutative 59.9%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   5. Simplified 59.9%

      \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -175:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 10: 57.3% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ x - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x z))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - z
end function

Java

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

Python

def code(x, y, z):
	return x - z

Julia

function code(x, y, z)
	return Float64(x - z)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

2. Taylor expanded in x around inf 61.2%

   \[\leadsto \color{blue}{x} - z \]

3. Final simplification 61.2%

   \[\leadsto x - z \]

Alternative 11: 30.4% accurate, 55.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -z
end function

Java

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

Python

def code(x, y, z):
	return -z

Julia

function code(x, y, z)
	return Float64(-z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.9%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+l+ 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. neg-sub0 99.9%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

   9. +-commutative 99.9%

      \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

   10. associate--r+ 99.9%

       \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

   11. metadata-eval 99.9%

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

   12. associate-+r- 99.9%

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

   13. +-commutative 99.9%

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

   14. associate-+r- 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in z around inf 27.0%

   \[\leadsto \color{blue}{-1 \cdot z} \]
5. Step-by-step derivation

   1. mul-1-neg 27.0%

      \[\leadsto \color{blue}{-z} \]

6. Simplified 27.0%

   \[\leadsto \color{blue}{-z} \]

7. Final simplification 27.0%

   \[\leadsto -z \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (- (+ y x) z) (* (+ y 0.5) (log y))))

C

double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * log(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y + x) - z) - ((y + 0.5d0) * log(y))
end function

Java

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

Python

def code(x, y, z):
	return ((y + x) - z) - ((y + 0.5) * math.log(y))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) - z) - Float64(Float64(y + 0.5) * log(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + x) - z) - ((y + 0.5) * log(y));
end

Wolfram

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

Reproduce

herbie shell --seed 2023199 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))