Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.6s

Alternatives: 13

Speedup: 0.5×

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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
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. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+38} \lor \neg \left(y \leq 5.8 \cdot 10^{+81}\right) \land y \leq 1.02 \cdot 10^{+126}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 2.7e+38) (and (not (<= y 5.8e+81)) (<= y 1.02e+126)))
   (- (+ x (* (log y) -0.5)) z)
   (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 2.7e+38) || (!(y <= 5.8e+81) && (y <= 1.02e+126))) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	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 <= 2.7d+38) .or. (.not. (y <= 5.8d+81)) .and. (y <= 1.02d+126)) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= 2.7e+38) || (!(y <= 5.8e+81) && (y <= 1.02e+126))) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 2.7e+38) or (not (y <= 5.8e+81) and (y <= 1.02e+126)):
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 2.7e+38) || (!(y <= 5.8e+81) && (y <= 1.02e+126)))
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 2.7e+38) || (~((y <= 5.8e+81)) && (y <= 1.02e+126)))
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 2.7e+38], And[N[Not[LessEqual[y, 5.8e+81]], $MachinePrecision], LessEqual[y, 1.02e+126]]], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.69999999999999996e38 or 5.7999999999999999e81 < y < 1.02e126

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 95.0%

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

   if 2.69999999999999996e38 < y < 5.7999999999999999e81 or 1.02e126 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.7%

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

      2. fma-undefine 99.5%

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

      3. associate-+r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-neg-in 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. *-commutative 99.5%

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

      10. sub-neg 99.5%

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

      11. associate-+r- 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r- 99.6%

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

      14. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 75.7%

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

      1. *-commutative 75.7%

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

      2. log-rec 75.7%

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

      3. cancel-sign-sub 75.7%

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

      4. *-commutative 75.7%

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

      5. neg-mul-1 75.7%

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

      6. sub-neg 75.7%

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

   9. Simplified 75.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+38} \lor \neg \left(y \leq 5.8 \cdot 10^{+81}\right) \land y \leq 1.02 \cdot 10^{+126}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+29}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+80} \lor \neg \left(y \leq 5.4 \cdot 10^{+117}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.2e+29)
   (+ x (- y z))
   (if (or (<= y 3.4e+80) (not (<= y 5.4e+117)))
     (* y (- 1.0 (log y)))
     (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.2e+29) {
		tmp = x + (y - z);
	} else if ((y <= 3.4e+80) || !(y <= 5.4e+117)) {
		tmp = y * (1.0 - log(y));
	} 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 (y <= 2.2d+29) then
        tmp = x + (y - z)
    else if ((y <= 3.4d+80) .or. (.not. (y <= 5.4d+117))) then
        tmp = y * (1.0d0 - log(y))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.2e+29) {
		tmp = x + (y - z);
	} else if ((y <= 3.4e+80) || !(y <= 5.4e+117)) {
		tmp = y * (1.0 - Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.2e+29:
		tmp = x + (y - z)
	elif (y <= 3.4e+80) or not (y <= 5.4e+117):
		tmp = y * (1.0 - math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.2e+29)
		tmp = Float64(x + Float64(y - z));
	elseif ((y <= 3.4e+80) || !(y <= 5.4e+117))
		tmp = Float64(y * Float64(1.0 - log(y)));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.2e+29)
		tmp = x + (y - z);
	elseif ((y <= 3.4e+80) || ~((y <= 5.4e+117)))
		tmp = y * (1.0 - log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.2e+29], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.4e+80], N[Not[LessEqual[y, 5.4e+117]], $MachinePrecision]], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.2000000000000001e29

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{\log y \cdot \left(0.5 + y\right)}{x}\right)} + \left(y - z\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. +-commutative 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \log y \cdot \frac{y + 0.5}{x}\right)} + \left(y - z\right) \]
   8. Step-by-step derivation

      1. expm1-log1p-u 72.1%

         \[\leadsto x \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y \cdot \frac{y + 0.5}{x}\right)\right)}\right) + \left(y - z\right) \]

      2. expm1-undefine 72.1%

         \[\leadsto x \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\log y \cdot \frac{y + 0.5}{x}\right)} - 1\right)}\right) + \left(y - z\right) \]

      3. log1p-undefine 72.1%

         \[\leadsto x \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \log y \cdot \frac{y + 0.5}{x}\right)}} - 1\right)\right) + \left(y - z\right) \]

      4. add-exp-log 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in x around inf 76.2%

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

   if 2.2000000000000001e29 < y < 3.39999999999999992e80 or 5.4000000000000005e117 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.7%

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

      2. fma-undefine 99.5%

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

      3. associate-+r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-neg-in 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. *-commutative 99.5%

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

      10. sub-neg 99.5%

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

      11. associate-+r- 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r- 99.6%

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

      14. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. log-rec 74.2%

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

      3. cancel-sign-sub 74.2%

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

      4. *-commutative 74.2%

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

      5. neg-mul-1 74.2%

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

      6. sub-neg 74.2%

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

   9. Simplified 74.2%

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

   if 3.39999999999999992e80 < y < 5.4000000000000005e117

   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. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.7%

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

      2. fma-undefine 99.7%

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

      3. associate-+r+ 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-rgt-neg-in 99.7%

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

      9. *-commutative 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-+r- 99.7%

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

      12. +-commutative 99.7%

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

      13. associate-+r- 99.7%

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

      14. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. remove-double-neg 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 69.3%

       \[\leadsto \color{blue}{x - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+29}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+80} \lor \neg \left(y \leq 5.4 \cdot 10^{+117}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-170}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-170)
   (+ x (- y z))
   (if (<= y 3.5e-119)
     (- (* (log y) -0.5) z)
     (if (<= y 1.6e+116) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-170) {
		tmp = x + (y - z);
	} else if (y <= 3.5e-119) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.6e+116) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	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 <= 4d-170) then
        tmp = x + (y - z)
    else if (y <= 3.5d-119) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.6d+116) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-170) {
		tmp = x + (y - z);
	} else if (y <= 3.5e-119) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.6e+116) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e-170:
		tmp = x + (y - z)
	elif y <= 3.5e-119:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.6e+116:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e-170)
		tmp = Float64(x + Float64(y - z));
	elseif (y <= 3.5e-119)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.6e+116)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e-170)
		tmp = x + (y - z);
	elseif (y <= 3.5e-119)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.6e+116)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e-170], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-119], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.6e+116], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.99999999999999993e-170

   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+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{\log y \cdot \left(0.5 + y\right)}{x}\right)} + \left(y - z\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \log y \cdot \frac{y + 0.5}{x}\right)} + \left(y - z\right) \]
   8. Step-by-step derivation

      1. expm1-log1p-u 68.6%

         \[\leadsto x \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y \cdot \frac{y + 0.5}{x}\right)\right)}\right) + \left(y - z\right) \]

      2. expm1-undefine 68.6%

         \[\leadsto x \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\log y \cdot \frac{y + 0.5}{x}\right)} - 1\right)}\right) + \left(y - z\right) \]

      3. log1p-undefine 68.6%

         \[\leadsto x \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \log y \cdot \frac{y + 0.5}{x}\right)}} - 1\right)\right) + \left(y - z\right) \]

      4. add-exp-log 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around inf 82.2%

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

   if 3.99999999999999993e-170 < y < 3.5e-119

   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+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{\log y \cdot \left(0.5 + y\right)}{x}\right)} + \left(y - z\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. +-commutative 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \log y \cdot \frac{y + 0.5}{x}\right)} + \left(y - z\right) \]
   8. Step-by-step derivation

      1. expm1-log1p-u 71.9%

         \[\leadsto x \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log y \cdot \frac{y + 0.5}{x}\right)\right)}\right) + \left(y - z\right) \]

      2. expm1-undefine 71.9%

         \[\leadsto x \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\log y \cdot \frac{y + 0.5}{x}\right)} - 1\right)}\right) + \left(y - z\right) \]

      3. log1p-undefine 71.9%

         \[\leadsto x \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \log y \cdot \frac{y + 0.5}{x}\right)}} - 1\right)\right) + \left(y - z\right) \]

      4. add-exp-log 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 99.9%

       \[\leadsto \color{blue}{x \cdot \left(1 - 0.5 \cdot \frac{\log y}{x}\right) - z} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto x \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \log y}{x}}\right) - z \]

       2. *-commutative 99.9%

          \[\leadsto x \cdot \left(1 - \frac{\color{blue}{\log y \cdot 0.5}}{x}\right) - z \]

       3. associate-*r/ 99.8%

          \[\leadsto x \cdot \left(1 - \color{blue}{\log y \cdot \frac{0.5}{x}}\right) - z \]

       4. cancel-sign-sub-inv 99.8%

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

       5. cancel-sign-sub-inv 99.8%

          \[\leadsto x \cdot \color{blue}{\left(1 - \log y \cdot \frac{0.5}{x}\right)} - z \]

   12. Simplified 99.8%

       \[\leadsto \color{blue}{x \cdot \left(1 - \log y \cdot \frac{0.5}{x}\right) - z} \]

   13. Taylor expanded in x around 0 87.0%

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

       1. *-commutative 87.0%

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

   15. Simplified 87.0%

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

   if 3.5e-119 < y < 1.6e116

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.9%

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

      2. fma-undefine 99.9%

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

      3. associate-+r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-rgt-neg-in 99.9%

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

      9. *-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-+r- 99.9%

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

      12. +-commutative 99.9%

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

      13. associate-+r- 99.9%

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

      14. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. log-rec 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. remove-double-neg 87.2%

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

   9. Simplified 87.2%

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

   10. Taylor expanded in y around 0 67.3%

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

   if 1.6e116 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-+r- 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.7%

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

      2. fma-undefine 99.5%

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

      3. associate-+r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-neg-in 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. *-commutative 99.5%

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

      10. sub-neg 99.5%

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

      11. associate-+r- 99.5%

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

      12. +-commutative 99.5%

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

      13. associate-+r- 99.5%

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

      14. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 79.7%

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

      1. *-commutative 79.7%

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

      2. log-rec 79.7%

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

      3. cancel-sign-sub 79.7%

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

      4. *-commutative 79.7%

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

      5. neg-mul-1 79.7%

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

      6. sub-neg 79.7%

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

   9. Simplified 79.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-170}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{+26}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;\left(x + y\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= y 8.5e+26)
     (- (+ x (* (log y) -0.5)) z)
     (if (<= y 3.7e+65) (- (+ x y) t_0) (- (- y z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (y <= 8.5e+26) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 3.7e+65) {
		tmp = (x + y) - t_0;
	} else {
		tmp = (y - z) - t_0;
	}
	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 * log(y)
    if (y <= 8.5d+26) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 3.7d+65) then
        tmp = (x + y) - t_0
    else
        tmp = (y - z) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (y <= 8.5e+26) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 3.7e+65) {
		tmp = (x + y) - t_0;
	} else {
		tmp = (y - z) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if y <= 8.5e+26:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 3.7e+65:
		tmp = (x + y) - t_0
	else:
		tmp = (y - z) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (y <= 8.5e+26)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 3.7e+65)
		tmp = Float64(Float64(x + y) - t_0);
	else
		tmp = Float64(Float64(y - z) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (y <= 8.5e+26)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 3.7e+65)
		tmp = (x + y) - t_0;
	else
		tmp = (y - z) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 8.5e+26], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 3.7e+65], N[(N[(x + y), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(y - z), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5e26

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

   if 8.5e26 < y < 3.69999999999999995e65

   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. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.8%

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

      2. fma-undefine 99.7%

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

      3. associate-+r+ 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-rgt-neg-in 99.7%

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

      9. *-commutative 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-+r- 99.7%

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

      12. +-commutative 99.7%

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

      13. associate-+r- 99.7%

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

      14. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. remove-double-neg 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in z around 0 87.6%

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

       1. +-commutative 87.6%

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

   12. Simplified 87.6%

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

   if 3.69999999999999995e65 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 89.3%

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

      1. *-commutative 89.3%

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

      2. log-rec 89.3%

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

      3. distribute-lft-neg-in 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in z around 0 89.3%

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

      1. mul-1-neg 89.3%

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

      2. associate-+r+ 89.3%

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

      3. sub-neg 89.3%

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

      4. neg-mul-1 89.3%

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

      5. unsub-neg 89.3%

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

   10. Simplified 89.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+26}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-14)
   (- (+ x (* (log y) -0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-14) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + ((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 <= 2.1d-14) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + ((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 <= 2.1e-14) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.1e-14:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e-14)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + 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 <= 2.1e-14)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e-14], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0999999999999999e-14

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 99.9%

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

   if 2.0999999999999999e-14 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. sub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\left(x - z\right) - \log y \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-14)
   (- (- x z) (* (log y) (+ y 0.5)))
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-14) {
		tmp = (x - z) - (log(y) * (y + 0.5));
	} else {
		tmp = x + ((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 <= 2.1d-14) then
        tmp = (x - z) - (log(y) * (y + 0.5d0))
    else
        tmp = x + ((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 <= 2.1e-14) {
		tmp = (x - z) - (Math.log(y) * (y + 0.5));
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.1e-14:
		tmp = (x - z) - (math.log(y) * (y + 0.5))
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e-14)
		tmp = Float64(Float64(x - z) - Float64(log(y) * Float64(y + 0.5)));
	else
		tmp = Float64(x + 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 <= 2.1e-14)
		tmp = (x - z) - (log(y) * (y + 0.5));
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e-14], N[(N[(x - z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0999999999999999e-14

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.9%

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

      2. fma-undefine 99.9%

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

      3. associate-+r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-rgt-neg-in 99.9%

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

      9. *-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-+r- 99.9%

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

      12. +-commutative 99.9%

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

      13. associate-+r- 100.0%

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

      14. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 100.0%

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

   if 2.0999999999999999e-14 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. sub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\left(x - z\right) - \log y \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+28) (- (+ x (* (log y) -0.5)) z) (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+28) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (x + y) - (y * log(y));
	}
	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 <= 2d+28) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+28) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+28:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+28)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+28)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+28], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e28

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

   if 1.99999999999999992e28 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.7%

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

      2. fma-undefine 99.6%

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

      3. associate-+r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-in 99.6%

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

      9. *-commutative 99.6%

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

      10. sub-neg 99.6%

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

      11. associate-+r- 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r- 99.6%

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

      14. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. log-rec 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. remove-double-neg 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in z around 0 81.9%

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

       1. +-commutative 81.9%

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

   12. Simplified 81.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - z) + (x - (log(y) * (y + 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - z)) - (log(y) * (y + 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
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. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r- 99.8%

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

   2. fma-undefine 99.8%

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

   3. associate-+r+ 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-neg-in 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-rgt-neg-in 99.8%

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

   9. *-commutative 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+r- 99.8%

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

   12. +-commutative 99.8%

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

   13. associate-+r- 99.8%

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

   14. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 11: 48.1% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e+19) x (if (<= x 1.7e+40) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e+19) {
		tmp = x;
	} else if (x <= 1.7e+40) {
		tmp = -z;
	} else {
		tmp = x;
	}
	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 <= (-1.6d+19)) then
        tmp = x
    else if (x <= 1.7d+40) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e+19) {
		tmp = x;
	} else if (x <= 1.7e+40) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e+19:
		tmp = x
	elif x <= 1.7e+40:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e+19)
		tmp = x;
	elseif (x <= 1.7e+40)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6e+19)
		tmp = x;
	elseif (x <= 1.7e+40)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e+19], x, If[LessEqual[x, 1.7e+40], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e19 or 1.69999999999999994e40 < x

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 61.3%

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

   if -1.6e19 < x < 1.69999999999999994e40

   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. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 38.5%

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

      1. neg-mul-1 38.5%

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

   7. Simplified 38.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.8% 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 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
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. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r- 99.8%

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

   2. fma-undefine 99.8%

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

   3. associate-+r+ 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-neg-in 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-rgt-neg-in 99.8%

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

   9. *-commutative 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+r- 99.8%

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

   12. +-commutative 99.8%

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

   13. associate-+r- 99.8%

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

   14. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in y around inf 86.3%

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

   1. mul-1-neg 86.3%

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

   2. log-rec 86.3%

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

   3. distribute-rgt-neg-in 86.3%

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

   4. remove-double-neg 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in y around 0 56.5%

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

11. Final simplification 56.5%

    \[\leadsto x - z \]
12. Add Preprocessing

Alternative 13: 30.1% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
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. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 28.5%

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

6. Final simplification 28.5%

   \[\leadsto x \]
7. Add Preprocessing

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 2024079 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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