Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.7s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.6032211712972844e+70
  2. y = 9.952387624403188e-194
  3. z = -3.5623363082465264e+70

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

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

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 + fma(log(y), Float64(-0.5 - y), Float64(y - z)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y - z), $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. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-144}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-74}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-31}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 7800000000 \lor \neg \left(y \leq 3.8 \cdot 10^{+79}\right) \land y \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e-144)
   (+ x (- y z))
   (if (<= y 1.25e-74)
     (- (* (log y) -0.5) z)
     (if (<= y 6e-31)
       (- x (* (log y) 0.5))
       (if (or (<= y 7800000000.0) (and (not (<= y 3.8e+79)) (<= y 2.4e+161)))
         (- x z)
         (+ x (* y (- 1.0 (log y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e-144) {
		tmp = x + (y - z);
	} else if (y <= 1.25e-74) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 6e-31) {
		tmp = x - (log(y) * 0.5);
	} else if ((y <= 7800000000.0) || (!(y <= 3.8e+79) && (y <= 2.4e+161))) {
		tmp = x - z;
	} else {
		tmp = x + (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 <= 1.25d-144) then
        tmp = x + (y - z)
    else if (y <= 1.25d-74) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 6d-31) then
        tmp = x - (log(y) * 0.5d0)
    else if ((y <= 7800000000.0d0) .or. (.not. (y <= 3.8d+79)) .and. (y <= 2.4d+161)) then
        tmp = x - z
    else
        tmp = x + (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 <= 1.25e-144) {
		tmp = x + (y - z);
	} else if (y <= 1.25e-74) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 6e-31) {
		tmp = x - (Math.log(y) * 0.5);
	} else if ((y <= 7800000000.0) || (!(y <= 3.8e+79) && (y <= 2.4e+161))) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.25e-144:
		tmp = x + (y - z)
	elif y <= 1.25e-74:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 6e-31:
		tmp = x - (math.log(y) * 0.5)
	elif (y <= 7800000000.0) or (not (y <= 3.8e+79) and (y <= 2.4e+161)):
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.25e-144)
		tmp = Float64(x + Float64(y - z));
	elseif (y <= 1.25e-74)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 6e-31)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif ((y <= 7800000000.0) || (!(y <= 3.8e+79) && (y <= 2.4e+161)))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.25e-144)
		tmp = x + (y - z);
	elseif (y <= 1.25e-74)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 6e-31)
		tmp = x - (log(y) * 0.5);
	elseif ((y <= 7800000000.0) || (~((y <= 3.8e+79)) && (y <= 2.4e+161)))
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.25e-144], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-74], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 6e-31], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7800000000.0], And[N[Not[LessEqual[y, 3.8e+79]], $MachinePrecision], LessEqual[y, 2.4e+161]]], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 1.2499999999999999e-144

   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}} - \left(y - z\right)\right) \]

      2. pow3 99.7%

         \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}} - \left(y - z\right)\right) \]

      3. *-commutative 99.7%

         \[\leadsto x - \left({\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3} - \left(y - z\right)\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}} - \left(y - z\right)\right) \]

   6. Taylor expanded in y around inf 77.4%

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

      1. neg-mul-1 77.4%

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

      2. sub-neg 77.4%

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

   8. Simplified 77.4%

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

   if 1.2499999999999999e-144 < y < 1.25e-74

   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. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 88.0%

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   if 1.25e-74 < y < 5.99999999999999962e-31

   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 100.0%

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

   if 5.99999999999999962e-31 < y < 7.8e9 or 3.8000000000000002e79 < y < 2.3999999999999999e161

   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+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 79.5%

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

   if 7.8e9 < y < 3.8000000000000002e79 or 2.3999999999999999e161 < 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. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 83.2%

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

      1. sub-neg 83.2%

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

      2. mul-1-neg 83.2%

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

      3. log-rec 83.2%

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

      4. remove-double-neg 83.2%

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

      5. metadata-eval 83.2%

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

   6. Simplified 83.2%

      \[\leadsto x - \color{blue}{y \cdot \left(\log y + -1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-144}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-74}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-31}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 7800000000 \lor \neg \left(y \leq 3.8 \cdot 10^{+79}\right) \land y \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-145}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-32}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-145)
   (+ x (- y z))
   (if (<= y 5.6e-75)
     (- (* (log y) -0.5) z)
     (if (<= y 1.8e-32)
       (- x (* (log y) 0.5))
       (if (<= y 1.5e+197) (- x z) (* y (- 1.0 (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-145) {
		tmp = x + (y - z);
	} else if (y <= 5.6e-75) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.8e-32) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 1.5e+197) {
		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 <= 7d-145) then
        tmp = x + (y - z)
    else if (y <= 5.6d-75) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.8d-32) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 1.5d+197) 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 <= 7e-145) {
		tmp = x + (y - z);
	} else if (y <= 5.6e-75) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.8e-32) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 1.5e+197) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7e-145:
		tmp = x + (y - z)
	elif y <= 5.6e-75:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.8e-32:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 1.5e+197:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e-145)
		tmp = Float64(x + Float64(y - z));
	elseif (y <= 5.6e-75)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.8e-32)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 1.5e+197)
		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 <= 7e-145)
		tmp = x + (y - z);
	elseif (y <= 5.6e-75)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.8e-32)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 1.5e+197)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e-145], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e-75], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.8e-32], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+197], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 6.99999999999999994e-145

   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}} - \left(y - z\right)\right) \]

      2. pow3 99.7%

         \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}} - \left(y - z\right)\right) \]

      3. *-commutative 99.7%

         \[\leadsto x - \left({\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3} - \left(y - z\right)\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}} - \left(y - z\right)\right) \]

   6. Taylor expanded in y around inf 77.4%

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

      1. neg-mul-1 77.4%

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

      2. sub-neg 77.4%

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

   8. Simplified 77.4%

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

   if 6.99999999999999994e-145 < y < 5.59999999999999996e-75

   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. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 88.0%

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   if 5.59999999999999996e-75 < y < 1.79999999999999996e-32

   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 100.0%

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

   if 1.79999999999999996e-32 < y < 1.5000000000000001e197

   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. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 69.4%

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

   if 1.5000000000000001e197 < 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. associate-+l- 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 99.6%

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

   5. Taylor expanded in z around 0 88.2%

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

      1. distribute-lft-out-- 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. sub-neg 88.1%

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

      4. associate-+l+ 88.1%

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

      5. sub-neg 88.1%

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

      6. associate--r+ 88.1%

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

      7. +-commutative 88.1%

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

      8. distribute-rgt-in 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in y around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. log-rec 81.6%

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

      3. remove-double-neg 81.6%

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

   10. Simplified 81.6%

       \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-145}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-32}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7800000000 \lor \neg \left(y \leq 5.5 \cdot 10^{+79}\right) \land y \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 7800000000.0) (and (not (<= y 5.5e+79)) (<= y 2.4e+161)))
   (- (+ x (* (log y) -0.5)) z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 7800000000.0) || (!(y <= 5.5e+79) && (y <= 2.4e+161))) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + (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 <= 7800000000.0d0) .or. (.not. (y <= 5.5d+79)) .and. (y <= 2.4d+161)) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + (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 <= 7800000000.0) || (!(y <= 5.5e+79) && (y <= 2.4e+161))) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 7800000000.0) or (not (y <= 5.5e+79) and (y <= 2.4e+161)):
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 7800000000.0) || (!(y <= 5.5e+79) && (y <= 2.4e+161)))
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 7800000000.0) || (~((y <= 5.5e+79)) && (y <= 2.4e+161)))
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 7800000000.0], And[N[Not[LessEqual[y, 5.5e+79]], $MachinePrecision], LessEqual[y, 2.4e+161]]], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.8e9 or 5.50000000000000007e79 < y < 2.3999999999999999e161

   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+ 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. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 93.2%

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

   if 7.8e9 < y < 5.50000000000000007e79 or 2.3999999999999999e161 < 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. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 83.2%

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

      1. sub-neg 83.2%

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

      2. mul-1-neg 83.2%

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

      3. log-rec 83.2%

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

      4. remove-double-neg 83.2%

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

      5. metadata-eval 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7800000000 \lor \neg \left(y \leq 5.5 \cdot 10^{+79}\right) \land y \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 5: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+28} \lor \neg \left(y \leq 1.66 \cdot 10^{+93}\right) \land y \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 8.8e+28) (and (not (<= y 1.66e+93)) (<= y 2.2e+138)))
   (- (+ x (* (log y) -0.5)) z)
   (- (- y z) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 8.8e+28) || (!(y <= 1.66e+93) && (y <= 2.2e+138))) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (y - z) - (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 <= 8.8d+28) .or. (.not. (y <= 1.66d+93)) .and. (y <= 2.2d+138)) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (y - z) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= 8.8e+28) || (!(y <= 1.66e+93) && (y <= 2.2e+138))) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (y - z) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 8.8e+28) or (not (y <= 1.66e+93) and (y <= 2.2e+138)):
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (y - z) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 8.8e+28) || (!(y <= 1.66e+93) && (y <= 2.2e+138)))
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(y - z) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 8.8e+28) || (~((y <= 1.66e+93)) && (y <= 2.2e+138)))
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (y - z) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 8.8e+28], And[N[Not[LessEqual[y, 1.66e+93]], $MachinePrecision], LessEqual[y, 2.2e+138]]], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.79999999999999946e28 or 1.65999999999999999e93 < y < 2.2000000000000001e138

   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. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 95.4%

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

   if 8.79999999999999946e28 < y < 1.65999999999999999e93 or 2.2000000000000001e138 < 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. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. log-rec 99.7%

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

      4. remove-double-neg 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 87.4%

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

      1. associate--r+ 87.4%

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

   9. Simplified 87.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+28} \lor \neg \left(y \leq 1.66 \cdot 10^{+93}\right) \land y \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \]

Alternative 6: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-103}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e-103)
   (+ x (- y z))
   (if (<= y 1.7e-32)
     (- x (* (log y) 0.5))
     (if (<= y 1.5e+197) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e-103) {
		tmp = x + (y - z);
	} else if (y <= 1.7e-32) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 1.5e+197) {
		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 <= 3.2d-103) then
        tmp = x + (y - z)
    else if (y <= 1.7d-32) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 1.5d+197) 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 <= 3.2e-103) {
		tmp = x + (y - z);
	} else if (y <= 1.7e-32) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 1.5e+197) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.2e-103:
		tmp = x + (y - z)
	elif y <= 1.7e-32:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 1.5e+197:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e-103)
		tmp = Float64(x + Float64(y - z));
	elseif (y <= 1.7e-32)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 1.5e+197)
		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 <= 3.2e-103)
		tmp = x + (y - z);
	elseif (y <= 1.7e-32)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 1.5e+197)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.2e-103], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e-32], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+197], 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 < 3.19999999999999976e-103

   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.6%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}} - \left(y - z\right)\right) \]

      2. pow3 99.6%

         \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}} - \left(y - z\right)\right) \]

      3. *-commutative 99.6%

         \[\leadsto x - \left({\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3} - \left(y - z\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}} - \left(y - z\right)\right) \]

   6. Taylor expanded in y around inf 74.9%

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

      1. neg-mul-1 74.9%

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

      2. sub-neg 74.9%

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

   8. Simplified 74.9%

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

   if 3.19999999999999976e-103 < y < 1.69999999999999989e-32

   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in z around 0 83.6%

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

      1. distribute-lft-out-- 83.6%

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

      2. *-rgt-identity 83.6%

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

      3. sub-neg 83.6%

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

      4. associate-+l+ 83.6%

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

      5. sub-neg 83.6%

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

      6. associate--r+ 83.6%

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

      7. +-commutative 83.6%

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

      8. distribute-rgt-in 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in y around 0 83.6%

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

   if 1.69999999999999989e-32 < y < 1.5000000000000001e197

   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. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 69.4%

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

   if 1.5000000000000001e197 < 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. associate-+l- 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 99.6%

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

   5. Taylor expanded in z around 0 88.2%

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

      1. distribute-lft-out-- 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. sub-neg 88.1%

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

      4. associate-+l+ 88.1%

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

      5. sub-neg 88.1%

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

      6. associate--r+ 88.1%

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

      7. +-commutative 88.1%

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

      8. distribute-rgt-in 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in y around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. log-rec 81.6%

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

      3. remove-double-neg 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-103}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e-30)
   (- (+ x (* (log y) -0.5)) z)
   (+ x (- (- y z) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e-30) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y - z) - (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 <= 1.15d-30) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + ((y - z) - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e-30) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y - z) - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e-30:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + ((y - z) - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e-30)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y - z) - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e-30)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + ((y - z) - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e-30], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999992e-30

   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. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   if 1.14999999999999992e-30 < y

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-in 98.9%

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

      3. log-rec 98.9%

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

      4. remove-double-neg 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) - y \cdot \log y\right)\\ \end{array} \]

Alternative 8: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \left(1 - \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e-30)
   (- (+ 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 <= 1.15e-30) {
		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 <= 1.15d-30) 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 <= 1.15e-30) {
		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 <= 1.15e-30:
		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 <= 1.15e-30)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(x + 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 <= 1.15e-30)
		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, 1.15e-30], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999992e-30

   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. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   if 1.14999999999999992e-30 < y

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-in 98.9%

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

      3. log-rec 98.9%

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

      4. remove-double-neg 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in y around 0 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \left(1 - \log y\right)\right) - z\\ \end{array} \]

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

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(x + Float64(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[(x + N[(N[(y - z), $MachinePrecision] - 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%

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

   2. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 10: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e+197) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e+197) {
		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 <= 1.5d+197) 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 <= 1.5e+197) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.5e+197:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e+197)
		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 <= 1.5e+197)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.5e+197], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5000000000000001e197

   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. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 69.5%

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

   if 1.5000000000000001e197 < 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. associate-+l- 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 99.6%

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

   5. Taylor expanded in z around 0 88.2%

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

      1. distribute-lft-out-- 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. sub-neg 88.1%

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

      4. associate-+l+ 88.1%

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

      5. sub-neg 88.1%

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

      6. associate--r+ 88.1%

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

      7. +-commutative 88.1%

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

      8. distribute-rgt-in 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in y around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. log-rec 81.6%

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

      3. remove-double-neg 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 11: 47.6% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+29}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+54}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8e+29) x (if (<= x 3.1e+54) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e+29) {
		tmp = x;
	} else if (x <= 3.1e+54) {
		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 <= (-5.8d+29)) then
        tmp = x
    else if (x <= 3.1d+54) 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 <= -5.8e+29) {
		tmp = x;
	} else if (x <= 3.1e+54) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8e+29:
		tmp = x
	elif x <= 3.1e+54:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.8e+29)
		tmp = x;
	elseif (x <= 3.1e+54)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.8e+29)
		tmp = x;
	elseif (x <= 3.1e+54)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8e+29], x, If[LessEqual[x, 3.1e+54], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999999e29 or 3.0999999999999999e54 < x

   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+ 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. unsub-neg 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 65.4%

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

   if -5.7999999999999999e29 < x < 3.0999999999999999e54

   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. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.9%

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

   5. Taylor expanded in z around inf 41.3%

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

      1. mul-1-neg 41.3%

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

   7. Simplified 41.3%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+29}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+54}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 57.0% 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. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 56.7%

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

5. Final simplification 56.7%

   \[\leadsto x - z \]

Alternative 13: 29.5% 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. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 28.0%

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

5. Final simplification 28.0%

   \[\leadsto x \]

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 2023291 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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