Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Percentage Accurate: 77.6% → 99.7%

Time: 11.7s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* (* x 3.0) (log (/ (cbrt x) (cbrt y)))) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * log((cbrt(x) / cbrt(y)))) - z;
}

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * Math.log((Math.cbrt(x) / Math.cbrt(y)))) - z;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * log(Float64(cbrt(x) / cbrt(y)))) - z)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. add-sqr-sqrt 37.6%

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

   2. pow2 37.6%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \log \left(\frac{x}{y}\right)}\right)}^{2}} - z \]

4. Applied egg-rr 37.6%

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

   1. unpow2 37.6%

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

   2. add-sqr-sqrt 75.7%

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

   3. frac-2neg 75.7%

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

   4. diff-log 47.8%

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

   5. diff-log 75.7%

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

   6. frac-2neg 75.7%

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

   7. add-cube-cbrt 75.7%

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

   8. unpow3 75.7%

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

   9. log-pow 75.6%

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

   10. associate-*r* 75.6%

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

6. Applied egg-rr 75.6%

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

   1. cbrt-div 99.6%

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

   2. div-inv 99.7%

      \[\leadsto \left(x \cdot 3\right) \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)} - z \]

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.6%

      \[\leadsto \left(x \cdot 3\right) \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot 1}{\sqrt[3]{y}}\right)} - z \]

   2. *-rgt-identity 99.6%

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

10. Simplified 99.6%

    \[\leadsto \left(x \cdot 3\right) \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} - z \]
11. Add Preprocessing

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 4e+306) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 4e+306) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 4e+306) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 4e+306:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 4e+306)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 4e+306)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 4e+306], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0

   1. Initial program 4.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.8%

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

      1. neg-mul-1 54.8%

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

   5. Simplified 54.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.00000000000000007e306

   1. Initial program 99.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   if 4.00000000000000007e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.8%

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

      1. add-cbrt-cube 5.8%

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

      2. pow3 5.8%

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

   4. Applied egg-rr 5.8%

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

   5. Taylor expanded in x around inf 60.7%

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

      1. distribute-lft-in 60.8%

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

      2. mul-1-neg 60.8%

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

      3. log-rec 60.8%

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

      4. remove-double-neg 60.8%

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

      5. +-commutative 60.8%

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

      6. distribute-lft-in 60.7%

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

      7. log-rec 60.7%

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

      8. neg-mul-1 60.7%

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

      9. neg-mul-1 60.7%

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

      10. sub-neg 60.7%

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

   7. Simplified 60.7%

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

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 4 \cdot 10^{+306}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+306))) (- z) (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 4e+306)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 4e+306)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 4e+306):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 4e+306))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 4e+306)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 4e+306]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.00000000000000007e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 45.8%

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

      1. neg-mul-1 45.8%

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

   5. Simplified 45.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.00000000000000007e306

   1. Initial program 99.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+306}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.1 \cdot 10^{-169}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e+138)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -7.1e-169)
     (- (* x (log (/ x y))) z)
     (if (<= x -1e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e+138) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -7.1e-169) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - 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 (x <= (-2.3d+138)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-7.1d-169)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-1d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e+138) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -7.1e-169) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.3e+138:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -7.1e-169:
		tmp = (x * math.log((x / y))) - z
	elif x <= -1e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e+138)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -7.1e-169)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e+138)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -7.1e-169)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -1e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e+138], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.1e-169], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.30000000000000008e138

   1. Initial program 64.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 62.5%

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

   4. Taylor expanded in y around -inf 92.2%

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

      1. metadata-eval 99.2%

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

      2. distribute-neg-frac 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. neg-mul-1 99.2%

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

      5. log-rec 99.2%

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

      6. sub-neg 99.2%

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

   6. Simplified 92.2%

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

   if -2.30000000000000008e138 < x < -7.0999999999999997e-169

   1. Initial program 97.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   if -7.0999999999999997e-169 < x < -9.9999999999999991e-309

   1. Initial program 62.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 84.9%

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

      1. neg-mul-1 84.9%

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

   5. Simplified 84.9%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 73.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

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

      1. log-rec 99.4%

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

      2. neg-mul-1 99.4%

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

      3. neg-mul-1 99.4%

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

      4. sub-neg 99.4%

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

   5. Simplified 99.4%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = ((x * log(x)) - (x * 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 <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = ((x * log(x)) - (x * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = ((x * math.log(x)) - (x * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = ((x * log(x)) - (x * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 78.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. distribute-neg-frac 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. neg-mul-1 99.4%

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

      5. log-rec 99.4%

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

      6. sub-neg 99.4%

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

   5. Simplified 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 73.4%

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

      1. *-un-lft-identity 73.4%

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

      2. add-cube-cbrt 73.4%

         \[\leadsto x \cdot \log \left(\frac{1 \cdot x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right) - z \]

      3. times-frac 73.4%

         \[\leadsto x \cdot \log \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)} - z \]

      4. log-prod 87.6%

         \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z \]

      5. pow2 87.6%

         \[\leadsto x \cdot \left(\log \left(\frac{1}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right) - z \]

   4. Applied egg-rr 87.6%

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

      1. +-commutative 87.6%

         \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{x}{\sqrt[3]{y}}\right) + \log \left(\frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right)\right)} - z \]

      2. log-rec 87.6%

         \[\leadsto x \cdot \left(\log \left(\frac{x}{\sqrt[3]{y}}\right) + \color{blue}{\left(-\log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)}\right) - z \]

      3. unsub-neg 87.6%

         \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{x}{\sqrt[3]{y}}\right) - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)} - z \]

   6. Simplified 87.6%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{x}{\sqrt[3]{y}}\right) - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)} - z \]
   7. Step-by-step derivation

      1. diff-log 73.4%

         \[\leadsto x \cdot \color{blue}{\log \left(\frac{\frac{x}{\sqrt[3]{y}}}{{\left(\sqrt[3]{y}\right)}^{2}}\right)} - z \]

      2. associate-/l/ 73.4%

         \[\leadsto x \cdot \log \color{blue}{\left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \sqrt[3]{y}}\right)} - z \]

      3. unpow2 73.4%

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

      4. add-cube-cbrt 73.4%

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

      5. diff-log 99.4%

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

      6. sub-neg 99.4%

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

      7. distribute-rgt-in 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - 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 <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 78.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. distribute-neg-frac 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. neg-mul-1 99.4%

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

      5. log-rec 99.4%

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

      6. sub-neg 99.4%

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

   5. Simplified 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 73.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

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

      1. log-rec 99.4%

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

      2. neg-mul-1 99.4%

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

      3. neg-mul-1 99.4%

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

      4. sub-neg 99.4%

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

   5. Simplified 99.4%

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

Alternative 7: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-81} \lor \neg \left(x \leq 1.16 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.5e-81) (not (<= x 1.16e-17))) (* x (- (log (/ y x)))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e-81) || !(x <= 1.16e-17)) {
		tmp = x * -log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-5.5d-81)) .or. (.not. (x <= 1.16d-17))) then
        tmp = x * -log((y / x))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e-81) || !(x <= 1.16e-17)) {
		tmp = x * -Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.5e-81) or not (x <= 1.16e-17):
		tmp = x * -math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.5e-81) || !(x <= 1.16e-17))
		tmp = Float64(x * Float64(-log(Float64(y / x))));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.5e-81) || ~((x <= 1.16e-17)))
		tmp = x * -log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.5e-81], N[Not[LessEqual[x, 1.16e-17]], $MachinePrecision]], N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -5.50000000000000026e-81 or 1.16e-17 < x

   1. Initial program 77.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 60.3%

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

      1. clear-num 60.3%

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

      2. neg-log 61.0%

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

   5. Applied egg-rr 61.0%

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

   if -5.50000000000000026e-81 < x < 1.16e-17

   1. Initial program 73.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.4%

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

      1. neg-mul-1 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-81} \lor \neg \left(x \leq 1.16 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-81} \lor \neg \left(x \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.8e-81) (not (<= x 3.4e-17))) (* x (log (/ x y))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.8e-81) || !(x <= 3.4e-17)) {
		tmp = x * log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-7.8d-81)) .or. (.not. (x <= 3.4d-17))) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.8e-81) || !(x <= 3.4e-17)) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.8e-81) or not (x <= 3.4e-17):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.8e-81) || !(x <= 3.4e-17))
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.8e-81) || ~((x <= 3.4e-17)))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.8e-81], N[Not[LessEqual[x, 3.4e-17]], $MachinePrecision]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -7.7999999999999997e-81 or 3.3999999999999998e-17 < x

   1. Initial program 77.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 60.3%

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

   if -7.7999999999999997e-81 < x < 3.3999999999999998e-17

   1. Initial program 73.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.4%

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

      1. neg-mul-1 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-81} \lor \neg \left(x \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around 0 45.5%

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

   1. neg-mul-1 45.5%

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

5. Simplified 45.5%

   \[\leadsto \color{blue}{-z} \]
6. Add Preprocessing

Alternative 10: 2.2% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 75.7%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around 0 45.5%

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

   1. neg-mul-1 45.5%

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

5. Simplified 45.5%

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

   1. neg-sub0 45.5%

      \[\leadsto \color{blue}{0 - z} \]

   2. sub-neg 45.5%

      \[\leadsto \color{blue}{0 + \left(-z\right)} \]

   3. add-sqr-sqrt 21.3%

      \[\leadsto 0 + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

   4. sqrt-unprod 12.9%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}} \]

   5. sqr-neg 12.9%

      \[\leadsto 0 + \sqrt{\color{blue}{z \cdot z}} \]

   6. sqrt-unprod 1.4%

      \[\leadsto 0 + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   7. add-sqr-sqrt 2.5%

      \[\leadsto 0 + \color{blue}{z} \]

7. Applied egg-rr 2.5%

   \[\leadsto \color{blue}{0 + z} \]
8. Step-by-step derivation

   1. +-lft-identity 2.5%

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

9. Simplified 2.5%

   \[\leadsto \color{blue}{z} \]
10. Add Preprocessing

Developer Target 1: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - 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 < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024177 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7595077799083773/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z)))

  (- (* x (log (/ x y))) z))