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

Percentage Accurate: 77.2% → 98.8%

Time: 10.9s

Alternatives: 9

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: 77.2% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (-
    (* x (+ (log (/ (pow (cbrt x) 2.0) (sqrt y))) (log (/ (cbrt x) (sqrt y)))))
    z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log((pow(cbrt(x), 2.0) / sqrt(y))) + log((cbrt(x) / sqrt(y))))) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log((Math.pow(Math.cbrt(x), 2.0) / Math.sqrt(y))) + Math.log((Math.cbrt(x) / Math.sqrt(y))))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(Float64((cbrt(x) ^ 2.0) / sqrt(y))) + log(Float64(cbrt(x) / sqrt(y))))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 75.5%

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

   3. Taylor expanded in y around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. neg-mul-1 99.6%

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

      5. log-rec 99.6%

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

      6. sub-neg 99.6%

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

   5. Simplified 99.6%

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

   if -3.999999999999988e-310 < y

   1. Initial program 78.4%

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

      1. add-cube-cbrt 78.4%

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

      2. add-sqr-sqrt 78.4%

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

      3. times-frac 78.4%

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

      4. log-prod 99.7%

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

      5. pow2 99.7%

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

   4. Applied egg-rr 99.7%

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

Alternative 2: 82.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 INFINITY)))
     (- (* x (log (* y x))) 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 <= ((double) INFINITY))) {
		tmp = (x * log((y * x))) - 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 <= Double.POSITIVE_INFINITY)) {
		tmp = (x * Math.log((y * x))) - 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 <= math.inf):
		tmp = (x * math.log((y * x))) - 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 <= Inf))
		tmp = Float64(Float64(x * log(Float64(y * x))) - 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 <= Inf)))
		tmp = (x * log((y * x))) - 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, Infinity]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 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 +inf.0 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.9%

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

      1. clear-num 4.9%

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

      2. neg-log 4.9%

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

   4. Applied egg-rr 4.9%

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

      1. *-commutative 4.9%

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

      2. fma-neg 4.9%

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

      3. neg-log 4.9%

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

      4. clear-num 4.9%

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

      5. div-inv 4.9%

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

      6. add-exp-log 1.5%

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

      7. rec-exp 1.5%

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

      8. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\color{blue}{\sqrt{-\log y} \cdot \sqrt{-\log y}}}\right), x, -z\right) \]

      9. sqrt-unprod 33.7%

         \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\color{blue}{\sqrt{\left(-\log y\right) \cdot \left(-\log y\right)}}}\right), x, -z\right) \]

      10. sqr-neg 33.7%

          \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\sqrt{\color{blue}{\log y \cdot \log y}}}\right), x, -z\right) \]

      11. sqrt-unprod 33.7%

          \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\color{blue}{\sqrt{\log y} \cdot \sqrt{\log y}}}\right), x, -z\right) \]

      12. add-sqr-sqrt 33.7%

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

      13. add-exp-log 48.1%

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

   6. Applied egg-rr 48.1%

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

      1. fma-undefine 48.1%

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

      2. unsub-neg 48.1%

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

   8. Simplified 48.1%

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

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

   1. Initial program 85.4%

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

4. Final simplification 81.5%

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

Alternative 3: 82.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 INFINITY))) (- 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 <= ((double) INFINITY))) {
		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 <= Double.POSITIVE_INFINITY)) {
		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 <= math.inf):
		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 <= Inf))
		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 <= Inf)))
		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, Infinity]], $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 +inf.0 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0 46.5%

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

      1. neg-mul-1 46.5%

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

   5. Simplified 46.5%

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

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

   1. Initial program 85.4%

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

4. Final simplification 81.3%

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

Alternative 4: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \log \left(y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(-t\_1\right) - z\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (log (* y x))))
   (if (<= t_0 (- INFINITY))
     (- (* x (- t_1)) z)
     (if (<= t_0 INFINITY) (- t_0 z) (- (* x t_1) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = log((y * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * -t_1) - z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 - z;
	} else {
		tmp = (x * t_1) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = Math.log((y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * -t_1) - z;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 - z;
	} else {
		tmp = (x * t_1) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = math.log((y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * -t_1) - z
	elif t_0 <= math.inf:
		tmp = t_0 - z
	else:
		tmp = (x * t_1) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = log(Float64(y * x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(-t_1)) - z);
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(Float64(x * t_1) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = log((y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * -t_1) - z;
	elseif (t_0 <= Inf)
		tmp = t_0 - z;
	else
		tmp = (x * t_1) - 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]}, Block[{t$95$1 = N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * (-t$95$1)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 - z), $MachinePrecision], N[(N[(x * t$95$1), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.9%

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

      1. clear-num 4.9%

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

      2. neg-log 4.9%

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

   4. Applied egg-rr 4.9%

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

      1. add-exp-log 4.9%

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

      2. add-sqr-sqrt 1.5%

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

      3. sqrt-unprod 2.0%

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

      4. sqr-neg 2.0%

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

      5. sqrt-unprod 0.5%

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

      6. add-sqr-sqrt 1.1%

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

      7. rec-exp 1.1%

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

      8. add-exp-log 1.1%

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

      9. associate-/r/ 1.1%

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

      10. add-exp-log 0.6%

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

      11. rec-exp 0.6%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 35.2%

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

      14. sqr-neg 35.2%

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

      15. sqrt-unprod 35.2%

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

      16. add-sqr-sqrt 35.2%

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

      17. add-exp-log 50.6%

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

   6. Applied egg-rr 50.6%

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

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

   1. Initial program 85.4%

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

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

   1. Initial program 76.9%

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

      1. clear-num 75.9%

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

      2. neg-log 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. fma-neg 76.6%

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

      3. neg-log 75.9%

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

      4. clear-num 76.9%

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

      5. div-inv 76.9%

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

      6. add-exp-log 37.9%

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

      7. rec-exp 37.9%

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

      8. add-sqr-sqrt 20.2%

         \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\color{blue}{\sqrt{-\log y} \cdot \sqrt{-\log y}}}\right), x, -z\right) \]

      9. sqrt-unprod 30.0%

         \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\color{blue}{\sqrt{\left(-\log y\right) \cdot \left(-\log y\right)}}}\right), x, -z\right) \]

      10. sqr-neg 30.0%

          \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\sqrt{\color{blue}{\log y \cdot \log y}}}\right), x, -z\right) \]

      11. sqrt-unprod 9.8%

          \[\leadsto \mathsf{fma}\left(\log \left(x \cdot e^{\color{blue}{\sqrt{\log y} \cdot \sqrt{\log y}}}\right), x, -z\right) \]

      12. add-sqr-sqrt 17.8%

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

      13. add-exp-log 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. fma-undefine 39.1%

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

      2. unsub-neg 39.1%

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

   8. Simplified 39.1%

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

4. Final simplification 81.7%

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

Alternative 5: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.7e+171)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.8e-163)
     (- (* x (log (/ x y))) z)
     (if (<= x -4e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.7e+171) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.8e-163) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -4e-309) {
		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 <= (-3.7d+171)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.8d-163)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-4d-309)) 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 <= -3.7e+171) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.8e-163) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -4e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.7e+171:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.8e-163:
		tmp = (x * math.log((x / y))) - z
	elif x <= -4e-309:
		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 <= -3.7e+171)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.8e-163)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -4e-309)
		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 <= -3.7e+171)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.8e-163)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -4e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.7e+171], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-163], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-309], (-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 < -3.69999999999999998e171

   1. Initial program 66.7%

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

   3. Taylor expanded in z around 0 63.7%

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

   4. Taylor expanded in y around -inf 90.9%

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

      1. metadata-eval 98.8%

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

      2. distribute-neg-frac 98.8%

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

      3. distribute-frac-neg2 98.8%

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

      4. neg-mul-1 98.8%

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

      5. log-rec 98.8%

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

      6. sub-neg 98.8%

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

   6. Simplified 90.9%

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

   if -3.69999999999999998e171 < x < -1.7999999999999999e-163

   1. Initial program 86.7%

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

   if -1.7999999999999999e-163 < x < -3.9999999999999977e-309

   1. Initial program 61.6%

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

   3. Taylor expanded in x around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   5. Simplified 91.5%

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

   if -3.9999999999999977e-309 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. neg-mul-1 99.6%

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

      3. neg-mul-1 99.6%

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

      4. sub-neg 99.6%

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

   5. Simplified 99.6%

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

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-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 <= -4e-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 <= (-4d-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 <= -4e-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 <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 75.5%

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

   3. Taylor expanded in y around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. neg-mul-1 99.6%

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

      5. log-rec 99.6%

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

      6. sub-neg 99.6%

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

   5. Simplified 99.6%

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

   if -3.999999999999988e-310 < y

   1. Initial program 78.4%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. neg-mul-1 99.6%

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

      3. neg-mul-1 99.6%

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

      4. sub-neg 99.6%

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

   5. Simplified 99.6%

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

Alternative 7: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-56} \lor \neg \left(z \leq 1.3 \cdot 10^{-22}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.1e-56) (not (<= z 1.3e-22))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.1e-56) || !(z <= 1.3e-22)) {
		tmp = -z;
	} else {
		tmp = x * log((x / 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 ((z <= (-6.1d-56)) .or. (.not. (z <= 1.3d-22))) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.1e-56) || !(z <= 1.3e-22)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.1e-56) or not (z <= 1.3e-22):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.1e-56) || !(z <= 1.3e-22))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.1e-56) || ~((z <= 1.3e-22)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.1e-56], N[Not[LessEqual[z, 1.3e-22]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.0999999999999998e-56 or 1.3e-22 < z

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0 75.2%

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

      1. neg-mul-1 75.2%

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

   5. Simplified 75.2%

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

   if -6.0999999999999998e-56 < z < 1.3e-22

   1. Initial program 77.5%

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

   3. Taylor expanded in z around 0 69.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-56} \lor \neg \left(z \leq 1.3 \cdot 10^{-22}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% 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 76.9%

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

3. Taylor expanded in x around 0 48.3%

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

   1. neg-mul-1 48.3%

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

5. Simplified 48.3%

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

Alternative 9: 2.3% 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 76.9%

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

3. Taylor expanded in x around 0 48.3%

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

   1. neg-mul-1 48.3%

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

5. Simplified 48.3%

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

   1. neg-sub0 48.3%

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

   2. sub-neg 48.3%

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

   3. add-sqr-sqrt 20.2%

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

   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 1.5%

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

   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 2024138 
(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))