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

Percentage Accurate: 76.9% → 99.8%

Time: 9.1s

Alternatives: 8

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: 76.9% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{-x}}{\sqrt{-y}}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311)
   (- (* x (* 2.0 (log (/ (sqrt (- x)) (sqrt (- y)))))) z)
   (- (* x (* 2.0 (log (/ (sqrt x) (sqrt y))))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = (x * (2.0 * log((sqrt(-x) / sqrt(-y))))) - z;
	} else {
		tmp = (x * (2.0 * log((sqrt(x) / sqrt(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-311)) then
        tmp = (x * (2.0d0 * log((sqrt(-x) / sqrt(-y))))) - z
    else
        tmp = (x * (2.0d0 * log((sqrt(x) / sqrt(y))))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = (x * (2.0 * Math.log((Math.sqrt(-x) / Math.sqrt(-y))))) - z;
	} else {
		tmp = (x * (2.0 * Math.log((Math.sqrt(x) / Math.sqrt(y))))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-311:
		tmp = (x * (2.0 * math.log((math.sqrt(-x) / math.sqrt(-y))))) - z
	else:
		tmp = (x * (2.0 * math.log((math.sqrt(x) / math.sqrt(y))))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-311)
		tmp = Float64(Float64(x * Float64(2.0 * log(Float64(sqrt(Float64(-x)) / sqrt(Float64(-y)))))) - z);
	else
		tmp = Float64(Float64(x * Float64(2.0 * log(Float64(sqrt(x) / sqrt(y))))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-311)
		tmp = (x * (2.0 * log((sqrt(-x) / sqrt(-y))))) - z;
	else
		tmp = (x * (2.0 * log((sqrt(x) / sqrt(y))))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-311], N[(N[(x * N[(2.0 * N[Log[N[(N[Sqrt[(-x)], $MachinePrecision] / N[Sqrt[(-y)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(2.0 * N[Log[N[(N[Sqrt[x], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000023e-311

   1. Initial program 84.8%

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

      1. add-sqr-sqrt 84.8%

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

      2. log-prod 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. count-2 84.8%

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

   6. Simplified 84.8%

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

      1. frac-2neg 84.8%

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

      2. sqrt-div 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -5.00000000000023e-311 < y

   1. Initial program 85.1%

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

      1. add-sqr-sqrt 85.1%

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

      2. log-prod 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. count-2 85.1%

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

   6. Simplified 85.1%

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

      1. sqrt-div 99.8%

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

      2. div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{-x}}{\sqrt{-y}}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% 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 8.5%

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

      1. remove-double-neg 8.5%

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

      2. sub-neg 8.5%

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

      3. distribute-neg-in 8.5%

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

      4. sub-neg 8.5%

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

      5. distribute-rgt-neg-in 8.5%

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

      6. fma-neg 8.5%

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

      7. log-div 39.1%

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

      8. sub-neg 39.1%

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

      9. distribute-neg-in 39.1%

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

      10. remove-double-neg 39.1%

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

      11. +-commutative 39.1%

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

      12. sub-neg 39.1%

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

      13. log-div 12.6%

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

      14. remove-double-neg 12.6%

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

   3. Simplified 12.6%

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

   5. Taylor expanded in x around 0 44.4%

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

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

   1. Initial program 92.5%

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

4. Final simplification 88.2%

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

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (* 2.0 (log (/ (sqrt x) (sqrt y))))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (2.0 * log((sqrt(x) / sqrt(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-311)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (2.0d0 * log((sqrt(x) / sqrt(y))))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (2.0 * Math.log((Math.sqrt(x) / Math.sqrt(y))))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-311:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (2.0 * math.log((math.sqrt(x) / math.sqrt(y))))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-311)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(2.0 * log(Float64(sqrt(x) / sqrt(y))))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-311)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (2.0 * log((sqrt(x) / sqrt(y))))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-311], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(2.0 * N[Log[N[(N[Sqrt[x], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000023e-311

   1. Initial program 84.8%

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

      1. frac-2neg 84.8%

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

      2. log-div 99.3%

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

   4. Applied egg-rr 99.3%

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

   if -5.00000000000023e-311 < y

   1. Initial program 85.1%

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

      1. add-sqr-sqrt 85.1%

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

      2. log-prod 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. count-2 85.1%

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

   6. Simplified 85.1%

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

      1. sqrt-div 99.8%

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

      2. div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\sqrt{\frac{x}{y}}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e-301)
   (- (* x (* 2.0 (log (sqrt (/ x y))))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e-301) {
		tmp = (x * (2.0 * log(sqrt((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 <= 1.9d-301) then
        tmp = (x * (2.0d0 * log(sqrt((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 <= 1.9e-301) {
		tmp = (x * (2.0 * Math.log(Math.sqrt((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 <= 1.9e-301:
		tmp = (x * (2.0 * math.log(math.sqrt((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 <= 1.9e-301)
		tmp = Float64(Float64(x * Float64(2.0 * log(sqrt(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 <= 1.9e-301)
		tmp = (x * (2.0 * log(sqrt((x / y))))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.9e-301], N[(N[(x * N[(2.0 * N[Log[N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]], $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 < 1.89999999999999998e-301

   1. Initial program 84.9%

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

      1. add-sqr-sqrt 84.9%

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

      2. log-prod 84.9%

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

   4. Applied egg-rr 84.9%

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

      1. count-2 84.9%

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

   6. Simplified 84.9%

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

   if 1.89999999999999998e-301 < y

   1. Initial program 85.0%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\sqrt{\frac{x}{y}}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311)
   (- (* 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-311) {
		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-311)) 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-311) {
		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-311:
		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-311)
		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-311)
		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-311], 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 < -5.00000000000023e-311

   1. Initial program 84.8%

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

      1. frac-2neg 84.8%

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

      2. log-div 99.3%

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

   4. Applied egg-rr 99.3%

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

   if -5.00000000000023e-311 < y

   1. Initial program 85.1%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e-301)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e-301) {
		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 <= 1.9d-301) 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 <= 1.9e-301) {
		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 <= 1.9e-301:
		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 <= 1.9e-301)
		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 <= 1.9e-301)
		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[LessEqual[y, 1.9e-301], 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]]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999998e-301

   1. Initial program 84.9%

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

   if 1.89999999999999998e-301 < y

   1. Initial program 85.0%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+91} \lor \neg \left(x \leq 8 \cdot 10^{-24}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e+91) (not (<= x 8e-24))) (* (- x) (log (/ y x))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e+91) || !(x <= 8e-24)) {
		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 <= (-5.8d+91)) .or. (.not. (x <= 8d-24))) 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 <= -5.8e+91) || !(x <= 8e-24)) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.8e+91) or not (x <= 8e-24):
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.8e+91) || !(x <= 8e-24))
		tmp = Float64(Float64(-x) * 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 <= -5.8e+91) || ~((x <= 8e-24)))
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.8e+91], N[Not[LessEqual[x, 8e-24]], $MachinePrecision]], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -5.80000000000000028e91 or 7.99999999999999939e-24 < x

   1. Initial program 83.8%

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

      1. remove-double-neg 83.8%

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

      2. sub-neg 83.8%

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

      3. distribute-neg-in 83.8%

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

      4. sub-neg 83.8%

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

      5. distribute-rgt-neg-in 83.8%

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

      6. fma-neg 83.8%

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

      7. log-div 59.9%

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

      8. sub-neg 59.9%

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

      9. distribute-neg-in 59.9%

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

      10. remove-double-neg 59.9%

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

      11. +-commutative 59.9%

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

      12. sub-neg 59.9%

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

      13. log-div 84.9%

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

      14. remove-double-neg 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around inf 44.6%

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

      1. log-rec 44.6%

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

      2. sub-neg 44.6%

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

      3. log-div 67.2%

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

   7. Simplified 67.2%

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

   if -5.80000000000000028e91 < x < 7.99999999999999939e-24

   1. Initial program 86.1%

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

      1. remove-double-neg 86.1%

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

      2. sub-neg 86.1%

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

      3. distribute-neg-in 86.1%

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

      4. sub-neg 86.1%

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

      5. distribute-rgt-neg-in 86.1%

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

      6. fma-neg 86.1%

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

      7. log-div 36.7%

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

      8. sub-neg 36.7%

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

      9. distribute-neg-in 36.7%

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

      10. remove-double-neg 36.7%

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

      11. +-commutative 36.7%

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

      12. sub-neg 36.7%

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

      13. log-div 81.9%

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

      14. remove-double-neg 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 75.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+91} \lor \neg \left(x \leq 8 \cdot 10^{-24}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% 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 85.0%

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

   1. remove-double-neg 85.0%

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

   2. sub-neg 85.0%

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

   3. distribute-neg-in 85.0%

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

   4. sub-neg 85.0%

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

   5. distribute-rgt-neg-in 85.0%

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

   6. fma-neg 85.0%

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

   7. log-div 48.6%

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

   8. sub-neg 48.6%

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

   9. distribute-neg-in 48.6%

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

   10. remove-double-neg 48.6%

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

   11. +-commutative 48.6%

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

   12. sub-neg 48.6%

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

   13. log-div 83.5%

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

   14. remove-double-neg 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in x around 0 47.4%

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

6. Final simplification 47.4%

   \[\leadsto -z \]
7. Add Preprocessing

Developer target: 88.3% 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 2024047 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

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