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

Percentage Accurate: 77.3% → 99.5%

Time: 6.2s

Alternatives: 5

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: 77.3% 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: 99.5% 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 78.4%

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

   3. Taylor expanded in y around -inf 99.6%

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

      1. neg-mul-1 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. distribute-frac-neg2 99.6%

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

      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%

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

   if -4.999999999999985e-310 < y

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 99.4%

      \[\leadsto x \cdot \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)} - z \]
   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. sub-neg 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 99.5%

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

Alternative 2: 88.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+306)))
     (- (* x (fabs (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 <= 4e+306)) {
		tmp = (x * fabs(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 <= 4e+306)) {
		tmp = (x * Math.abs(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 <= 4e+306):
		tmp = (x * math.fabs(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 <= 4e+306))
		tmp = Float64(Float64(x * abs(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 <= 4e+306)))
		tmp = (x * abs(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, 4e+306]], $MachinePrecision]], N[(N[(x * N[Abs[N[Log[N[(y * x), $MachinePrecision]], $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 4.00000000000000007e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.4%

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

      1. add-cube-cbrt 6.4%

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

      2. associate-/l* 6.4%

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

      3. log-prod 67.8%

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

      4. pow2 67.8%

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

   4. Applied egg-rr 67.8%

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

      1. log-div 45.5%

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

      2. associate-+r- 45.5%

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

      3. log-prod 45.5%

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

      4. unpow2 45.5%

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

      5. add-cube-cbrt 45.5%

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

      6. add-sqr-sqrt 22.7%

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

      7. fma-neg 22.7%

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

      8. add-sqr-sqrt 22.7%

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

      9. sqrt-unprod 22.7%

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

      10. pow2 22.7%

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

   6. Applied egg-rr 55.3%

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

      1. unpow2 55.3%

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

      2. rem-sqrt-square 55.3%

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

   8. Simplified 55.3%

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

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

   1. Initial program 99.9%

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

4. Final simplification 89.9%

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

Alternative 3: 88.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+306)))
     (- (* 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 <= 4e+306)) {
		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 <= 4e+306)) {
		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 <= 4e+306):
		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 <= 4e+306))
		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 <= 4e+306)))
		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, 4e+306]], $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 4.00000000000000007e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.4%

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

      1. add-cube-cbrt 6.4%

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

      2. associate-/l* 6.4%

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

      3. log-prod 67.8%

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

      4. pow2 67.8%

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

   4. Applied egg-rr 67.8%

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

      1. log-div 45.5%

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

      2. associate-+r- 45.5%

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

      3. log-prod 45.5%

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

      4. unpow2 45.5%

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

      5. add-cube-cbrt 45.5%

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

      6. *-un-lft-identity 45.5%

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

      7. add-sqr-sqrt 22.7%

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

      8. fma-neg 22.7%

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

      9. *-commutative 22.7%

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

   6. Applied egg-rr 52.9%

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

      1. *-rgt-identity 52.9%

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

   8. Simplified 52.9%

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

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

   1. Initial program 99.9%

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

4. Final simplification 89.4%

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

Alternative 4: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-88} \lor \neg \left(z \leq 4 \cdot 10^{-13}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.2e-88) (not (<= z 4e-13))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.2e-88) || !(z <= 4e-13)) {
		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 <= (-4.2d-88)) .or. (.not. (z <= 4d-13))) 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 <= -4.2e-88) || !(z <= 4e-13)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.2e-88) or not (z <= 4e-13):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.2e-88) || !(z <= 4e-13))
		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 <= -4.2e-88) || ~((z <= 4e-13)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.2e-88], N[Not[LessEqual[z, 4e-13]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1999999999999999e-88 or 4.0000000000000001e-13 < z

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. neg-mul-1 75.8%

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

   5. Simplified 75.8%

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

   if -4.1999999999999999e-88 < z < 4.0000000000000001e-13

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0 68.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-88} \lor \neg \left(z \leq 4 \cdot 10^{-13}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% 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 79.0%

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

3. Taylor expanded in x around 0 54.4%

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

   1. neg-mul-1 54.4%

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

5. Simplified 54.4%

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

6. Final simplification 54.4%

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

Developer target: 88.7% 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 2024077 
(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))