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

Percentage Accurate: 77.7% → 99.7%

Time: 13.7s

Alternatives: 7

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: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. add-cube-cbrt 74.4%

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

   2. associate-*l* 74.4%

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

   3. log-prod 74.4%

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

   4. pow2 74.4%

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

   5. metadata-eval 74.4%

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

   6. log-pow 74.4%

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

   7. metadata-eval 74.4%

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

3. Applied egg-rr 74.4%

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

   1. distribute-rgt1-in 74.4%

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

   2. metadata-eval 74.4%

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

   3. *-commutative 74.4%

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

5. Simplified 74.4%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 5e+303) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 5e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -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) {
		tmp = -z;
	} else if (t_0 <= 5e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 5e+303:
		tmp = t_0 - z
	else:
		tmp = -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))
		tmp = Float64(-z);
	elseif (t_0 <= 5e+303)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-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)
		tmp = -z;
	elseif (t_0 <= 5e+303)
		tmp = t_0 - z;
	else
		tmp = -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[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 5e+303], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.9999999999999997e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 45.3%

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

      1. neg-mul-1 45.3%

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

   4. Simplified 45.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.9999999999999997e303

   1. Initial program 99.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 -2e+307)
     (* x (- (log (- x)) (log (- y))))
     (if (<= t_0 5e+303) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -2e+307) {
		tmp = x * (log(-x) - log(-y));
	} else if (t_0 <= 5e+303) {
		tmp = t_0 - z;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * log((x / y))
    if (t_0 <= (-2d+307)) then
        tmp = x * (log(-x) - log(-y))
    else if (t_0 <= 5d+303) then
        tmp = t_0 - z
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -2e+307) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (t_0 <= 5e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -2e+307:
		tmp = x * (math.log(-x) - math.log(-y))
	elif t_0 <= 5e+303:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= -2e+307)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (t_0 <= 5e+303)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-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 <= -2e+307)
		tmp = x * (log(-x) - log(-y));
	elseif (t_0 <= 5e+303)
		tmp = t_0 - z;
	else
		tmp = -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[LessEqual[t$95$0, -2e+307], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+303], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -1.99999999999999997e307

   1. Initial program 12.8%

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

      1. add-cube-cbrt 12.7%

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

      2. pow3 12.7%

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

   3. Applied egg-rr 12.7%

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

   4. Taylor expanded in z around 0 12.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log \left(\frac{x}{y}\right)\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 12.8%

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

      2. *-lft-identity 12.8%

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

   6. Simplified 12.8%

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

      1. frac-2neg 12.8%

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

      2. log-div 57.5%

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

   8. Applied egg-rr 57.5%

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

   if -1.99999999999999997e307 < (*.f64 x (log.f64 (/.f64 x y))) < 4.9999999999999997e303

   1. Initial program 99.8%

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

   if 4.9999999999999997e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.2%

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

   2. Taylor expanded in x around 0 47.7%

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

      1. neg-mul-1 47.7%

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

   4. Simplified 47.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -2 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-136}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.9e+115)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -2.6e-136)
     (- (* x (log (/ x y))) z)
     (if (<= x -2e-307) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.9e+115) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -2.6e-136) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-307) {
		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 <= (-6.9d+115)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-2.6d-136)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-307)) 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 <= -6.9e+115) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -2.6e-136) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.9e+115:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -2.6e-136:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-307:
		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 <= -6.9e+115)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -2.6e-136)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-307)
		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 <= -6.9e+115)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -2.6e-136)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-307)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.9e+115], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-136], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-307], (-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 < -6.89999999999999967e115

   1. Initial program 64.1%

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

      1. add-cube-cbrt 63.1%

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

      2. pow3 63.1%

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

   3. Applied egg-rr 63.1%

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

   4. Taylor expanded in z around 0 54.4%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log \left(\frac{x}{y}\right)\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 54.4%

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

      2. *-lft-identity 54.4%

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

   6. Simplified 54.4%

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

      1. frac-2neg 54.4%

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

      2. log-div 83.5%

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

   8. Applied egg-rr 83.5%

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

   if -6.89999999999999967e115 < x < -2.59999999999999997e-136

   1. Initial program 95.7%

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

   if -2.59999999999999997e-136 < x < -1.99999999999999982e-307

   1. Initial program 58.0%

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

   2. Taylor expanded in x around 0 83.9%

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

      1. neg-mul-1 83.9%

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

   4. Simplified 83.9%

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

   if -1.99999999999999982e-307 < x

   1. Initial program 75.5%

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

      1. log-div 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-136}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-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 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-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 <= -1e-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, -1e-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 < -9.999999999999969e-311

   1. Initial program 73.4%

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

      1. frac-2neg 39.7%

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

      2. log-div 54.5%

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

   3. Applied egg-rr 99.4%

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

   if -9.999999999999969e-311 < y

   1. Initial program 75.5%

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

      1. log-div 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+22}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e+22) (- z) (if (<= z 1.6e+16) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+22) {
		tmp = -z;
	} else if (z <= 1.6e+16) {
		tmp = x * log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.3d+22)) then
        tmp = -z
    else if (z <= 1.6d+16) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+22) {
		tmp = -z;
	} else if (z <= 1.6e+16) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e+22:
		tmp = -z
	elif z <= 1.6e+16:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e+22)
		tmp = Float64(-z);
	elseif (z <= 1.6e+16)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.3e+22)
		tmp = -z;
	elseif (z <= 1.6e+16)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e+22], (-z), If[LessEqual[z, 1.6e+16], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e22 or 1.6e16 < z

   1. Initial program 73.6%

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

   2. Taylor expanded in x around 0 77.3%

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

      1. neg-mul-1 77.3%

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

   4. Simplified 77.3%

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

   if -1.3e22 < z < 1.6e16

   1. Initial program 75.4%

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

      1. add-cube-cbrt 74.2%

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

      2. pow3 74.1%

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

   3. Applied egg-rr 74.1%

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

   4. Taylor expanded in z around 0 64.1%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log \left(\frac{x}{y}\right)\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 64.1%

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

      2. *-lft-identity 64.1%

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

   6. Simplified 64.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+22}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 49.6% 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 74.5%

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

2. Taylor expanded in x around 0 49.4%

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

   1. neg-mul-1 49.4%

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

4. Simplified 49.4%

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

5. Final simplification 49.4%

   \[\leadsto -z \]

Developer target: 88.8% 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 2023277 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

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

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