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

Percentage Accurate: 76.8% → 99.7%

Time: 7.6s

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: 76.8% 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 80.0%

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

   1. add-cube-cbrt 80.0%

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

   2. log-prod 80.0%

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

   3. pow2 80.0%

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

   4. metadata-eval 80.0%

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

   5. log-pow 80.0%

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

   6. metadata-eval 80.0%

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

3. Applied egg-rr 80.0%

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

   1. distribute-lft1-in 80.0%

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

   2. metadata-eval 80.0%

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

   3. *-commutative 80.0%

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

5. Simplified 80.0%

   \[\leadsto x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot 3\right)} - z \]
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: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 4 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+305))) (- 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+305)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 4e+305)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 4e+305):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 4e+305))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 4e+305)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 4e+305]], $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 3.9999999999999998e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 9.5%

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

      1. remove-double-neg 9.5%

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

      2. sub0-neg 9.5%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 9.5%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 9.5%

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

      5. distribute-rgt-neg-in 9.5%

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

      6. neg-sub0 9.5%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 44.5%

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

      8. associate-+l- 44.5%

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

      9. neg-sub0 44.5%

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

      10. +-commutative 44.5%

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

      11. sub-neg 44.5%

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

      12. log-div 16.4%

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

      13. fma-udef 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in x around 0 51.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 3.9999999999999998e305

   1. Initial program 99.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]

Alternative 3: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e+169)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -4.9e-180)
     (- (* x (log (/ x y))) z)
     (if (<= x -4e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e+169) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.9e-180) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9.5d+169)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-4.9d-180)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-4d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e+169) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -4.9e-180) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e+169:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -4.9e-180:
		tmp = (x * math.log((x / y))) - z
	elif x <= -4e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e+169)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.9e-180)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -4e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e+169)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -4.9e-180)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -4e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e+169], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.9e-180], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.4999999999999995e169

   1. Initial program 62.6%

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

      1. remove-double-neg 62.6%

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

      2. sub0-neg 62.6%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 62.6%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 62.6%

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

      5. distribute-rgt-neg-in 62.6%

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

      6. neg-sub0 62.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 62.6%

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

      13. fma-udef 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 53.5%

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

   6. Simplified 53.5%

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

      1. frac-2neg 53.5%

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

      2. log-div 78.2%

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

   8. Applied egg-rr 78.2%

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

   if -9.4999999999999995e169 < x < -4.9000000000000003e-180

   1. Initial program 87.4%

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

   if -4.9000000000000003e-180 < x < -4.00000000000000013e-308

   1. Initial program 62.0%

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

      1. remove-double-neg 62.0%

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

      2. sub0-neg 62.0%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 62.0%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 62.0%

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

      5. distribute-rgt-neg-in 62.0%

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

      6. neg-sub0 62.0%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 62.0%

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

      13. fma-udef 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around 0 96.4%

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

   if -4.00000000000000013e-308 < x

   1. Initial program 83.7%

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

      1. log-div 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.35e-179)
   (- (* x (log (/ x y))) z)
   (if (<= x -1e-308) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.35e-179) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.35d-179)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-1d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.35e-179) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.35e-179:
		tmp = (x * math.log((x / y))) - z
	elif x <= -1e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.35e-179)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.35e-179)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -1e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.35e-179], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999994e-179

   1. Initial program 80.0%

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

   if -1.34999999999999994e-179 < x < -9.9999999999999991e-309

   1. Initial program 62.0%

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

      1. remove-double-neg 62.0%

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

      2. sub0-neg 62.0%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 62.0%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 62.0%

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

      5. distribute-rgt-neg-in 62.0%

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

      6. neg-sub0 62.0%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 62.0%

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

      13. fma-udef 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around 0 96.4%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 83.7%

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

      1. log-div 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

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 76.3%

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

      1. frac-2neg 76.3%

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

      2. log-div 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -9.999999999999969e-311 < y

   1. Initial program 83.7%

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

      1. log-div 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

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

Alternative 6: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+73} \lor \neg \left(x \leq 2.8 \cdot 10^{+39}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.5e+73) (not (<= x 2.8e+39))) (* (- x) (log (/ y x))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5e+73) || !(x <= 2.8e+39)) {
		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 <= (-2.5d+73)) .or. (.not. (x <= 2.8d+39))) 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 <= -2.5e+73) || !(x <= 2.8e+39)) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.5e+73) or not (x <= 2.8e+39):
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.5e+73) || !(x <= 2.8e+39))
		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 <= -2.5e+73) || ~((x <= 2.8e+39)))
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.5e+73], N[Not[LessEqual[x, 2.8e+39]], $MachinePrecision]], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999988e73 or 2.80000000000000001e39 < x

   1. Initial program 77.8%

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

      1. remove-double-neg 77.8%

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

      2. sub0-neg 77.8%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 77.8%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 77.8%

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

      5. distribute-rgt-neg-in 77.8%

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

      6. neg-sub0 77.8%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 57.1%

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

      8. associate-+l- 57.1%

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

      9. neg-sub0 57.1%

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

      10. +-commutative 57.1%

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

      11. sub-neg 57.1%

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

      12. log-div 79.0%

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

      13. fma-udef 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in x around inf 49.6%

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

      1. log-rec 49.6%

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

      2. sub-neg 49.6%

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

      3. log-div 66.3%

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

   6. Simplified 66.3%

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

   if -2.49999999999999988e73 < x < 2.80000000000000001e39

   1. Initial program 81.7%

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

      1. remove-double-neg 81.7%

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

      2. sub0-neg 81.7%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 81.7%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 81.7%

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

      5. distribute-rgt-neg-in 81.7%

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

      6. neg-sub0 81.7%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 45.4%

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

      8. associate-+l- 45.4%

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

      9. neg-sub0 45.4%

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

      10. +-commutative 45.4%

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

      11. sub-neg 45.4%

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

      12. log-div 80.7%

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

      13. fma-udef 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in x around 0 79.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+73} \lor \neg \left(x \leq 2.8 \cdot 10^{+39}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 50.4% 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 80.0%

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

   1. remove-double-neg 80.0%

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

   2. sub0-neg 80.0%

      \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

   3. associate--r- 80.0%

      \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

   4. neg-sub0 80.0%

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

   5. distribute-rgt-neg-in 80.0%

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

   6. neg-sub0 80.0%

      \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

   7. log-div 50.5%

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

   8. associate-+l- 50.5%

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

   9. neg-sub0 50.5%

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

   10. +-commutative 50.5%

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

   11. sub-neg 50.5%

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

   12. log-div 80.0%

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

   13. fma-udef 80.0%

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

3. Simplified 80.0%

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

4. Taylor expanded in x around 0 52.3%

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

5. Final simplification 52.3%

   \[\leadsto -z \]

Developer target: 88.6% 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 2023298 
(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))