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

Percentage Accurate: 77.5% → 99.5%

Time: 15.0s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% 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

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 74.7%

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

   3. Taylor expanded in y around -inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-neg-frac 99.4%

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

      4. distribute-frac-neg2 99.4%

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

      5. log-rec 99.4%

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

      6. sub-neg 99.4%

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

   5. Simplified 99.4%

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

   if -1.999999999999994e-310 < y

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

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

Alternative 2: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+221}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+221)))
     (- (- z) (* x (log (* y x))))
     (- 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 <= 5e+221)) {
		tmp = -z - (x * log((y * x)));
	} 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 <= 5e+221)) {
		tmp = -z - (x * Math.log((y * x)));
	} 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 <= 5e+221):
		tmp = -z - (x * math.log((y * x)))
	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 <= 5e+221))
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y * x))));
	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 <= 5e+221)))
		tmp = -z - (x * log((y * x)));
	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, 5e+221]], $MachinePrecision]], N[((-z) - N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 5.0000000000000002e221 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 12.9%

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

   3. Taylor expanded in z around inf 10.2%

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

      1. sub-neg 10.2%

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

      2. associate-/l* 10.2%

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

      3. metadata-eval 10.2%

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

   5. Simplified 10.2%

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

      1. clear-num 12.9%

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

      2. log-div 16.6%

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

      3. metadata-eval 16.6%

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

   7. Applied egg-rr 13.9%

      \[\leadsto z \cdot \left(x \cdot \frac{\color{blue}{0 - \log \left(\frac{y}{x}\right)}}{z} + -1\right) \]
   8. Step-by-step derivation

      1. neg-sub0 16.6%

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

   9. Simplified 13.9%

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

       1. distribute-frac-neg 13.9%

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

       2. distribute-frac-neg2 13.9%

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

       3. div-inv 13.9%

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

   11. Applied egg-rr 47.0%

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

   12. Taylor expanded in z around 0 47.6%

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

       1. distribute-lft-out 47.6%

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

   14. Simplified 47.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000002e221

   1. Initial program 99.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+221}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(y \cdot x\right)\\ t_1 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\frac{z}{t\_0}} + -1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+221}:\\ \;\;\;\;t\_1 - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (* y x))) (t_1 (* x (log (/ x y)))))
   (if (<= t_1 (- INFINITY))
     (* z (+ (/ x (/ z t_0)) -1.0))
     (if (<= t_1 5e+221) (- t_1 z) (- (- z) (* x t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = log((y * x));
	double t_1 = x * log((x / y));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * ((x / (z / t_0)) + -1.0);
	} else if (t_1 <= 5e+221) {
		tmp = t_1 - z;
	} else {
		tmp = -z - (x * t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((y * x));
	double t_1 = x * Math.log((x / y));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z * ((x / (z / t_0)) + -1.0);
	} else if (t_1 <= 5e+221) {
		tmp = t_1 - z;
	} else {
		tmp = -z - (x * t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((y * x))
	t_1 = x * math.log((x / y))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z * ((x / (z / t_0)) + -1.0)
	elif t_1 <= 5e+221:
		tmp = t_1 - z
	else:
		tmp = -z - (x * t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = log(Float64(y * x))
	t_1 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(Float64(x / Float64(z / t_0)) + -1.0));
	elseif (t_1 <= 5e+221)
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(Float64(-z) - Float64(x * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((y * x));
	t_1 = x * log((x / y));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z * ((x / (z / t_0)) + -1.0);
	elseif (t_1 <= 5e+221)
		tmp = t_1 - z;
	else
		tmp = -z - (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+221], N[(t$95$1 - z), $MachinePrecision], N[((-z) - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.4%

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

   3. Taylor expanded in z around inf 11.4%

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

      1. sub-neg 11.4%

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

      2. associate-/l* 11.4%

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

      3. metadata-eval 11.4%

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

   5. Simplified 11.4%

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

      1. clear-num 11.4%

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

      2. associate-/r/ 11.4%

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

   7. Applied egg-rr 11.4%

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

      1. *-commutative 11.4%

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

      2. clear-num 11.4%

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

      3. neg-log 12.6%

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

      4. div-inv 12.6%

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

      5. clear-num 12.6%

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

      6. un-div-inv 12.6%

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

      7. neg-log 11.4%

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

      8. clear-num 11.4%

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

      9. log-div 37.9%

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

      10. sub-neg 37.9%

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

      11. add-log-exp 37.9%

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

      12. sum-log 1.2%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 37.9%

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

      15. sqr-neg 37.9%

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

      16. sqrt-unprod 37.9%

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

      17. add-sqr-sqrt 37.9%

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

      18. add-exp-log 50.4%

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

   9. Applied egg-rr 50.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000002e221

   1. Initial program 99.5%

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

   if 5.0000000000000002e221 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 13.9%

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

   3. Taylor expanded in z around inf 9.4%

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

      1. sub-neg 9.4%

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

      2. associate-/l* 9.4%

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

      3. metadata-eval 9.4%

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

   5. Simplified 9.4%

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

      1. clear-num 13.9%

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

      2. log-div 19.4%

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

      3. metadata-eval 19.4%

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

   7. Applied egg-rr 14.9%

      \[\leadsto z \cdot \left(x \cdot \frac{\color{blue}{0 - \log \left(\frac{y}{x}\right)}}{z} + -1\right) \]
   8. Step-by-step derivation

      1. neg-sub0 19.4%

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

   9. Simplified 14.9%

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

       1. distribute-frac-neg 14.9%

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

       2. distribute-frac-neg2 14.9%

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

       3. div-inv 14.9%

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

   11. Applied egg-rr 47.4%

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

   12. Taylor expanded in z around 0 48.0%

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

       1. distribute-lft-out 48.0%

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

   14. Simplified 48.0%

       \[\leadsto \color{blue}{-1 \cdot \left(z + x \cdot \log \left(x \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\frac{z}{\log \left(y \cdot x\right)}} + -1\right)\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+221}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+244}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+244))) (- 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 <= 1e+244)) {
		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 <= 1e+244)) {
		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 <= 1e+244):
		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 <= 1e+244))
		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 <= 1e+244)))
		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, 1e+244]], $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 1.00000000000000007e244 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 11.6%

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

   3. Taylor expanded in x around 0 45.0%

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

      1. neg-mul-1 45.0%

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

   5. Simplified 45.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000007e244

   1. Initial program 99.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+244}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-222}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+219}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e+144)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -2.8e-197)
     (- (* x (log (/ x y))) z)
     (if (<= x 3.9e-222)
       (- z)
       (if (<= x 2.3e+219)
         (- (- z) (* x (log (/ y x))))
         (* x (- (log x) (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e+144) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -2.8e-197) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= 3.9e-222) {
		tmp = -z;
	} else if (x <= 2.3e+219) {
		tmp = -z - (x * log((y / x)));
	} else {
		tmp = x * (log(x) - log(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 (x <= (-1.9d+144)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-2.8d-197)) then
        tmp = (x * log((x / y))) - z
    else if (x <= 3.9d-222) then
        tmp = -z
    else if (x <= 2.3d+219) then
        tmp = -z - (x * log((y / x)))
    else
        tmp = x * (log(x) - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e+144) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -2.8e-197) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= 3.9e-222) {
		tmp = -z;
	} else if (x <= 2.3e+219) {
		tmp = -z - (x * Math.log((y / x)));
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e+144:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -2.8e-197:
		tmp = (x * math.log((x / y))) - z
	elif x <= 3.9e-222:
		tmp = -z
	elif x <= 2.3e+219:
		tmp = -z - (x * math.log((y / x)))
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e+144)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -2.8e-197)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= 3.9e-222)
		tmp = Float64(-z);
	elseif (x <= 2.3e+219)
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y / x))));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.9e+144)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -2.8e-197)
		tmp = (x * log((x / y))) - z;
	elseif (x <= 3.9e-222)
		tmp = -z;
	elseif (x <= 2.3e+219)
		tmp = -z - (x * log((y / x)));
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e+144], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.8e-197], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 3.9e-222], (-z), If[LessEqual[x, 2.3e+219], N[((-z) - N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.90000000000000013e144

   1. Initial program 69.3%

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

   3. Taylor expanded in z around 0 62.2%

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

   4. Taylor expanded in y around -inf 90.7%

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

      1. neg-mul-1 98.6%

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

      2. metadata-eval 98.6%

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

      3. distribute-neg-frac 98.6%

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

      4. distribute-frac-neg2 98.6%

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

      5. log-rec 98.6%

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

      6. sub-neg 98.6%

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

   6. Simplified 90.7%

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

   if -1.90000000000000013e144 < x < -2.8000000000000002e-197

   1. Initial program 88.0%

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

   if -2.8000000000000002e-197 < x < 3.9000000000000001e-222

   1. Initial program 52.2%

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

   3. Taylor expanded in x around 0 93.2%

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

      1. neg-mul-1 93.2%

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

   5. Simplified 93.2%

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

   if 3.9000000000000001e-222 < x < 2.3000000000000001e219

   1. Initial program 87.7%

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

      1. clear-num 87.0%

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

      2. log-div 89.5%

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

      3. metadata-eval 89.5%

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

   4. Applied egg-rr 89.5%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]
   5. Step-by-step derivation

      1. neg-sub0 89.5%

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

   6. Simplified 89.5%

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

   if 2.3000000000000001e219 < x

   1. Initial program 39.1%

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

   3. Taylor expanded in z around 0 39.1%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. log-rec 98.8%

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

      2. sub-neg 98.8%

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

   6. Simplified 98.8%

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-222}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+219}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 10^{-294}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e+139)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -9.8e-197)
     (- (* x (log (/ x y))) z)
     (if (<= x 1e-294) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+139) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -9.8e-197) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= 1e-294) {
		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 <= (-1.3d+139)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-9.8d-197)) then
        tmp = (x * log((x / y))) - z
    else if (x <= 1d-294) 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 <= -1.3e+139) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -9.8e-197) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= 1e-294) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e+139:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -9.8e-197:
		tmp = (x * math.log((x / y))) - z
	elif x <= 1e-294:
		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 <= -1.3e+139)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -9.8e-197)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= 1e-294)
		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 <= -1.3e+139)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -9.8e-197)
		tmp = (x * log((x / y))) - z;
	elseif (x <= 1e-294)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e+139], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.8e-197], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1e-294], (-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 < -1.30000000000000011e139

   1. Initial program 69.3%

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

   3. Taylor expanded in z around 0 62.2%

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

   4. Taylor expanded in y around -inf 90.7%

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

      1. neg-mul-1 98.6%

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

      2. metadata-eval 98.6%

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

      3. distribute-neg-frac 98.6%

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

      4. distribute-frac-neg2 98.6%

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

      5. log-rec 98.6%

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

      6. sub-neg 98.6%

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

   6. Simplified 90.7%

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

   if -1.30000000000000011e139 < x < -9.8000000000000004e-197

   1. Initial program 88.0%

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

   if -9.8000000000000004e-197 < x < 1.00000000000000002e-294

   1. Initial program 47.9%

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

   3. Taylor expanded in x around 0 93.0%

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

      1. neg-mul-1 93.0%

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

   5. Simplified 93.0%

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

   if 1.00000000000000002e-294 < x

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

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

Alternative 7: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+84} \lor \neg \left(z \leq -8.5 \cdot 10^{+23} \lor \neg \left(z \leq -1.2 \cdot 10^{-47}\right) \land z \leq 0.78\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.4e+84)
         (not (or (<= z -8.5e+23) (and (not (<= z -1.2e-47)) (<= z 0.78)))))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e+84) || !((z <= -8.5e+23) || (!(z <= -1.2e-47) && (z <= 0.78)))) {
		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 <= (-1.4d+84)) .or. (.not. (z <= (-8.5d+23)) .or. (.not. (z <= (-1.2d-47))) .and. (z <= 0.78d0))) 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 <= -1.4e+84) || !((z <= -8.5e+23) || (!(z <= -1.2e-47) && (z <= 0.78)))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.4e+84) or not ((z <= -8.5e+23) or (not (z <= -1.2e-47) and (z <= 0.78))):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.4e+84) || !((z <= -8.5e+23) || (!(z <= -1.2e-47) && (z <= 0.78))))
		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 <= -1.4e+84) || ~(((z <= -8.5e+23) || (~((z <= -1.2e-47)) && (z <= 0.78)))))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.4e+84], N[Not[Or[LessEqual[z, -8.5e+23], And[N[Not[LessEqual[z, -1.2e-47]], $MachinePrecision], LessEqual[z, 0.78]]]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999991e84 or -8.5000000000000001e23 < z < -1.2e-47 or 0.78000000000000003 < z

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 79.1%

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

      1. neg-mul-1 79.1%

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

   5. Simplified 79.1%

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

   if -1.39999999999999991e84 < z < -8.5000000000000001e23 or -1.2e-47 < z < 0.78000000000000003

   1. Initial program 74.6%

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

   3. Taylor expanded in z around 0 61.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+84} \lor \neg \left(z \leq -8.5 \cdot 10^{+23} \lor \neg \left(z \leq -1.2 \cdot 10^{-47}\right) \land z \leq 0.78\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% 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 75.8%

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

3. Taylor expanded in x around 0 47.3%

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

   1. neg-mul-1 47.3%

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

5. Simplified 47.3%

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

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 2024107 
(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))