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

Percentage Accurate: 77.4% → 99.5%

Time: 9.1s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 80.7%

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

      1. frac-2neg 80.7%

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

      2. log-div 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 68.0%

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

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

Alternative 2: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+300) (- t_0 z) (* x (- (log x) (log y)))))))

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 <= 2e+300) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	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 <= 2e+300) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	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 <= 2e+300:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	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 <= 2e+300)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	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 <= 2e+300)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	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, 2e+300], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.5%

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

   2. Taylor expanded in x around 0 58.8%

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

      1. neg-mul-1 58.8%

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

   4. Simplified 58.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.0000000000000001e300

   1. Initial program 99.8%

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

   if 2.0000000000000001e300 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.8%

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

   2. Taylor expanded in z around 0 7.8%

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

      1. log-div 62.5%

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

   4. Applied egg-rr 54.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+300}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 3: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+300) (- 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 <= 2e+300) {
		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 <= 2e+300) {
		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 <= 2e+300:
		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 <= 2e+300)
		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 <= 2e+300)
		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, 2e+300], 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 2.0000000000000001e300 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.2%

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

   2. Taylor expanded in x around 0 50.1%

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

      1. neg-mul-1 50.1%

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

   4. Simplified 50.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.0000000000000001e300

   1. Initial program 99.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+300}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-134}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (- x) (log (/ y x))) z)))
   (if (<= x -1.42e+145)
     (* x (- (log (- x)) (log (- y))))
     (if (<= x -4.8e-157)
       t_0
       (if (<= x 1.7e-134)
         (- z)
         (if (<= x 2.1e+208) t_0 (* x (- (log x) (log y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-x * log((y / x))) - z;
	double tmp;
	if (x <= -1.42e+145) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.8e-157) {
		tmp = t_0;
	} else if (x <= 1.7e-134) {
		tmp = -z;
	} else if (x <= 2.1e+208) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (-x * log((y / x))) - z
    if (x <= (-1.42d+145)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-4.8d-157)) then
        tmp = t_0
    else if (x <= 1.7d-134) then
        tmp = -z
    else if (x <= 2.1d+208) then
        tmp = t_0
    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 t_0 = (-x * Math.log((y / x))) - z;
	double tmp;
	if (x <= -1.42e+145) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -4.8e-157) {
		tmp = t_0;
	} else if (x <= 1.7e-134) {
		tmp = -z;
	} else if (x <= 2.1e+208) {
		tmp = t_0;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-x * math.log((y / x))) - z
	tmp = 0
	if x <= -1.42e+145:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -4.8e-157:
		tmp = t_0
	elif x <= 1.7e-134:
		tmp = -z
	elif x <= 2.1e+208:
		tmp = t_0
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z)
	tmp = 0.0
	if (x <= -1.42e+145)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.8e-157)
		tmp = t_0;
	elseif (x <= 1.7e-134)
		tmp = Float64(-z);
	elseif (x <= 2.1e+208)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-x * log((y / x))) - z;
	tmp = 0.0;
	if (x <= -1.42e+145)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -4.8e-157)
		tmp = t_0;
	elseif (x <= 1.7e-134)
		tmp = -z;
	elseif (x <= 2.1e+208)
		tmp = t_0;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -1.42e+145], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-157], t$95$0, If[LessEqual[x, 1.7e-134], (-z), If[LessEqual[x, 2.1e+208], t$95$0, N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.41999999999999991e145

   1. Initial program 72.1%

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

   2. Taylor expanded in z around 0 69.3%

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

      1. frac-2neg 72.1%

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

      2. log-div 99.0%

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

   4. Applied egg-rr 92.4%

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

   if -1.41999999999999991e145 < x < -4.8e-157 or 1.69999999999999988e-134 < x < 2.0999999999999998e208

   1. Initial program 90.5%

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

      1. clear-num 42.8%

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

      2. neg-log 44.0%

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

   3. Applied egg-rr 91.7%

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

   if -4.8e-157 < x < 1.69999999999999988e-134

   1. Initial program 59.3%

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

   2. Taylor expanded in x around 0 91.3%

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

      1. neg-mul-1 91.3%

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

   4. Simplified 91.3%

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

   if 2.0999999999999998e208 < x

   1. Initial program 40.2%

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

   2. Taylor expanded in z around 0 40.2%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 95.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-157}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-134}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+208}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 5: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-154}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+149)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -9e-154)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -2e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+149) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -9e-154) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -2e-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 <= (-1.8d+149)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-9d-154)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-2d-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 <= -1.8e+149) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -9e-154) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -2e-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 <= -1.8e+149:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -9e-154:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -2e-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 <= -1.8e+149)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -9e-154)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-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 <= -1.8e+149)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -9e-154)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -2e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+149], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e-154], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-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 < -1.79999999999999997e149

   1. Initial program 72.1%

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

   2. Taylor expanded in z around 0 69.3%

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

      1. frac-2neg 72.1%

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

      2. log-div 99.0%

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

   4. Applied egg-rr 92.4%

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

   if -1.79999999999999997e149 < x < -8.9999999999999994e-154

   1. Initial program 92.4%

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

      1. clear-num 45.6%

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

      2. neg-log 46.4%

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

   3. Applied egg-rr 93.2%

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

   if -8.9999999999999994e-154 < x < -1.9999999999999998e-308

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0 90.8%

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

      1. neg-mul-1 90.8%

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

   4. Simplified 90.8%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 68.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-154}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+138}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x) (log (/ y x)))))
   (if (<= x -8e+34)
     t_0
     (if (<= x -4e-32)
       (- z)
       (if (<= x -2.6e-47)
         (* x (log (/ x y)))
         (if (<= x 6.2e+138) (- z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -x * log((y / x));
	double tmp;
	if (x <= -8e+34) {
		tmp = t_0;
	} else if (x <= -4e-32) {
		tmp = -z;
	} else if (x <= -2.6e-47) {
		tmp = x * log((x / y));
	} else if (x <= 6.2e+138) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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((y / x))
    if (x <= (-8d+34)) then
        tmp = t_0
    else if (x <= (-4d-32)) then
        tmp = -z
    else if (x <= (-2.6d-47)) then
        tmp = x * log((x / y))
    else if (x <= 6.2d+138) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -x * Math.log((y / x));
	double tmp;
	if (x <= -8e+34) {
		tmp = t_0;
	} else if (x <= -4e-32) {
		tmp = -z;
	} else if (x <= -2.6e-47) {
		tmp = x * Math.log((x / y));
	} else if (x <= 6.2e+138) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x * math.log((y / x))
	tmp = 0
	if x <= -8e+34:
		tmp = t_0
	elif x <= -4e-32:
		tmp = -z
	elif x <= -2.6e-47:
		tmp = x * math.log((x / y))
	elif x <= 6.2e+138:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) * log(Float64(y / x)))
	tmp = 0.0
	if (x <= -8e+34)
		tmp = t_0;
	elseif (x <= -4e-32)
		tmp = Float64(-z);
	elseif (x <= -2.6e-47)
		tmp = Float64(x * log(Float64(x / y)));
	elseif (x <= 6.2e+138)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x * log((y / x));
	tmp = 0.0;
	if (x <= -8e+34)
		tmp = t_0;
	elseif (x <= -4e-32)
		tmp = -z;
	elseif (x <= -2.6e-47)
		tmp = x * log((x / y));
	elseif (x <= 6.2e+138)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+34], t$95$0, If[LessEqual[x, -4e-32], (-z), If[LessEqual[x, -2.6e-47], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e+138], (-z), t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999956e34 or 6.1999999999999995e138 < x

   1. Initial program 71.2%

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

   2. Taylor expanded in z around 0 62.3%

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

      1. clear-num 62.3%

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

      2. neg-log 63.8%

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

   4. Applied egg-rr 63.8%

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

   if -7.99999999999999956e34 < x < -4.00000000000000022e-32 or -2.6e-47 < x < 6.1999999999999995e138

   1. Initial program 76.7%

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

   2. Taylor expanded in x around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   4. Simplified 79.6%

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

   if -4.00000000000000022e-32 < x < -2.6e-47

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 100.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+34}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+138}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]

Alternative 7: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+34} \lor \neg \left(x \leq -5.2 \cdot 10^{-32} \lor \neg \left(x \leq -2.7 \cdot 10^{-43}\right) \land x \leq 8.5 \cdot 10^{+134}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+34)
         (not
          (or (<= x -5.2e-32) (and (not (<= x -2.7e-43)) (<= x 8.5e+134)))))
   (* x (log (/ x y)))
   (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+34) || !((x <= -5.2e-32) || (!(x <= -2.7e-43) && (x <= 8.5e+134)))) {
		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 ((x <= (-6.8d+34)) .or. (.not. (x <= (-5.2d-32)) .or. (.not. (x <= (-2.7d-43))) .and. (x <= 8.5d+134))) 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 ((x <= -6.8e+34) || !((x <= -5.2e-32) || (!(x <= -2.7e-43) && (x <= 8.5e+134)))) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e+34) or not ((x <= -5.2e-32) or (not (x <= -2.7e-43) and (x <= 8.5e+134))):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e+34) || !((x <= -5.2e-32) || (!(x <= -2.7e-43) && (x <= 8.5e+134))))
		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 ((x <= -6.8e+34) || ~(((x <= -5.2e-32) || (~((x <= -2.7e-43)) && (x <= 8.5e+134)))))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e+34], N[Not[Or[LessEqual[x, -5.2e-32], And[N[Not[LessEqual[x, -2.7e-43]], $MachinePrecision], LessEqual[x, 8.5e+134]]]], $MachinePrecision]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -6.7999999999999999e34 or -5.1999999999999995e-32 < x < -2.69999999999999991e-43 or 8.50000000000000024e134 < x

   1. Initial program 72.5%

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

   2. Taylor expanded in z around 0 64.0%

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

   if -6.7999999999999999e34 < x < -5.1999999999999995e-32 or -2.69999999999999991e-43 < x < 8.50000000000000024e134

   1. Initial program 76.7%

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

   2. Taylor expanded in x around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   4. Simplified 79.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+34} \lor \neg \left(x \leq -5.2 \cdot 10^{-32} \lor \neg \left(x \leq -2.7 \cdot 10^{-43}\right) \land x \leq 8.5 \cdot 10^{+134}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 50.3% 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 74.9%

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

2. Taylor expanded in x around 0 51.1%

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

   1. neg-mul-1 51.1%

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

4. Simplified 51.1%

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

5. Final simplification 51.1%

   \[\leadsto -z \]

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 2023229 
(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))