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

Percentage Accurate: 77.1% → 99.5%

Time: 14.7s

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: 77.1% 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 73.6%

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

   3. Taylor expanded in y around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. neg-mul-1 99.6%

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

      5. log-rec 99.6%

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

      6. sub-neg 99.6%

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

   5. Simplified 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   5. Simplified 99.4%

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

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+308}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+308)))
     (- z)
     (fma x t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+308)) {
		tmp = -z;
	} else {
		tmp = fma(x, t_0, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+308))
		tmp = Float64(-z);
	else
		tmp = fma(x, t_0, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+308]], $MachinePrecision]], (-z), N[(x * t$95$0 + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1e308 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.5%

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

   3. Taylor expanded in x around 0 55.1%

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

      1. mul-1-neg 55.1%

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

   5. Simplified 55.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e308

   1. Initial program 99.6%

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

      1. fma-neg 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+308}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+308}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+308))) (- 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+308)) {
		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+308)) {
		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+308):
		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+308))
		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+308)))
		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+308]], $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 1e308 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.5%

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

   3. Taylor expanded in x around 0 55.1%

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

      1. mul-1-neg 55.1%

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

   5. Simplified 55.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e308

   1. Initial program 99.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+308}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.04 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e+142)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.04e-138)
     (- (* x (log (/ x y))) z)
     (if (<= x -1e-305) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e+142) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.04e-138) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-305) {
		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+142)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.04d-138)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-1d-305)) 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+142) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.04e-138) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -1e-305) {
		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+142:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.04e-138:
		tmp = (x * math.log((x / y))) - z
	elif x <= -1e-305:
		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+142)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.04e-138)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-305)
		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+142)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.04e-138)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -1e-305)
		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+142], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.04e-138], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-305], (-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 < -9.50000000000000001e142

   1. Initial program 65.0%

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

   3. Taylor expanded in y around -inf 99.1%

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

      1. metadata-eval 99.1%

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

      2. distribute-neg-frac 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. neg-mul-1 99.1%

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

      5. log-rec 99.1%

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

      6. sub-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in z around 0 79.5%

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

   if -9.50000000000000001e142 < x < -1.0399999999999999e-138

   1. Initial program 93.1%

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

   if -1.0399999999999999e-138 < x < -9.99999999999999996e-306

   1. Initial program 59.9%

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

   3. Taylor expanded in x around 0 88.4%

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

      1. mul-1-neg 88.4%

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

   5. Simplified 88.4%

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

   if -9.99999999999999996e-306 < x

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   5. Simplified 99.4%

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

Alternative 5: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+173}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-136}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-55}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x) (log (/ y x)))))
   (if (<= x -9.5e+173)
     t_0
     (if (<= x -1.5e-23)
       (- z)
       (if (<= x -9.8e-136)
         (* x (log (/ x y)))
         (if (<= x 6e-55) (- z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -x * log((y / x));
	double tmp;
	if (x <= -9.5e+173) {
		tmp = t_0;
	} else if (x <= -1.5e-23) {
		tmp = -z;
	} else if (x <= -9.8e-136) {
		tmp = x * log((x / y));
	} else if (x <= 6e-55) {
		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 <= (-9.5d+173)) then
        tmp = t_0
    else if (x <= (-1.5d-23)) then
        tmp = -z
    else if (x <= (-9.8d-136)) then
        tmp = x * log((x / y))
    else if (x <= 6d-55) 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 <= -9.5e+173) {
		tmp = t_0;
	} else if (x <= -1.5e-23) {
		tmp = -z;
	} else if (x <= -9.8e-136) {
		tmp = x * Math.log((x / y));
	} else if (x <= 6e-55) {
		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 <= -9.5e+173:
		tmp = t_0
	elif x <= -1.5e-23:
		tmp = -z
	elif x <= -9.8e-136:
		tmp = x * math.log((x / y))
	elif x <= 6e-55:
		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 <= -9.5e+173)
		tmp = t_0;
	elseif (x <= -1.5e-23)
		tmp = Float64(-z);
	elseif (x <= -9.8e-136)
		tmp = Float64(x * log(Float64(x / y)));
	elseif (x <= 6e-55)
		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 <= -9.5e+173)
		tmp = t_0;
	elseif (x <= -1.5e-23)
		tmp = -z;
	elseif (x <= -9.8e-136)
		tmp = x * log((x / y));
	elseif (x <= 6e-55)
		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, -9.5e+173], t$95$0, If[LessEqual[x, -1.5e-23], (-z), If[LessEqual[x, -9.8e-136], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-55], (-z), t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5000000000000005e173 or 6.00000000000000033e-55 < x

   1. Initial program 71.5%

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

   3. Taylor expanded in z around inf 56.8%

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

      1. sub-neg 56.8%

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

      2. associate-/l* 56.7%

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

      3. metadata-eval 56.7%

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

   5. Simplified 56.7%

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

      1. *-commutative 56.7%

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

      2. frac-2neg 56.7%

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

      3. associate-*l/ 56.8%

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

      4. neg-log 56.8%

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

      5. clear-num 59.0%

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

   7. Applied egg-rr 59.0%

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

   8. Taylor expanded in z around 0 57.5%

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

      1. associate-*r* 57.5%

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

      2. neg-mul-1 57.5%

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

   10. Simplified 57.5%

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

   if -9.5000000000000005e173 < x < -1.50000000000000001e-23 or -9.7999999999999999e-136 < x < 6.00000000000000033e-55

   1. Initial program 62.0%

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

   3. Taylor expanded in x around 0 76.3%

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

      1. mul-1-neg 76.3%

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

   5. Simplified 76.3%

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

   if -1.50000000000000001e-23 < x < -9.7999999999999999e-136

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 72.9%

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

Alternative 6: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+29} \lor \neg \left(z \leq 4.4 \cdot 10^{-12}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.25e+29) (not (<= z 4.4e-12))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e+29) || !(z <= 4.4e-12)) {
		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 <= (-2.25d+29)) .or. (.not. (z <= 4.4d-12))) 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 <= -2.25e+29) || !(z <= 4.4e-12)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.25e+29) or not (z <= 4.4e-12):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.25e+29) || !(z <= 4.4e-12))
		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 <= -2.25e+29) || ~((z <= 4.4e-12)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.25e+29], N[Not[LessEqual[z, 4.4e-12]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2500000000000001e29 or 4.39999999999999983e-12 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0 78.2%

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

      1. mul-1-neg 78.2%

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

   5. Simplified 78.2%

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

   if -2.2500000000000001e29 < z < 4.39999999999999983e-12

   1. Initial program 64.7%

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

   3. Taylor expanded in z around 0 56.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+29} \lor \neg \left(z \leq 4.4 \cdot 10^{-12}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% 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 68.4%

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

3. Taylor expanded in x around 0 50.7%

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

   1. mul-1-neg 50.7%

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

5. Simplified 50.7%

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

Alternative 8: 2.2% accurate, 107.0× 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 z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 68.4%

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

3. Taylor expanded in x around 0 50.7%

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

   1. mul-1-neg 50.7%

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

5. Simplified 50.7%

   \[\leadsto \color{blue}{-z} \]
6. Step-by-step derivation

   1. neg-sub0 50.7%

      \[\leadsto \color{blue}{0 - z} \]

   2. sub-neg 50.7%

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

   3. add-sqr-sqrt 25.8%

      \[\leadsto 0 + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

   4. sqrt-unprod 12.4%

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

   5. sqr-neg 12.4%

      \[\leadsto 0 + \sqrt{\color{blue}{z \cdot z}} \]

   6. sqrt-unprod 1.2%

      \[\leadsto 0 + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   7. add-sqr-sqrt 2.3%

      \[\leadsto 0 + \color{blue}{z} \]

7. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{0 + z} \]
8. Step-by-step derivation

   1. +-lft-identity 2.3%

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

9. Simplified 2.3%

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

Developer target: 87.9% 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 2024110 
(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))