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

Percentage Accurate: 76.9% → 99.5%

Time: 9.7s

Alternatives: 7

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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 78.8%

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

   3. Taylor expanded in y around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. distribute-neg-frac 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. neg-mul-1 99.5%

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

      5. log-rec 99.5%

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

      6. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 73.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \end{array} \end{array} \]

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 5e+305)))
     (- (* x (log (* y x))) 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 <= 5e+305)) {
		tmp = (x * log((y * x))) - 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 <= 5e+305))
		tmp = Float64(Float64(x * log(Float64(y * x))) - 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, 5e+305]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 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 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.6%

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

      1. clear-num 5.6%

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

      2. log-div 9.0%

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

      3. metadata-eval 9.0%

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

   4. Applied egg-rr 9.0%

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

      1. neg-sub0 9.0%

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

   6. Simplified 9.0%

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

      1. log1p-expm1-u 5.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-\log \left(\frac{y}{x}\right)\right)\right)} - z \]

      2. neg-log 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log \left(\frac{1}{\frac{y}{x}}\right)}\right)\right) - z \]

      3. clear-num 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{\left(\frac{x}{y}\right)}\right)\right) - z \]

      4. diff-log 2.1%

         \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log x - \log y}\right)\right) - z \]

      5. expm1-define 2.1%

         \[\leadsto x \cdot \mathsf{log1p}\left(\color{blue}{e^{\log x - \log y} - 1}\right) - z \]

      6. diff-log 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{x}{y}\right)}} - 1\right) - z \]

      7. add-exp-log 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(\color{blue}{\frac{x}{y}} - 1\right) - z \]

      8. pow1 5.6%

         \[\leadsto \color{blue}{{\left(x \cdot \mathsf{log1p}\left(\frac{x}{y} - 1\right)\right)}^{1}} - z \]

   8. Applied egg-rr 47.1%

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

      1. unpow1 47.1%

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

      2. *-commutative 47.1%

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

   10. Simplified 47.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

      1. fma-neg 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+305)))
     (- (* x (log (* y x))) 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 <= 5e+305)) {
		tmp = (x * log((y * x))) - 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 <= 5e+305)) {
		tmp = (x * Math.log((y * x))) - 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 <= 5e+305):
		tmp = (x * math.log((y * x))) - 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 <= 5e+305))
		tmp = Float64(Float64(x * log(Float64(y * x))) - 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 <= 5e+305)))
		tmp = (x * log((y * x))) - 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, 5e+305]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $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 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.6%

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

      1. clear-num 5.6%

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

      2. log-div 9.0%

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

      3. metadata-eval 9.0%

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

   4. Applied egg-rr 9.0%

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

      1. neg-sub0 9.0%

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

   6. Simplified 9.0%

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

      1. log1p-expm1-u 5.6%

         \[\leadsto x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-\log \left(\frac{y}{x}\right)\right)\right)} - z \]

      2. neg-log 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log \left(\frac{1}{\frac{y}{x}}\right)}\right)\right) - z \]

      3. clear-num 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{\left(\frac{x}{y}\right)}\right)\right) - z \]

      4. diff-log 2.1%

         \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log x - \log y}\right)\right) - z \]

      5. expm1-define 2.1%

         \[\leadsto x \cdot \mathsf{log1p}\left(\color{blue}{e^{\log x - \log y} - 1}\right) - z \]

      6. diff-log 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{x}{y}\right)}} - 1\right) - z \]

      7. add-exp-log 5.6%

         \[\leadsto x \cdot \mathsf{log1p}\left(\color{blue}{\frac{x}{y}} - 1\right) - z \]

      8. pow1 5.6%

         \[\leadsto \color{blue}{{\left(x \cdot \mathsf{log1p}\left(\frac{x}{y} - 1\right)\right)}^{1}} - z \]

   8. Applied egg-rr 47.1%

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

      1. unpow1 47.1%

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

      2. *-commutative 47.1%

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

   10. Simplified 47.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(x * N[Log[N[(x / y), $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 78.8%

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

      1. fma-neg 78.8%

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

   3. Simplified 78.8%

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

   if -4.999999999999985e-310 < y

   1. Initial program 73.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 5: 64.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+81}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-121} \lor \neg \left(z \leq 3.1 \cdot 10^{-79}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e+81)
   (- z)
   (if (<= z -6.6e-18)
     (* x (log (/ x y)))
     (if (or (<= z -9e-121) (not (<= z 3.1e-79)))
       (- z)
       (* (- x) (log (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+81) {
		tmp = -z;
	} else if (z <= -6.6e-18) {
		tmp = x * log((x / y));
	} else if ((z <= -9e-121) || !(z <= 3.1e-79)) {
		tmp = -z;
	} else {
		tmp = -x * log((y / x));
	}
	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 <= (-5.2d+81)) then
        tmp = -z
    else if (z <= (-6.6d-18)) then
        tmp = x * log((x / y))
    else if ((z <= (-9d-121)) .or. (.not. (z <= 3.1d-79))) then
        tmp = -z
    else
        tmp = -x * log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+81) {
		tmp = -z;
	} else if (z <= -6.6e-18) {
		tmp = x * Math.log((x / y));
	} else if ((z <= -9e-121) || !(z <= 3.1e-79)) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.2e+81:
		tmp = -z
	elif z <= -6.6e-18:
		tmp = x * math.log((x / y))
	elif (z <= -9e-121) or not (z <= 3.1e-79):
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e+81)
		tmp = Float64(-z);
	elseif (z <= -6.6e-18)
		tmp = Float64(x * log(Float64(x / y)));
	elseif ((z <= -9e-121) || !(z <= 3.1e-79))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.2e+81)
		tmp = -z;
	elseif (z <= -6.6e-18)
		tmp = x * log((x / y));
	elseif ((z <= -9e-121) || ~((z <= 3.1e-79)))
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e+81], (-z), If[LessEqual[z, -6.6e-18], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -9e-121], N[Not[LessEqual[z, 3.1e-79]], $MachinePrecision]], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.19999999999999984e81 or -6.6000000000000003e-18 < z < -9.0000000000000007e-121 or 3.0999999999999999e-79 < z

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 70.6%

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

      1. mul-1-neg 70.6%

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

   5. Simplified 70.6%

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

   if -5.19999999999999984e81 < z < -6.6000000000000003e-18

   1. Initial program 83.5%

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

   3. Taylor expanded in z around 0 62.3%

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

   if -9.0000000000000007e-121 < z < 3.0999999999999999e-79

   1. Initial program 75.5%

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

      1. clear-num 75.5%

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

      2. log-div 76.3%

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

      3. metadata-eval 76.3%

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

   4. Applied egg-rr 76.3%

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

      1. neg-sub0 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in z around 0 64.9%

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

      1. associate-*r* 64.9%

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

      2. neg-mul-1 64.9%

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

   9. Simplified 64.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+81}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-121} \lor \neg \left(z \leq 3.1 \cdot 10^{-79}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+82} \lor \neg \left(z \leq -2.7 \cdot 10^{-18}\right) \land \left(z \leq -6.8 \cdot 10^{-68} \lor \neg \left(z \leq 3.2 \cdot 10^{-78}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.7e+82)
         (and (not (<= z -2.7e-18)) (or (<= z -6.8e-68) (not (<= z 3.2e-78)))))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.7e+82) || (!(z <= -2.7e-18) && ((z <= -6.8e-68) || !(z <= 3.2e-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 <= (-2.7d+82)) .or. (.not. (z <= (-2.7d-18))) .and. (z <= (-6.8d-68)) .or. (.not. (z <= 3.2d-78))) 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 <= -2.7e+82) || (!(z <= -2.7e-18) && ((z <= -6.8e-68) || !(z <= 3.2e-78)))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.7e+82) or (not (z <= -2.7e-18) and ((z <= -6.8e-68) or not (z <= 3.2e-78))):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.7e+82) || (!(z <= -2.7e-18) && ((z <= -6.8e-68) || !(z <= 3.2e-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 <= -2.7e+82) || (~((z <= -2.7e-18)) && ((z <= -6.8e-68) || ~((z <= 3.2e-78)))))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.7e+82], And[N[Not[LessEqual[z, -2.7e-18]], $MachinePrecision], Or[LessEqual[z, -6.8e-68], N[Not[LessEqual[z, 3.2e-78]], $MachinePrecision]]]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6999999999999999e82 or -2.69999999999999989e-18 < z < -6.80000000000000037e-68 or 3.2e-78 < z

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0 71.8%

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

      1. mul-1-neg 71.8%

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

   5. Simplified 71.8%

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

   if -2.6999999999999999e82 < z < -2.69999999999999989e-18 or -6.80000000000000037e-68 < z < 3.2e-78

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0 62.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+82} \lor \neg \left(z \leq -2.7 \cdot 10^{-18}\right) \land \left(z \leq -6.8 \cdot 10^{-68} \lor \neg \left(z \leq 3.2 \cdot 10^{-78}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% 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 76.2%

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

3. Taylor expanded in x around 0 50.0%

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

   1. mul-1-neg 50.0%

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

5. Simplified 50.0%

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

Developer target: 88.1% 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 2024089 
(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))