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

Percentage Accurate: 77.8% → 99.5%

Time: 12.7s

Alternatives: 6

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: 77.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\log x - \log y}} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (/ x (/ 1.0 (- (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 / (1.0 / (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 / (1.0d0 / (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 / (1.0 / (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 / (1.0 / (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(1.0 / 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 / (1.0 / (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[(1.0 / N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 79.6%

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

      1. frac-2neg 79.6%

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

      2. distribute-frac-neg 79.6%

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

      3. neg-mul-1 79.6%

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

      4. metadata-eval 79.6%

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

      5. log-prod 0.0%

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

      6. metadata-eval 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. log-div 0.0%

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

      2. associate-+r- 0.0%

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

      3. log-prod 99.6%

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

      4. mul-1-neg 99.6%

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

   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 78.3%

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

      1. add-exp-log_binary64 35.2%

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

   3. Applied rewrite-once 35.2%

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

      1. rem-exp-log 78.3%

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

      2. /-rgt-identity 78.3%

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

      3. associate-/l* 78.3%

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

   5. Applied egg-rr 78.3%

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

      1. log-div 99.5%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\log x - \log y}}} - z \]

   7. Applied egg-rr 99.5%

      \[\leadsto \frac{x}{\frac{1}{\color{blue}{\log x - \log y}}} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\log x - \log y}} - z\\ \end{array} \]

Alternative 2: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 5e+303) (- 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 <= 5e+303) {
		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 <= 5e+303) {
		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 <= 5e+303:
		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 <= 5e+303)
		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 <= 5e+303)
		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, 5e+303], 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 4.9999999999999997e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.9%

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

   2. Taylor expanded in x around 0 45.4%

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

      1. mul-1-neg 45.4%

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

   4. Simplified 45.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.9999999999999997e303

   1. Initial program 99.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-239}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4e-239)
   (- (* x (log (/ x y))) z)
   (if (<= x -4e-308) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e-239) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4d-239)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-4d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e-239) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4e-239:
		tmp = (x * math.log((x / y))) - z
	elif x <= -4e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4e-239)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -4e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4e-239)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -4e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4e-239], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000003e-239

   1. Initial program 81.7%

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

   if -4.0000000000000003e-239 < x < -4.00000000000000013e-308

   1. Initial program 65.5%

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

   2. Taylor expanded in x around 0 91.3%

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

      1. mul-1-neg 91.3%

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

   4. Simplified 91.3%

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

   if -4.00000000000000013e-308 < x

   1. Initial program 78.3%

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

      1. log-div 99.5%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\log x - \log y}}} - z \]

   3. Applied egg-rr 99.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-239}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 4: 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 79.6%

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

      1. frac-2neg 79.6%

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

      2. distribute-frac-neg 79.6%

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

      3. neg-mul-1 79.6%

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

      4. metadata-eval 79.6%

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

      5. log-prod 0.0%

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

      6. metadata-eval 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. log-div 0.0%

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

      2. associate-+r- 0.0%

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

      3. log-prod 99.6%

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

      4. mul-1-neg 99.6%

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

   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 78.3%

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

      1. log-div 99.5%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\log x - \log y}}} - z \]

   3. Applied egg-rr 99.5%

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]
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 5: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e-9) (- z) (if (<= z 4.3e-16) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e-9) {
		tmp = -z;
	} else if (z <= 4.3e-16) {
		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 (z <= (-3.4d-9)) then
        tmp = -z
    else if (z <= 4.3d-16) 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 (z <= -3.4e-9) {
		tmp = -z;
	} else if (z <= 4.3e-16) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.4e-9:
		tmp = -z
	elif z <= 4.3e-16:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e-9)
		tmp = Float64(-z);
	elseif (z <= 4.3e-16)
		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 (z <= -3.4e-9)
		tmp = -z;
	elseif (z <= 4.3e-16)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e-9], (-z), If[LessEqual[z, 4.3e-16], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999998e-9 or 4.2999999999999999e-16 < z

   1. Initial program 82.9%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. mul-1-neg 81.3%

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

   4. Simplified 81.3%

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

   if -3.3999999999999998e-9 < z < 4.2999999999999999e-16

   1. Initial program 75.6%

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

      1. remove-double-div 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in x around inf 37.3%

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

      1. log-rec 37.3%

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

      2. distribute-rgt-in 37.2%

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

      3. log-rec 37.2%

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

      4. mul-1-neg 37.2%

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

      5. remove-double-neg 37.2%

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

      6. +-commutative 37.2%

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

      7. distribute-rgt-in 37.3%

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

      8. sub-neg 37.3%

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

      9. log-div 59.4%

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

   6. Simplified 59.4%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 50.1% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 78.9%

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

2. Taylor expanded in x around 0 50.7%

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

   1. mul-1-neg 50.7%

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

4. Simplified 50.7%

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

5. Final simplification 50.7%

   \[\leadsto -z \]

Developer target: 88.3% 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 2023297 
(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))