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

Percentage Accurate: 76.8% → 99.3%

Time: 9.8s

Alternatives: 9

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: 76.8% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = ((x * log(x)) - (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 <= (-1d-309)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = ((x * log(x)) - (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 <= -1e-309) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-309:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = ((x * math.log(x)) - (x * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-309)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-309)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = ((x * log(x)) - (x * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-309], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 83.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 -1.000000000000002e-309 < y

   1. Initial program 71.5%

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

   3. Taylor expanded in y around -inf 0.0%

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

      1. metadata-eval 0.0%

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

      2. distribute-neg-frac 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. neg-mul-1 0.0%

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

      5. log-rec 0.0%

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

      6. sub-neg 0.0%

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

   5. Simplified 0.0%

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

      1. sub-neg 0.0%

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

      2. neg-log 0.0%

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

      3. metadata-eval 0.0%

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

      4. frac-2neg 0.0%

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

      5. +-commutative 0.0%

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

      6. distribute-rgt-in 0.0%

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

      7. frac-2neg 0.0%

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

      8. metadata-eval 0.0%

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

      9. neg-log 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. sqr-neg 0.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 5e+295) (- 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 <= 5e+295) {
		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 <= 5e+295) {
		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 <= 5e+295:
		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 <= 5e+295)
		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 <= 5e+295)
		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, 5e+295], 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 3.8%

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

   3. Taylor expanded in x around 0 61.0%

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

      1. mul-1-neg 61.0%

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

   5. Simplified 61.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.99999999999999991e295

   1. Initial program 99.7%

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

   if 4.99999999999999991e295 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 13.6%

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

   3. Taylor expanded in z around 0 13.6%

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

   4. Taylor expanded in x around 0 54.1%

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

      1. log-rec 54.1%

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

      2. sub-neg 54.1%

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

   6. Simplified 54.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+304))) (- 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 <= 2e+304)) {
		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 <= 2e+304)) {
		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 <= 2e+304):
		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 <= 2e+304))
		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 <= 2e+304)))
		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, 2e+304]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1.9999999999999999e304 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.5%

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

   3. Taylor expanded in x around 0 56.3%

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

      1. mul-1-neg 56.3%

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

   5. Simplified 56.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.9999999999999999e304

   1. Initial program 99.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-185}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-281}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-311}:\\ \;\;\;\;z \cdot \left(x \cdot \frac{\log \left(\frac{x}{y}\right)}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.5e+207)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -7e-185)
     (- (* x (- (log (/ y x)))) z)
     (if (<= x -2.7e-281)
       (- z)
       (if (<= x -4e-311)
         (* z (+ (* x (/ (log (/ x y)) z)) -1.0))
         (- (* x (- (log x) (log y))) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e+207) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -7e-185) {
		tmp = (x * -log((y / x))) - z;
	} else if (x <= -2.7e-281) {
		tmp = -z;
	} else if (x <= -4e-311) {
		tmp = z * ((x * (log((x / y)) / z)) + -1.0);
	} 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 <= (-8.5d+207)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-7d-185)) then
        tmp = (x * -log((y / x))) - z
    else if (x <= (-2.7d-281)) then
        tmp = -z
    else if (x <= (-4d-311)) then
        tmp = z * ((x * (log((x / y)) / z)) + (-1.0d0))
    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 <= -8.5e+207) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -7e-185) {
		tmp = (x * -Math.log((y / x))) - z;
	} else if (x <= -2.7e-281) {
		tmp = -z;
	} else if (x <= -4e-311) {
		tmp = z * ((x * (Math.log((x / y)) / z)) + -1.0);
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.5e+207:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -7e-185:
		tmp = (x * -math.log((y / x))) - z
	elif x <= -2.7e-281:
		tmp = -z
	elif x <= -4e-311:
		tmp = z * ((x * (math.log((x / y)) / z)) + -1.0)
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.5e+207)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -7e-185)
		tmp = Float64(Float64(x * Float64(-log(Float64(y / x)))) - z);
	elseif (x <= -2.7e-281)
		tmp = Float64(-z);
	elseif (x <= -4e-311)
		tmp = Float64(z * Float64(Float64(x * Float64(log(Float64(x / y)) / z)) + -1.0));
	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 <= -8.5e+207)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -7e-185)
		tmp = (x * -log((y / x))) - z;
	elseif (x <= -2.7e-281)
		tmp = -z;
	elseif (x <= -4e-311)
		tmp = z * ((x * (log((x / y)) / z)) + -1.0);
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.5e+207], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-185], N[(N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2.7e-281], (-z), If[LessEqual[x, -4e-311], N[(z * N[(N[(x * N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.4999999999999996e207

   1. Initial program 48.5%

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

   3. Taylor expanded in y around -inf 99.0%

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

      1. metadata-eval 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. neg-mul-1 99.0%

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

      5. log-rec 99.0%

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

      6. sub-neg 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in z around 0 93.9%

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

   if -8.4999999999999996e207 < x < -6.9999999999999996e-185

   1. Initial program 92.6%

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

      1. clear-num 52.4%

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

      2. log-div 52.3%

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

      3. metadata-eval 52.3%

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

   4. Applied egg-rr 93.8%

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

      1. neg-sub0 52.3%

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

   6. Simplified 93.8%

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

   if -6.9999999999999996e-185 < x < -2.6999999999999999e-281

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -2.6999999999999999e-281 < x < -3.99999999999979e-311

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-/l* 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if -3.99999999999979e-311 < x

   1. Initial program 71.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 46.6%

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

      2. sub-neg 46.6%

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

   5. Simplified 99.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-185}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-281}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-311}:\\ \;\;\;\;z \cdot \left(x \cdot \frac{\log \left(\frac{x}{y}\right)}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* 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 <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		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 <= -1e-309:
		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 <= -1e-309)
		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 <= -1e-309)
		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, -1e-309], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 83.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 -1.000000000000002e-309 < y

   1. Initial program 71.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 46.6%

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

      2. sub-neg 46.6%

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

   5. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-40} \lor \neg \left(z \leq 8.2 \cdot 10^{+14}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.1e-40) (not (<= z 8.2e+14))) (- z) (* x (- (log (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e-40) || !(z <= 8.2e+14)) {
		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 <= (-2.1d-40)) .or. (.not. (z <= 8.2d+14))) 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 <= -2.1e-40) || !(z <= 8.2e+14)) {
		tmp = -z;
	} else {
		tmp = x * -Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.1e-40) or not (z <= 8.2e+14):
		tmp = -z
	else:
		tmp = x * -math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.1e-40) || !(z <= 8.2e+14))
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(-log(Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.1e-40) || ~((z <= 8.2e+14)))
		tmp = -z;
	else
		tmp = x * -log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.1e-40], N[Not[LessEqual[z, 8.2e+14]], $MachinePrecision]], (-z), N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000018e-40 or 8.2e14 < z

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0 77.6%

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

      1. mul-1-neg 77.6%

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

   5. Simplified 77.6%

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

   if -2.10000000000000018e-40 < z < 8.2e14

   1. Initial program 80.2%

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

   3. Taylor expanded in z around 0 64.7%

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

      1. clear-num 64.7%

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

      2. log-div 65.5%

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

      3. metadata-eval 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. neg-sub0 65.5%

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

   7. Simplified 65.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-40} \lor \neg \left(z \leq 8.2 \cdot 10^{+14}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-38} \lor \neg \left(z \leq 22000000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e-38) (not (<= z 22000000000.0)))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e-38) || !(z <= 22000000000.0)) {
		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 <= (-5.2d-38)) .or. (.not. (z <= 22000000000.0d0))) 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 <= -5.2e-38) || !(z <= 22000000000.0)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.2e-38) or not (z <= 22000000000.0):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e-38) || !(z <= 22000000000.0))
		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 <= -5.2e-38) || ~((z <= 22000000000.0)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e-38], N[Not[LessEqual[z, 22000000000.0]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000022e-38 or 2.2e10 < z

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0 77.6%

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

      1. mul-1-neg 77.6%

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

   5. Simplified 77.6%

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

   if -5.20000000000000022e-38 < z < 2.2e10

   1. Initial program 80.2%

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

   3. Taylor expanded in z around 0 64.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-38} \lor \neg \left(z \leq 22000000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.0% 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 77.3%

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

3. Taylor expanded in x around 0 52.4%

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

   1. mul-1-neg 52.4%

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

5. Simplified 52.4%

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

6. Final simplification 52.4%

   \[\leadsto -z \]
7. Add Preprocessing

Alternative 9: 2.3% 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 77.3%

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

   1. flip-- 43.1%

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

   2. div-inv 42.9%

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

   3. difference-of-squares 43.2%

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

   4. add-sqr-sqrt 18.3%

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

   5. sqrt-unprod 30.5%

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

   6. sqr-neg 30.5%

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

   7. sqrt-unprod 12.2%

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

   8. add-sqr-sqrt 24.2%

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

   9. sub-neg 24.2%

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

   10. pow2 24.2%

       \[\leadsto \color{blue}{{\left(x \cdot \log \left(\frac{x}{y}\right) - z\right)}^{2}} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   11. fmm-def 24.2%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\right)}}^{2} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   12. add-sqr-sqrt 12.2%

       \[\leadsto {\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), \color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right)\right)}^{2} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   13. sqrt-unprod 24.2%

       \[\leadsto {\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right)\right)}^{2} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   14. sqr-neg 24.2%

       \[\leadsto {\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), \sqrt{\color{blue}{z \cdot z}}\right)\right)}^{2} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   15. sqrt-unprod 12.1%

       \[\leadsto {\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), \color{blue}{\sqrt{z} \cdot \sqrt{z}}\right)\right)}^{2} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   16. add-sqr-sqrt 24.2%

       \[\leadsto {\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), \color{blue}{z}\right)\right)}^{2} \cdot \frac{1}{x \cdot \log \left(\frac{x}{y}\right) + z} \]

   17. fma-define 24.2%

       \[\leadsto {\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), z\right)\right)}^{2} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), z\right)}} \]

4. Applied egg-rr 24.2%

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), z\right)\right)}^{2} \cdot \frac{1}{\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), z\right)}} \]

5. Taylor expanded in x around 0 2.3%

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

6. Final simplification 2.3%

   \[\leadsto z \]
7. Add Preprocessing

Developer target: 88.0% 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 2024095 
(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))