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

Percentage Accurate: 76.6% → 99.4%

Time: 11.5s

Alternatives: 10

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: 76.6% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 77.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -4.999999999999985e-310 < y

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. mul-1-neg N/A

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

      14. associate-*r/ N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+r+ N/A

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

      17. distribute-rgt-in N/A

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.4%

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

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := t\_0 \cdot x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* t_0 x)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 1e+303) (fma t_0 x (- z)) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = t_0 * x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 1e+303) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(t_0 * x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 1e+303)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = 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[(t$95$0 * x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 1e+303], N[(t$95$0 * x + (-z)), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e303

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 89.1%

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

Alternative 3: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+303}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+303) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+303:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+303)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+303)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+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 1e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e303

   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 89.1%

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

Alternative 4: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+169}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-170}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e+169)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -3.3e-170)
     (- (* (log (/ y x)) (- x)) z)
     (if (<= x -2e-307) (- z) (- (fma (- (log y) (log x)) x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+169) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -3.3e-170) {
		tmp = (log((y / x)) * -x) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = -fma((log(y) - log(x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e+169)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -3.3e-170)
		tmp = Float64(Float64(log(Float64(y / x)) * Float64(-x)) - z);
	elseif (x <= -2e-307)
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(Float64(log(y) - log(x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e+169], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -3.3e-170], N[(N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-307], (-z), (-N[(N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * x + z), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.15e169

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. log-rec N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 92.4%

      \[\leadsto \left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x \]

   if -1.15e169 < x < -3.30000000000000004e-170

   1. Initial program 95.3%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

   if -3.30000000000000004e-170 < x < -1.99999999999999982e-307

   1. Initial program 52.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -1.99999999999999982e-307 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. mul-1-neg N/A

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

      14. associate-*r/ N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+r+ N/A

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

      17. distribute-rgt-in N/A

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

   5. Applied rewrites 99.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+169}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-170}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-170}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.3e-170)
   (- (* (log (/ y x)) (- x)) z)
   (if (<= x -2e-307) (- z) (- (fma (- (log y) (log x)) x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.3e-170) {
		tmp = (log((y / x)) * -x) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = -fma((log(y) - log(x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.3e-170)
		tmp = Float64(Float64(log(Float64(y / x)) * Float64(-x)) - z);
	elseif (x <= -2e-307)
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(Float64(log(y) - log(x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.3e-170], N[(N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-307], (-z), (-N[(N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * x + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -3.30000000000000004e-170

   1. Initial program 84.9%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

   if -3.30000000000000004e-170 < x < -1.99999999999999982e-307

   1. Initial program 52.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -1.99999999999999982e-307 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. mul-1-neg N/A

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

      14. associate-*r/ N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+r+ N/A

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

      17. distribute-rgt-in N/A

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

   5. Applied rewrites 99.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-170}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 77.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if -4.999999999999985e-310 < y

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. mul-1-neg N/A

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

      14. associate-*r/ N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+r+ N/A

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

      17. distribute-rgt-in N/A

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.4%

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

Alternative 7: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ y x)) (- x))))
   (if (<= x -1.6e-32) t_0 (if (<= x 9.5e+72) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log((y / x)) * -x;
	double tmp;
	if (x <= -1.6e-32) {
		tmp = t_0;
	} else if (x <= 9.5e+72) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((y / x)) * -x
    if (x <= (-1.6d-32)) then
        tmp = t_0
    else if (x <= 9.5d+72) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((y / x)) * -x;
	double tmp;
	if (x <= -1.6e-32) {
		tmp = t_0;
	} else if (x <= 9.5e+72) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((y / x)) * -x
	tmp = 0
	if x <= -1.6e-32:
		tmp = t_0
	elif x <= 9.5e+72:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(y / x)) * Float64(-x))
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = t_0;
	elseif (x <= 9.5e+72)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((y / x)) * -x;
	tmp = 0.0;
	if (x <= -1.6e-32)
		tmp = t_0;
	elseif (x <= 9.5e+72)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision]}, If[LessEqual[x, -1.6e-32], t$95$0, If[LessEqual[x, 9.5e+72], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001e-32 or 9.50000000000000054e72 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. log-rec N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 34.4

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

   5. Applied rewrites 34.4%

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

   7. Applied rewrites 65.4%

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

   if -1.6000000000000001e-32 < x < 9.50000000000000054e72

   1. Initial program 78.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= x -1.6e-32) t_0 (if (<= x 9.5e+72) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (x <= -1.6e-32) {
		tmp = t_0;
	} else if (x <= 9.5e+72) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((x / y)) * x
    if (x <= (-1.6d-32)) then
        tmp = t_0
    else if (x <= 9.5d+72) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (x <= -1.6e-32) {
		tmp = t_0;
	} else if (x <= 9.5e+72) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if x <= -1.6e-32:
		tmp = t_0
	elif x <= 9.5e+72:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = t_0;
	elseif (x <= 9.5e+72)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (x <= -1.6e-32)
		tmp = t_0;
	elseif (x <= 9.5e+72)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.6e-32], t$95$0, If[LessEqual[x, 9.5e+72], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001e-32 or 9.50000000000000054e72 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -1.6000000000000001e-32 < x < 9.50000000000000054e72

   1. Initial program 78.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 77.6

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

   5. Applied rewrites 77.6%

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

Alternative 9: 49.5% accurate, 40.0× 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 Float64(-z)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 47.1

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

5. Applied rewrites 47.1%

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

Alternative 10: 2.3% accurate, 120.0× 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 78.1%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 47.1

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

5. Applied rewrites 47.1%

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

7. Applied rewrites 26.9%

   \[\leadsto \frac{\left(-z\right) \cdot z}{\color{blue}{0 + z}} \]
8. Step-by-step derivation

9. Applied rewrites 2.3%

   \[\leadsto z \]
10. Add Preprocessing

Developer Target 1: 88.1% accurate, 0.6× 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 2024268 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7595077799083773/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z)))

  (- (* x (log (/ x y))) z))