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

Percentage Accurate: 77.0% → 99.5%

Time: 11.4s

Alternatives: 12

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (fma (- (log (- x)) (log (- y))) x (- z))
   (- (fma (log x) x (- (* x (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = fma((log(-x) - log(-y)), x, -z);
	} else {
		tmp = fma(log(x), x, -(x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = fma(Float64(log(Float64(-x)) - log(Float64(-y))), x, Float64(-z));
	else
		tmp = Float64(fma(log(x), x, Float64(-Float64(x * log(y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * x + (-N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 70.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 70.8

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

   4. Applied rewrites 70.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -1.999999999999994e-310 < y

   1. Initial program 74.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-log.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 2e+296) (fma t_0 x (- z)) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 2e+296) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 2e+296)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 2e+296], N[(t$95$0 * x + (-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 9.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.2

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

   5. Applied rewrites 44.2%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.99999999999999996e296

   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%

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

   if 1.99999999999999996e296 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 9.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-*.f64 N/A

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

      8. log-rec N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 52.8

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

   5. Applied rewrites 52.8%

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

Alternative 3: 87.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 1e+308) (fma t_0 x (- z)) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 1e+308) {
		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(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 1e+308)
		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[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 1e+308], 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 1e308 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 45.3

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

   5. Applied rewrites 45.3%

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

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

   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. Add Preprocessing

Alternative 4: 87.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+308) (- t_0 z) (- z)))))

C

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

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+308) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+308:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+308)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+308], 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 1e308 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 45.3

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

   5. Applied rewrites 45.3%

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

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

   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. Add Preprocessing

Alternative 5: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-123}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.1e+184)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -3.8e-123)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -4e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+184) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -3.8e-123) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.1d+184)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-3.8d-123)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-4d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+184) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -3.8e-123) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.1e+184:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -3.8e-123:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -4e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.1e+184)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -3.8e-123)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -4e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.1e+184)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -3.8e-123)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -4e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.1e+184], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.8e-123], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.1e184

   1. Initial program 49.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. unpow1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. inv-pow N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. unpow-1 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 49.5

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

   4. Applied rewrites 49.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow1 49.5

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. log-div N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. distribute-neg-frac2 N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-neg.f64 94.8

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

   9. Applied rewrites 94.8%

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

   if -1.1e184 < x < -3.79999999999999996e-123

   1. Initial program 82.8%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 83.6

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

   4. Applied rewrites 83.6%

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

   if -3.79999999999999996e-123 < x < -4.00000000000000013e-308

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -4.00000000000000013e-308 < x

   1. Initial program 74.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-123}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-123}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.8e-123)
   (- (* (- x) (log (/ y x))) z)
   (if (<= x -4e-308) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e-123) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.8d-123)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-4d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e-123) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.8e-123:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -4e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.8e-123)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -4e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.8e-123)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -4e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.8e-123], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.79999999999999996e-123

   1. Initial program 73.4%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 74.5

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

   4. Applied rewrites 74.5%

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

   if -3.79999999999999996e-123 < x < -4.00000000000000013e-308

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -4.00000000000000013e-308 < x

   1. Initial program 74.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-123}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (fma (- (log (- x)) (log (- y))) x (- z))
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = fma((log(-x) - log(-y)), x, -z);
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = fma(Float64(log(Float64(-x)) - log(Float64(-y))), x, Float64(-z));
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x + (-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 < -1.999999999999994e-310

   1. Initial program 70.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 70.8

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

   4. Applied rewrites 70.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -1.999999999999994e-310 < y

   1. Initial program 74.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 8: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 70.8%

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

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1.999999999999994e-310 < y

   1. Initial program 74.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 9: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 58000000000000:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e-13)
   (- z)
   (if (<= z 58000000000000.0) (* (- x) (log (/ y x))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e-13) {
		tmp = -z;
	} else if (z <= 58000000000000.0) {
		tmp = -x * log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.7d-13)) then
        tmp = -z
    else if (z <= 58000000000000.0d0) then
        tmp = -x * log((y / x))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e-13) {
		tmp = -z;
	} else if (z <= 58000000000000.0) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.7e-13:
		tmp = -z
	elif z <= 58000000000000.0:
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e-13)
		tmp = Float64(-z);
	elseif (z <= 58000000000000.0)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.7e-13)
		tmp = -z;
	elseif (z <= 58000000000000.0)
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e-13], (-z), If[LessEqual[z, 58000000000000.0], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.70000000000000011e-13 or 5.8e13 < z

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -2.70000000000000011e-13 < z < 5.8e13

   1. Initial program 73.3%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 60.8

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

   7. Applied rewrites 60.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 58000000000000:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.5e-13) (- z) (if (<= z 8.8e-135) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e-13) {
		tmp = -z;
	} else if (z <= 8.8e-135) {
		tmp = x * log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.5d-13)) then
        tmp = -z
    else if (z <= 8.8d-135) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e-13) {
		tmp = -z;
	} else if (z <= 8.8e-135) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.5e-13:
		tmp = -z
	elif z <= 8.8e-135:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.5e-13)
		tmp = Float64(-z);
	elseif (z <= 8.8e-135)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.5e-13)
		tmp = -z;
	elseif (z <= 8.8e-135)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.5e-13], (-z), If[LessEqual[z, 8.8e-135], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999995e-13 or 8.7999999999999999e-135 < z

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if -2.49999999999999995e-13 < z < 8.7999999999999999e-135

   1. Initial program 77.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 63.7

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

   5. Applied rewrites 63.7%

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

Alternative 11: 50.8% 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 73.0%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 49.8

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

5. Applied rewrites 49.8%

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

Alternative 12: 2.2% 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 73.0%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 49.8

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

5. Applied rewrites 49.8%

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

7. Applied rewrites 1.1%

   \[\leadsto \left(\left(z \cdot z\right) \cdot z\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(z, z, 0\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 2.5%

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

11. Applied rewrites 2.1%

    \[\leadsto z \]
12. Add Preprocessing

Developer Target 1: 88.2% 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 2024220 
(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))