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

Percentage Accurate: 77.6% → 99.5%

Time: 10.8s

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: 77.6% 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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-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 <= -5e-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 <= -5e-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, -5e-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 < -4.999999999999985e-310

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 77.1%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+303) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+303) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+303:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+303)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+303)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+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 2e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.6%

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

   3. 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 53.3

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

   5. Applied rewrites 53.3%

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

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

   1. Initial program 99.7%

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

Alternative 3: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+172}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-168}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x - \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.5e+172)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.05e-168)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -5e-309) (- z) (fma x (- (log x) (log y)) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e+172) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.05e-168) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = fma(x, (log(x) - log(y)), -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.5e+172)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.05e-168)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = fma(x, Float64(log(x) - log(y)), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.5e+172], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-168], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.50000000000000053e172

   1. Initial program 59.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 0.0

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

   5. Applied rewrites 0.0%

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

      1. diff-log N/A

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

      2. frac-2neg N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 85.4

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

   7. Applied rewrites 85.4%

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

   if -8.50000000000000053e172 < x < -1.04999999999999997e-168

   1. Initial program 90.5%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 90.5

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

   4. Applied rewrites 90.5%

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

   if -1.04999999999999997e-168 < x < -4.9999999999999995e-309

   1. Initial program 48.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. log-rec N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+172}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-168}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x - \log y, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-168}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x - \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e-168)
   (- (* (- x) (log (/ y x))) z)
   (if (<= x -5e-309) (- z) (fma x (- (log x) (log y)) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-168) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = fma(x, (log(x) - log(y)), -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e-168)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = fma(x, Float64(log(x) - log(y)), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.05e-168], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04999999999999997e-168

   1. Initial program 77.8%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 77.8

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

   4. Applied rewrites 77.8%

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

   if -1.04999999999999997e-168 < x < -4.9999999999999995e-309

   1. Initial program 48.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. log-rec N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-168}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x - \log y, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. log-rec N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 71.2%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. log-rec N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 7: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -820000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -820000000000.0)
   (- z)
   (if (<= z 1.05e-47) (* (- x) (log (/ y x))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000000000.0) {
		tmp = -z;
	} else if (z <= 1.05e-47) {
		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 <= (-820000000000.0d0)) then
        tmp = -z
    else if (z <= 1.05d-47) 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 <= -820000000000.0) {
		tmp = -z;
	} else if (z <= 1.05e-47) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -820000000000.0:
		tmp = -z
	elif z <= 1.05e-47:
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -820000000000.0)
		tmp = Float64(-z);
	elseif (z <= 1.05e-47)
		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 <= -820000000000.0)
		tmp = -z;
	elseif (z <= 1.05e-47)
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -820000000000.0], (-z), If[LessEqual[z, 1.05e-47], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e11 or 1.05e-47 < z

   1. Initial program 68.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   if -8.2e11 < z < 1.05e-47

   1. Initial program 80.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 40.8

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

   5. Applied rewrites 40.8%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -820000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -820000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -820000000000.0)
   (- z)
   (if (<= z 1.05e-47) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000000000.0) {
		tmp = -z;
	} else if (z <= 1.05e-47) {
		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 <= (-820000000000.0d0)) then
        tmp = -z
    else if (z <= 1.05d-47) 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 <= -820000000000.0) {
		tmp = -z;
	} else if (z <= 1.05e-47) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -820000000000.0:
		tmp = -z
	elif z <= 1.05e-47:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -820000000000.0)
		tmp = Float64(-z);
	elseif (z <= 1.05e-47)
		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 <= -820000000000.0)
		tmp = -z;
	elseif (z <= 1.05e-47)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -820000000000.0], (-z), If[LessEqual[z, 1.05e-47], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e11 or 1.05e-47 < z

   1. Initial program 68.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   if -8.2e11 < z < 1.05e-47

   1. Initial program 80.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

Alternative 9: 49.3% 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 74.0%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 47.7

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

5. Applied rewrites 47.7%

   \[\leadsto \color{blue}{-z} \]
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 74.0%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 47.7

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

5. Applied rewrites 47.7%

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

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - z} \]

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)} \]

   4. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)} \]

   5. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)} \]

   6. sqr-pow N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   7. pow-prod-down N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   8. sqr-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   9. lift-neg.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   10. lift-neg.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   11. pow-prod-down N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   12. sqr-pow N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   13. lift-neg.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   14. cube-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{0 - {z}^{3}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{0}^{3}} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right) \]

   17. flip3-- N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 - z\right)}\right) \]

   18. neg-sub0 N/A

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

   19. remove-double-neg 2.2

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

7. Applied rewrites 2.2%

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

Developer Target 1: 88.5% 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 2024219 
(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))