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

Percentage Accurate: 77.6% → 99.5%

Time: 10.1s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-311)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (fma (- (log y) (log x)) x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-311) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = -fma((log(y) - log(x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-311)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(-fma(Float64(log(y) - log(x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-311], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-N[(N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * x + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999979e-311

   1. Initial program 74.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -3.99999999999979e-311 < y

   1. Initial program 76.6%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l- N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. distribute-rgt-out-- N/A

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

      10. unsub-neg N/A

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

      11. log-rec N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unsub-neg N/A

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

      15. log-rec N/A

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

      16. distribute-lft-neg-out N/A

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

      17. distribute-neg-out N/A

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

   5. Applied rewrites 99.4%

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

Alternative 2: 87.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+308))) (- z) (- t_0 z))))

C

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

Java

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

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e+308):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e+308]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 39.8

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

   5. Applied rewrites 39.8%

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

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

   1. Initial program 99.9%

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

4. Final simplification 83.7%

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

Alternative 3: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+187}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.3e+187)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -3.9e-89)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -2e-308) (- z) (- (fma (- (log y) (log x)) x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.3e+187) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -3.9e-89) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = -fma((log(y) - log(x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.3e+187)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -3.9e-89)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(Float64(log(y) - log(x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.3e+187], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -3.9e-89], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), (-N[(N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * x + z), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.3000000000000001e187

   1. Initial program 45.8%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 45.7

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 45.7

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

   4. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 45.7

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. log-rec N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 0.0

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

   9. Applied rewrites 0.0%

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

   11. Applied rewrites 94.9%

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

   if -3.3000000000000001e187 < x < -3.89999999999999978e-89

   1. Initial program 88.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if -3.89999999999999978e-89 < x < -1.9999999999999998e-308

   1. Initial program 73.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 95.3

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

   5. Applied rewrites 95.3%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 76.6%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l- N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. distribute-rgt-out-- N/A

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

      10. unsub-neg N/A

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

      11. log-rec N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unsub-neg N/A

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

      15. log-rec N/A

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

      16. distribute-lft-neg-out N/A

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

      17. distribute-neg-out N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+187}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-185}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+207}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (- x) (log (/ y x))) z)))
   (if (<= x -3.9e-89)
     t_0
     (if (<= x 6.5e-185)
       (- z)
       (if (<= x 2.6e+207) t_0 (* (- (log x) (log y)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (-x * log((y / x))) - z;
	double tmp;
	if (x <= -3.9e-89) {
		tmp = t_0;
	} else if (x <= 6.5e-185) {
		tmp = -z;
	} else if (x <= 2.6e+207) {
		tmp = t_0;
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-x * log((y / x))) - z
    if (x <= (-3.9d-89)) then
        tmp = t_0
    else if (x <= 6.5d-185) then
        tmp = -z
    else if (x <= 2.6d+207) then
        tmp = t_0
    else
        tmp = (log(x) - log(y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-x * Math.log((y / x))) - z;
	double tmp;
	if (x <= -3.9e-89) {
		tmp = t_0;
	} else if (x <= 6.5e-185) {
		tmp = -z;
	} else if (x <= 2.6e+207) {
		tmp = t_0;
	} else {
		tmp = (Math.log(x) - Math.log(y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-x * math.log((y / x))) - z
	tmp = 0
	if x <= -3.9e-89:
		tmp = t_0
	elif x <= 6.5e-185:
		tmp = -z
	elif x <= 2.6e+207:
		tmp = t_0
	else:
		tmp = (math.log(x) - math.log(y)) * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z)
	tmp = 0.0
	if (x <= -3.9e-89)
		tmp = t_0;
	elseif (x <= 6.5e-185)
		tmp = Float64(-z);
	elseif (x <= 2.6e+207)
		tmp = t_0;
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-x * log((y / x))) - z;
	tmp = 0.0;
	if (x <= -3.9e-89)
		tmp = t_0;
	elseif (x <= 6.5e-185)
		tmp = -z;
	elseif (x <= 2.6e+207)
		tmp = t_0;
	else
		tmp = (log(x) - log(y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -3.9e-89], t$95$0, If[LessEqual[x, 6.5e-185], (-z), If[LessEqual[x, 2.6e+207], t$95$0, N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999978e-89 or 6.49999999999999946e-185 < x < 2.5999999999999998e207

   1. Initial program 81.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 83.4

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

   4. Applied rewrites 83.4%

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

   if -3.89999999999999978e-89 < x < 6.49999999999999946e-185

   1. Initial program 67.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   if 2.5999999999999998e207 < x

   1. Initial program 56.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. log-rec N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-185}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+207}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e-89)
   (- (* (- x) (log (/ y x))) z)
   (if (<= x -2e-308) (- z) (- (fma (- (log y) (log x)) x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-89) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = -fma((log(y) - log(x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e-89)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(Float64(log(y) - log(x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e-89], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), (-N[(N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * x + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999978e-89

   1. Initial program 74.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 77.0

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

   4. Applied rewrites 77.0%

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

   if -3.89999999999999978e-89 < x < -1.9999999999999998e-308

   1. Initial program 73.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 95.3

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

   5. Applied rewrites 95.3%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 76.6%

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

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l- N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. distribute-rgt-out-- N/A

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

      10. unsub-neg N/A

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

      11. log-rec N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unsub-neg N/A

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

      15. log-rec N/A

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

      16. distribute-lft-neg-out N/A

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

      17. distribute-neg-out N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-89}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-26} \lor \neg \left(z \leq 1.05 \cdot 10^{-76}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.25e-26) (not (<= z 1.05e-76)))
   (- z)
   (* (log (/ y x)) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e-26) || !(z <= 1.05e-76)) {
		tmp = -z;
	} else {
		tmp = log((y / x)) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.25d-26)) .or. (.not. (z <= 1.05d-76))) then
        tmp = -z
    else
        tmp = log((y / x)) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e-26) || !(z <= 1.05e-76)) {
		tmp = -z;
	} else {
		tmp = Math.log((y / x)) * -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.25e-26) or not (z <= 1.05e-76):
		tmp = -z
	else:
		tmp = math.log((y / x)) * -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.25e-26) || !(z <= 1.05e-76))
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(y / x)) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.25e-26) || ~((z <= 1.05e-76)))
		tmp = -z;
	else
		tmp = log((y / x)) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.25e-26], N[Not[LessEqual[z, 1.05e-76]], $MachinePrecision]], (-z), N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2499999999999999e-26 or 1.04999999999999996e-76 < z

   1. Initial program 74.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 69.8

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

   5. Applied rewrites 69.8%

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

   if -2.2499999999999999e-26 < z < 1.04999999999999996e-76

   1. Initial program 76.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 78.5

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

   4. Applied rewrites 78.5%

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

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 70.2

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

   7. Applied rewrites 70.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-26} \lor \neg \left(z \leq 1.05 \cdot 10^{-76}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-26} \lor \neg \left(z \leq 1.05 \cdot 10^{-76}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.25e-26) (not (<= z 1.05e-76))) (- z) (* (log (/ x y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e-26) || !(z <= 1.05e-76)) {
		tmp = -z;
	} else {
		tmp = log((x / y)) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.25d-26)) .or. (.not. (z <= 1.05d-76))) then
        tmp = -z
    else
        tmp = log((x / y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e-26) || !(z <= 1.05e-76)) {
		tmp = -z;
	} else {
		tmp = Math.log((x / y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.25e-26) or not (z <= 1.05e-76):
		tmp = -z
	else:
		tmp = math.log((x / y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.25e-26) || !(z <= 1.05e-76))
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(x / y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.25e-26) || ~((z <= 1.05e-76)))
		tmp = -z;
	else
		tmp = log((x / y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.25e-26], N[Not[LessEqual[z, 1.05e-76]], $MachinePrecision]], (-z), N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2499999999999999e-26 or 1.04999999999999996e-76 < z

   1. Initial program 74.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 69.8

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

   5. Applied rewrites 69.8%

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

   if -2.2499999999999999e-26 < z < 1.04999999999999996e-76

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 68.3

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

   5. Applied rewrites 68.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-26} \lor \neg \left(z \leq 1.05 \cdot 10^{-76}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% 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 75.4%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 50.7

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

5. Applied rewrites 50.7%

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

Alternative 9: 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 75.4%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 31.2%

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

9. Applied rewrites 2.3%

   \[\leadsto z \]
10. Add Preprocessing

Developer Target 1: 88.9% 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 2024313 
(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))