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

Percentage Accurate: 77.4% → 99.5%

Time: 6.6s

Alternatives: 6

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: 77.4% 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

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (fma 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 = fma(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 = fma(x, Float64(log(Float64(-x)) - log(Float64(-y))), 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[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $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 82.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-neg-in N/A

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

      4. *-commutative N/A

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

      5. unsub-neg N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-rgt-identity N/A

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around -inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. neg-mul-1 N/A

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

      4. *-commutative N/A

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

      5. remove-double-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. metadata-eval N/A

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

      15. associate-/r* N/A

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

      16. neg-mul-1 N/A

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

      17. log-rec N/A

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

   8. Simplified 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 75.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.3

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

   5. Simplified 99.3%

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

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+297) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+297) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+297) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+297:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+297)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+297)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+297], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 53.8

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

   5. Simplified 53.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2e297

   1. Initial program 99.8%

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

   if 2e297 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 13.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 48.3

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

   5. Simplified 48.3%

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

Alternative 3: 82.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 INFINITY) (- 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 <= ((double) INFINITY)) {
		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 <= Double.POSITIVE_INFINITY) {
		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 <= math.inf:
		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 <= Inf)
		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 <= Inf)
		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, Infinity], 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 +inf.0 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 53.8

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

   5. Simplified 53.8%

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

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

   1. Initial program 88.1%

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

Alternative 4: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x - \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-202)
   (- (* x (log (/ x y))) z)
   (if (<= x -1e-307) (- z) (fma x (- (log x) (log y)) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-202) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-307) {
		tmp = -z;
	} else {
		tmp = fma(x, (log(x) - log(y)), -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-202)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-307)
		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, -2e-202], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-307], (-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 < -2.0000000000000001e-202

   1. Initial program 87.2%

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

   if -2.0000000000000001e-202 < x < -9.99999999999999909e-308

   1. Initial program 59.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 91.0

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

   5. Simplified 91.0%

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

   if -9.99999999999999909e-308 < x

   1. Initial program 75.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.3

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

   5. Simplified 99.3%

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

Alternative 5: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= x -8.5e+48) t_0 (if (<= x 2.3e-7) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (x <= -8.5e+48) {
		tmp = t_0;
	} else if (x <= 2.3e-7) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * log((x / y))
    if (x <= (-8.5d+48)) then
        tmp = t_0
    else if (x <= 2.3d-7) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (x <= -8.5e+48) {
		tmp = t_0;
	} else if (x <= 2.3e-7) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if x <= -8.5e+48:
		tmp = t_0
	elif x <= 2.3e-7:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (x <= -8.5e+48)
		tmp = t_0;
	elseif (x <= 2.3e-7)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (x <= -8.5e+48)
		tmp = t_0;
	elseif (x <= 2.3e-7)
		tmp = -z;
	else
		tmp = t_0;
	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[x, -8.5e+48], t$95$0, If[LessEqual[x, 2.3e-7], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000001e48 or 2.29999999999999995e-7 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 63.1

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

   5. Simplified 63.1%

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

   if -8.5000000000000001e48 < x < 2.29999999999999995e-7

   1. Initial program 77.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 75.0

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

   5. Simplified 75.0%

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

Alternative 6: 50.4% 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 79.3%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 51.1

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

5. Simplified 51.1%

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

Developer Target 1: 88.6% 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 2024208 
(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))