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

Percentage Accurate: 77.6% → 99.4%

Time: 10.3s

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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{\log x}^{3} - {\log y}^{3}}{\mathsf{fma}\left(\log x, \log x, {\log y}^{2} + \log x \cdot \log y\right)} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (fma (- (log (- x)) (log (- y))) x (- z))
   (-
    (*
     x
     (/
      (- (pow (log x) 3.0) (pow (log y) 3.0))
      (fma (log x) (log x) (+ (pow (log y) 2.0) (* (log x) (log y))))))
    z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = fma((log(-x) - log(-y)), x, -z);
	} else {
		tmp = (x * ((pow(log(x), 3.0) - pow(log(y), 3.0)) / fma(log(x), log(x), (pow(log(y), 2.0) + (log(x) * log(y)))))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = fma(Float64(log(Float64(-x)) - log(Float64(-y))), x, Float64(-z));
	else
		tmp = Float64(Float64(x * Float64(Float64((log(x) ^ 3.0) - (log(y) ^ 3.0)) / fma(log(x), log(x), Float64((log(y) ^ 2.0) + Float64(log(x) * log(y)))))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision], N[(N[(x * N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[y], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 83.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -1.999999999999994e-310 < y

   1. Initial program 78.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. sub-neg N/A

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

      5. flip3-+ N/A

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

      6. cube-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto x \cdot \frac{{\log x}^{3} - {\log y}^{3}}{\color{blue}{\mathsf{fma}\left(\log x, \log x, \left(\mathsf{neg}\left(\log y\right)\right) \cdot \left(\mathsf{neg}\left(\log y\right)\right) - \log x \cdot \left(\mathsf{neg}\left(\log y\right)\right)\right)}} - z \]

   4. Applied rewrites 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{\log x}^{3} - {\log y}^{3}}{\mathsf{fma}\left(\log x, \log x, {\log y}^{2} + \log x \cdot \log y\right)} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+277) (- t_0 z) (* (- (log x) (log y)) x)))))

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+277) {
		tmp = t_0 - z;
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	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+277) {
		tmp = t_0 - z;
	} else {
		tmp = (Math.log(x) - Math.log(y)) * x;
	}
	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+277:
		tmp = t_0 - z
	else:
		tmp = (math.log(x) - math.log(y)) * x
	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+277)
		tmp = Float64(t_0 - z);
	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((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+277)
		tmp = t_0 - z;
	else
		tmp = (log(x) - log(y)) * x;
	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+277], N[(t$95$0 - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 53.7

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

   5. Applied rewrites 53.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.00000000000000001e277

   1. Initial program 99.8%

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

   if 2.00000000000000001e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 12.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 56.9

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

   5. Applied rewrites 56.9%

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

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+296}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+296))) (- 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+296)) {
		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+296)) {
		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+296):
		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+296))
		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+296)))
		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+296]], $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 9.99999999999999981e295 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.99999999999999981e295

   1. Initial program 99.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+296}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+296)))
     (- z)
     (- (fma (log (/ y x)) x 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+296)) {
		tmp = -z;
	} else {
		tmp = -fma(log((y / x)), x, 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+296))
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	end
	return 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+296]], $MachinePrecision]], (-z), (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 9.99999999999999981e295 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.99999999999999981e295

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. lift-/.f64 N/A

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

      9. diff-log N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. distribute-neg-in N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-neg.f64 50.9

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

   6. Applied rewrites 98.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+296}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y - \log x, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.55e-174)
   (fma (log (/ x y)) x (- z))
   (if (<= x -1e-307) (- z) (- (fma (- (log y) (log x)) x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.55e-174) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -1e-307) {
		tmp = -z;
	} else {
		tmp = -fma((log(y) - log(x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.55e-174)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -1e-307)
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(Float64(log(y) - log(x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.55e-174], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -1e-307], (-z), (-N[(N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * x + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -2.55000000000000016e-174

   1. Initial program 86.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 86.5

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

   4. Applied rewrites 86.5%

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

   if -2.55000000000000016e-174 < x < -9.99999999999999909e-308

   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 96.5

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

   5. Applied rewrites 96.5%

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

   if -9.99999999999999909e-308 < x

   1. Initial program 78.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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 99.5%

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

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 83.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -1.999999999999994e-310 < y

   1. Initial program 78.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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 99.5%

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

Alternative 7: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (* 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 <= -2e-310) {
		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 <= -2e-310)
		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, -2e-310], 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 < -1.999999999999994e-310

   1. Initial program 83.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -1.999999999999994e-310 < y

   1. Initial program 78.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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 99.5%

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

Alternative 8: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-33} \lor \neg \left(x \leq 2.4 \cdot 10^{+57}\right):\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.6e-33) (not (<= x 2.4e+57))) (* (log (/ x y)) x) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.6e-33) || !(x <= 2.4e+57)) {
		tmp = log((x / 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 ((x <= (-9.6d-33)) .or. (.not. (x <= 2.4d+57))) then
        tmp = log((x / 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 ((x <= -9.6e-33) || !(x <= 2.4e+57)) {
		tmp = Math.log((x / y)) * x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.6e-33) or not (x <= 2.4e+57):
		tmp = math.log((x / y)) * x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.6e-33) || !(x <= 2.4e+57))
		tmp = Float64(log(Float64(x / y)) * x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.6e-33) || ~((x <= 2.4e+57)))
		tmp = log((x / y)) * x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.6e-33], N[Not[LessEqual[x, 2.4e+57]], $MachinePrecision]], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -9.6e-33 or 2.40000000000000005e57 < x

   1. Initial program 80.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 66.0

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

   5. Applied rewrites 66.0%

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

   if -9.6e-33 < x < 2.40000000000000005e57

   1. Initial program 81.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 76.7

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

   5. Applied rewrites 76.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-33} \lor \neg \left(x \leq 2.4 \cdot 10^{+57}\right):\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.1% 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 81.2%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 49.2

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

5. Applied rewrites 49.2%

   \[\leadsto \color{blue}{-z} \]
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 81.2%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 49.2

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

5. Applied rewrites 49.2%

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

7. Applied rewrites 27.3%

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

9. Applied rewrites 2.3%

   \[\leadsto z \]
10. Add Preprocessing

Developer Target 1: 88.3% 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 2024324 
(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))