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

Percentage Accurate: 77.6% → 99.3%

Time: 9.1s

Alternatives: 12

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% 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: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-y\right)\\ t_1 := \log \left(-x\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{t\_1}^{3} - {t\_0}^{3}}{{t\_1}^{2} + \left({t\_0}^{2} + t\_1 \cdot t\_0\right)} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\log x}^{3} - {\log y}^{3}}{\mathsf{fma}\left(\log x, \log x, \mathsf{fma}\left(\log y, \log y, \log x \cdot \log y\right)\right)}, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- y))) (t_1 (log (- x))))
   (if (<= y -5e-310)
     (-
      (*
       x
       (/
        (- (pow t_1 3.0) (pow t_0 3.0))
        (+ (pow t_1 2.0) (+ (pow t_0 2.0) (* t_1 t_0)))))
      z)
     (fma
      (/
       (- (pow (log x) 3.0) (pow (log y) 3.0))
       (fma (log x) (log x) (fma (log y) (log y) (* (log x) (log y)))))
      x
      (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-y);
	double t_1 = log(-x);
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * ((pow(t_1, 3.0) - pow(t_0, 3.0)) / (pow(t_1, 2.0) + (pow(t_0, 2.0) + (t_1 * t_0))))) - z;
	} else {
		tmp = fma(((pow(log(x), 3.0) - pow(log(y), 3.0)) / fma(log(x), log(x), fma(log(y), log(y), (log(x) * log(y))))), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-y))
	t_1 = log(Float64(-x))
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((t_1 ^ 3.0) - (t_0 ^ 3.0)) / Float64((t_1 ^ 2.0) + Float64((t_0 ^ 2.0) + Float64(t_1 * t_0))))) - z);
	else
		tmp = fma(Float64(Float64((log(x) ^ 3.0) - (log(y) ^ 3.0)) / fma(log(x), log(x), fma(log(y), log(y), Float64(log(x) * log(y))))), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-y)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x)], $MachinePrecision]}, If[LessEqual[y, -5e-310], N[(N[(x * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(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[Log[y], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 73.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{3} - {\log \left(-y\right)}^{3}}{{\log \left(-x\right)}^{2} + \left({\log \left(-y\right)}^{2} + \log \left(-x\right) \cdot \log \left(-y\right)\right)}} - z \]

   if -4.999999999999985e-310 < y

   1. Initial program 80.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-y\right)\\ t_1 := \log \left(-x\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{t\_1}^{2} - {t\_0}^{2}}{t\_0 + t\_1} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\log x}^{3} - {\log y}^{3}}{\mathsf{fma}\left(\log x, \log x, \mathsf{fma}\left(\log y, \log y, \log x \cdot \log y\right)\right)}, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- y))) (t_1 (log (- x))))
   (if (<= y -5e-310)
     (- (* x (/ (- (pow t_1 2.0) (pow t_0 2.0)) (+ t_0 t_1))) z)
     (fma
      (/
       (- (pow (log x) 3.0) (pow (log y) 3.0))
       (fma (log x) (log x) (fma (log y) (log y) (* (log x) (log y)))))
      x
      (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-y);
	double t_1 = log(-x);
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * ((pow(t_1, 2.0) - pow(t_0, 2.0)) / (t_0 + t_1))) - z;
	} else {
		tmp = fma(((pow(log(x), 3.0) - pow(log(y), 3.0)) / fma(log(x), log(x), fma(log(y), log(y), (log(x) * log(y))))), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-y))
	t_1 = log(Float64(-x))
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((t_1 ^ 2.0) - (t_0 ^ 2.0)) / Float64(t_0 + t_1))) - z);
	else
		tmp = fma(Float64(Float64((log(x) ^ 3.0) - (log(y) ^ 3.0)) / fma(log(x), log(x), fma(log(y), log(y), Float64(log(x) * log(y))))), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-y)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x)], $MachinePrecision]}, If[LessEqual[y, -5e-310], N[(N[(x * N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(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[Log[y], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 73.6%

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

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      9. lower-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      10. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\color{blue}{\log \left(\mathsf{neg}\left(x\right)\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      11. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \color{blue}{\left(-x\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      12. pow2 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      14. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\color{blue}{\log \left(\mathsf{neg}\left(y\right)\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      15. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \color{blue}{\left(-y\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      16. sum-log N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      17. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      18. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      19. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      20. lower-neg.f64 88.1

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \color{blue}{\left(-y\right)}\right)} - z \]

   4. Applied rewrites 88.1%

      \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]
   5. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(-y\right) \cdot \left(-x\right)\right)}} - z \]

      4. log-prod N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right) + \log \left(-x\right)}} - z \]

      5. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right)} + \log \left(-x\right)} - z \]

      6. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(-y\right) + \color{blue}{\log \left(-x\right)}} - z \]

      7. lower-+.f64 99.4

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right) + \log \left(-x\right)}} - z \]

   6. Applied rewrites 99.4%

      \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right) + \log \left(-x\right)}} - z \]

   if -4.999999999999985e-310 < y

   1. Initial program 80.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-y\right)\\ t_1 := \log \left(-x\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{t\_1}^{2} - {t\_0}^{2}}{t\_0 + t\_1} - z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- y))) (t_1 (log (- x))))
   (if (<= y -5e-310)
     (- (* x (/ (- (pow t_1 2.0) (pow t_0 2.0)) (+ t_0 t_1))) z)
     (- (* (* (sqrt x) (- (log x) (log y))) (sqrt x)) z))))

C

double code(double x, double y, double z) {
	double t_0 = log(-y);
	double t_1 = log(-x);
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * ((pow(t_1, 2.0) - pow(t_0, 2.0)) / (t_0 + t_1))) - z;
	} else {
		tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(-y)
    t_1 = log(-x)
    if (y <= (-5d-310)) then
        tmp = (x * (((t_1 ** 2.0d0) - (t_0 ** 2.0d0)) / (t_0 + t_1))) - z
    else
        tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(-y);
	double t_1 = Math.log(-x);
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * ((Math.pow(t_1, 2.0) - Math.pow(t_0, 2.0)) / (t_0 + t_1))) - z;
	} else {
		tmp = ((Math.sqrt(x) * (Math.log(x) - Math.log(y))) * Math.sqrt(x)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(-y)
	t_1 = math.log(-x)
	tmp = 0
	if y <= -5e-310:
		tmp = (x * ((math.pow(t_1, 2.0) - math.pow(t_0, 2.0)) / (t_0 + t_1))) - z
	else:
		tmp = ((math.sqrt(x) * (math.log(x) - math.log(y))) * math.sqrt(x)) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = log(Float64(-y))
	t_1 = log(Float64(-x))
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((t_1 ^ 2.0) - (t_0 ^ 2.0)) / Float64(t_0 + t_1))) - z);
	else
		tmp = Float64(Float64(Float64(sqrt(x) * Float64(log(x) - log(y))) * sqrt(x)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(-y);
	t_1 = log(-x);
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (((t_1 ^ 2.0) - (t_0 ^ 2.0)) / (t_0 + t_1))) - z;
	else
		tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-y)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x)], $MachinePrecision]}, If[LessEqual[y, -5e-310], N[(N[(x * N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 73.6%

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

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      9. lower-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      10. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\color{blue}{\log \left(\mathsf{neg}\left(x\right)\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      11. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \color{blue}{\left(-x\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      12. pow2 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      14. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\color{blue}{\log \left(\mathsf{neg}\left(y\right)\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      15. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \color{blue}{\left(-y\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      16. sum-log N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      17. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      18. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      19. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      20. lower-neg.f64 88.1

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \color{blue}{\left(-y\right)}\right)} - z \]

   4. Applied rewrites 88.1%

      \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]
   5. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(-y\right) \cdot \left(-x\right)\right)}} - z \]

      4. log-prod N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right) + \log \left(-x\right)}} - z \]

      5. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right)} + \log \left(-x\right)} - z \]

      6. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(-y\right) + \color{blue}{\log \left(-x\right)}} - z \]

      7. lower-+.f64 99.4

         \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right) + \log \left(-x\right)}} - z \]

   6. Applied rewrites 99.4%

      \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(-y\right) + \log \left(-x\right)}} - z \]

   if -4.999999999999985e-310 < y

   1. Initial program 80.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - z \]

      3. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{2}}} - z \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      17. lower-sqrt.f64 66.4

          \[\leadsto \sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \color{blue}{\sqrt{x}} - z \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \sqrt{x}} - z \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 4: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right)\\ t_1 := \log \left(-y\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{t\_0 - t\_1}\right)}^{2}, x, -z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{t\_0}^{2} - {t\_1}^{2}}{\log \left(x \cdot y\right)} - z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- x))) (t_1 (log (- y))))
   (if (<= x -4.4e+147)
     (fma (pow (sqrt (- t_0 t_1)) 2.0) x (- z))
     (if (<= x -5e-310)
       (- (* x (/ (- (pow t_0 2.0) (pow t_1 2.0)) (log (* x y)))) z)
       (- (* (* (sqrt x) (- (log x) (log y))) (sqrt x)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x);
	double t_1 = log(-y);
	double tmp;
	if (x <= -4.4e+147) {
		tmp = fma(pow(sqrt((t_0 - t_1)), 2.0), x, -z);
	} else if (x <= -5e-310) {
		tmp = (x * ((pow(t_0, 2.0) - pow(t_1, 2.0)) / log((x * y)))) - z;
	} else {
		tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-x))
	t_1 = log(Float64(-y))
	tmp = 0.0
	if (x <= -4.4e+147)
		tmp = fma((sqrt(Float64(t_0 - t_1)) ^ 2.0), x, Float64(-z));
	elseif (x <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((t_0 ^ 2.0) - (t_1 ^ 2.0)) / log(Float64(x * y)))) - z);
	else
		tmp = Float64(Float64(Float64(sqrt(x) * Float64(log(x) - log(y))) * sqrt(x)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-y)], $MachinePrecision]}, If[LessEqual[x, -4.4e+147], N[(N[Power[N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -5e-310], N[(N[(x * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000003e147

   1. Initial program 58.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 52.3%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(\frac{x}{y}\right)}\right)}^{2}, x, -z\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 92.7%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(-x\right) - \log \left(-y\right)}\right)}^{2}, x, -z\right) \]

   if -4.4000000000000003e147 < x < -4.999999999999985e-310

   1. Initial program 78.1%

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

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      9. lower-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      10. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\color{blue}{\log \left(\mathsf{neg}\left(x\right)\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      11. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \color{blue}{\left(-x\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      12. pow2 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      14. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\color{blue}{\log \left(\mathsf{neg}\left(y\right)\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      15. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \color{blue}{\left(-y\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      16. sum-log N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      17. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      18. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      19. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      20. lower-neg.f64 95.1

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \color{blue}{\left(-y\right)}\right)} - z \]

   4. Applied rewrites 95.1%

      \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

   if -4.999999999999985e-310 < x

   1. Initial program 80.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - z \]

      3. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{2}}} - z \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      17. lower-sqrt.f64 66.4

          \[\leadsto \sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \color{blue}{\sqrt{x}} - z \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \sqrt{x}} - z \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{\log \left(-x\right) - \log \left(-y\right)}\right)}^{2}, x, -z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(x \cdot y\right)} - z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right) - \log \left(-y\right)\\ t_1 := \log \left(y \cdot x\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{t\_0}\right)}^{2}, x, -z\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\left(t\_1 \cdot t\_0\right) \cdot x}{t\_1} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log (- x)) (log (- y)))) (t_1 (log (* y x))))
   (if (<= x -1.25e+190)
     (fma (pow (sqrt t_0) 2.0) x (- z))
     (if (<= x -3.2e-105)
       (- (/ (* (* t_1 t_0) x) t_1) z)
       (if (<= x -2e-308)
         (- z)
         (- (* (* (sqrt x) (- (log x) (log y))) (sqrt x)) z))))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x) - log(-y);
	double t_1 = log((y * x));
	double tmp;
	if (x <= -1.25e+190) {
		tmp = fma(pow(sqrt(t_0), 2.0), x, -z);
	} else if (x <= -3.2e-105) {
		tmp = (((t_1 * t_0) * x) / t_1) - z;
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(-x)) - log(Float64(-y)))
	t_1 = log(Float64(y * x))
	tmp = 0.0
	if (x <= -1.25e+190)
		tmp = fma((sqrt(t_0) ^ 2.0), x, Float64(-z));
	elseif (x <= -3.2e-105)
		tmp = Float64(Float64(Float64(Float64(t_1 * t_0) * x) / t_1) - z);
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(sqrt(x) * Float64(log(x) - log(y))) * sqrt(x)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.25e+190], N[(N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -3.2e-105], N[(N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] * x), $MachinePrecision] / t$95$1), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), N[(N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25000000000000009e190

   1. Initial program 60.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 60.9

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

   5. Applied rewrites 60.9%

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

   7. Applied rewrites 56.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(\frac{x}{y}\right)}\right)}^{2}, x, -z\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 94.7%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(-x\right) - \log \left(-y\right)}\right)}^{2}, x, -z\right) \]

   if -1.25000000000000009e190 < x < -3.19999999999999981e-105

   1. Initial program 85.2%

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

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      9. lower-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{2}} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      10. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\color{blue}{\log \left(\mathsf{neg}\left(x\right)\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      11. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \color{blue}{\left(-x\right)}}^{2} - \log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      12. pow2 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      13. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{2}}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      14. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\color{blue}{\log \left(\mathsf{neg}\left(y\right)\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      15. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \color{blue}{\left(-y\right)}}^{2}}{\log \left(\mathsf{neg}\left(x\right)\right) + \log \left(\mathsf{neg}\left(y\right)\right)} - z \]

      16. sum-log N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      17. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\color{blue}{\log \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      18. lower-*.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      19. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      20. lower-neg.f64 93.7

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \color{blue}{\left(-y\right)}\right)} - z \]

   4. Applied rewrites 93.7%

      \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      2. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left({\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}\right)}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left({\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}\right)}{\log \left(\left(-x\right) \cdot \left(-y\right)\right)}} - z \]

   6. Applied rewrites 79.4%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. log-div N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 93.8

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

   8. Applied rewrites 93.8%

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

   if -3.19999999999999981e-105 < x < -1.9999999999999998e-308

   1. Initial program 62.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 80.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - z \]

      3. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{2}}} - z \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      17. lower-sqrt.f64 66.4

          \[\leadsto \sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \color{blue}{\sqrt{x}} - z \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \sqrt{x}} - z \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{\log \left(-x\right) - \log \left(-y\right)}\right)}^{2}, x, -z\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\left(\log \left(y \cdot x\right) \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\right) \cdot x}{\log \left(y \cdot x\right)} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+308)))
     (- z)
     (fma t_0 x (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+308)) {
		tmp = -z;
	} else {
		tmp = fma(t_0, x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+308))
		tmp = Float64(-z);
	else
		tmp = fma(t_0, x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+308]], $MachinePrecision]], (-z), N[(t$95$0 * x + (-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 8.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.4

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

   5. Applied rewrites 51.4%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 87.5%

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

Alternative 7: 87.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (t_1 <= 1e+308) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (t_1 <= 1e+308)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.5%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 9.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

Alternative 8: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{\log \left(-x\right) - \log \left(-y\right)}\right)}^{2}, x, -z\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.8e+143)
   (fma (pow (sqrt (- (log (- x)) (log (- y)))) 2.0) x (- z))
   (if (<= x -3.2e-105)
     (fma (log (/ x y)) x (- z))
     (if (<= x -2e-308)
       (- z)
       (- (* (* (sqrt x) (- (log x) (log y))) (sqrt x)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8e+143) {
		tmp = fma(pow(sqrt((log(-x) - log(-y))), 2.0), x, -z);
	} else if (x <= -3.2e-105) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.8e+143)
		tmp = fma((sqrt(Float64(log(Float64(-x)) - log(Float64(-y)))) ^ 2.0), x, Float64(-z));
	elseif (x <= -3.2e-105)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(sqrt(x) * Float64(log(x) - log(y))) * sqrt(x)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.8e+143], N[(N[Power[N[Sqrt[N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -3.2e-105], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), N[(N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.79999999999999998e143

   1. Initial program 58.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 52.2%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(\frac{x}{y}\right)}\right)}^{2}, x, -z\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 93.1%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(-x\right) - \log \left(-y\right)}\right)}^{2}, x, -z\right) \]

   if -2.79999999999999998e143 < x < -3.19999999999999981e-105

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -3.19999999999999981e-105 < x < -1.9999999999999998e-308

   1. Initial program 62.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 80.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - z \]

      3. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{2}}} - z \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      17. lower-sqrt.f64 66.4

          \[\leadsto \sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \color{blue}{\sqrt{x}} - z \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \sqrt{x}} - z \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 9: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+144}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\log x - \log y\right)\right) \cdot \sqrt{x} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e+144)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -3.2e-105)
     (fma (log (/ x y)) x (- z))
     (if (<= x -2e-308)
       (- z)
       (- (* (* (sqrt x) (- (log x) (log y))) (sqrt x)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e+144) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -3.2e-105) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = ((sqrt(x) * (log(x) - log(y))) * sqrt(x)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e+144)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -3.2e-105)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(sqrt(x) * Float64(log(x) - log(y))) * sqrt(x)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e+144], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -3.2e-105], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), N[(N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.40000000000000003e144

   1. Initial program 58.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -1.40000000000000003e144 < x < -3.19999999999999981e-105

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -3.19999999999999981e-105 < x < -1.9999999999999998e-308

   1. Initial program 62.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 80.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - z \]

      3. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \log \left(\frac{x}{y}\right)\right)}^{2}}} - z \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{\color{blue}{{\log \left(\frac{x}{y}\right)}^{2}} \cdot x} \cdot \sqrt{x} - z \]

      17. lower-sqrt.f64 66.4

          \[\leadsto \sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \color{blue}{\sqrt{x}} - z \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\sqrt{{\log \left(\frac{x}{y}\right)}^{2} \cdot x} \cdot \sqrt{x}} - z \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 10: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x - \log y\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+144}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \frac{x}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log x) (log y))))
   (if (<= x -1.4e+144)
     (* (- (log (- x)) (log (- y))) x)
     (if (<= x -3.2e-105)
       (fma (log (/ x y)) x (- z))
       (if (<= x -2e-308)
         (- z)
         (if (<= x 1e+178) (fma (* t_0 (/ x z)) z (- z)) (* t_0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) - log(y);
	double tmp;
	if (x <= -1.4e+144) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -3.2e-105) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -2e-308) {
		tmp = -z;
	} else if (x <= 1e+178) {
		tmp = fma((t_0 * (x / z)), z, -z);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) - log(y))
	tmp = 0.0
	if (x <= -1.4e+144)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -3.2e-105)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	elseif (x <= 1e+178)
		tmp = fma(Float64(t_0 * Float64(x / z)), z, Float64(-z));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+144], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -3.2e-105], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), If[LessEqual[x, 1e+178], N[(N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.40000000000000003e144

   1. Initial program 58.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -1.40000000000000003e144 < x < -3.19999999999999981e-105

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -3.19999999999999981e-105 < x < -1.9999999999999998e-308

   1. Initial program 62.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   if -1.9999999999999998e-308 < x < 1.0000000000000001e178

   1. Initial program 83.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

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

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 79.2

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 94.6%

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

   if 1.0000000000000001e178 < x

   1. Initial program 70.3%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.3

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\log x - \log y\right) \cdot x} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-19} \lor \neg \left(z \leq 5.2 \cdot 10^{-88}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5e-19) (not (<= z 5.2e-88))) (- z) (* (log (/ x y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e-19) || !(z <= 5.2e-88)) {
		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 <= (-6.5d-19)) .or. (.not. (z <= 5.2d-88))) 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 <= -6.5e-19) || !(z <= 5.2e-88)) {
		tmp = -z;
	} else {
		tmp = Math.log((x / y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5e-19) or not (z <= 5.2e-88):
		tmp = -z
	else:
		tmp = math.log((x / y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5e-19) || !(z <= 5.2e-88))
		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 <= -6.5e-19) || ~((z <= 5.2e-88)))
		tmp = -z;
	else
		tmp = log((x / y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5e-19], N[Not[LessEqual[z, 5.2e-88]], $MachinePrecision]], (-z), N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000001e-19 or 5.20000000000000027e-88 < z

   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 77.2

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

   5. Applied rewrites 77.2%

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

   if -6.5000000000000001e-19 < z < 5.20000000000000027e-88

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. *-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 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-19} \lor \neg \left(z \leq 5.2 \cdot 10^{-88}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.8% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 76.8%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 54.4

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

5. Applied rewrites 54.4%

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

Developer Target 1: 88.4% 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 2024329 
(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))