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

Percentage Accurate: 76.4% → 99.3%

Time: 10.7s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (- (+ (* x (log x)) (* x (log (/ 1.0 y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	} else {
		tmp = ((x * log(x)) + (x * log((1.0 / 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 <= (-5d-310)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - y)))) - z
    else
        tmp = ((x * log(x)) + (x * log((1.0d0 / y)))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else {
		tmp = ((x * Math.log(x)) + (x * Math.log((1.0 / y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - y)))) - z
	else:
		tmp = ((x * math.log(x)) + (x * math.log((1.0 / y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) + Float64(x * log(Float64(1.0 / y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	else
		tmp = ((x * log(x)) + (x * log((1.0 / y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 79.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log \left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(y\right)}\right)\right), z\right) \]

      2. log-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\log \left(\mathsf{neg}\left(x\right)\right) - \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log \left(\mathsf{neg}\left(x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\log 1 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), z\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(0 - y\right)\right)\right)\right), z\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\log 1 - y\right)\right)\right)\right), z\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, y\right)\right)\right)\right), z\right) \]

      13. metadata-eval 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(0 - x\right) - \log \left(0 - y\right)\right)} - z \]
   5. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

      2. neg-lowering-neg.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto x \cdot \left(\log \color{blue}{\left(-x\right)} - \log \left(0 - y\right)\right) - z \]

   if -4.999999999999985e-310 < y

   1. Initial program 75.2%

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

      1. log-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\log x - \log y\right)\right), z\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\log x + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right), z\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log x \cdot x + \left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right), z\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\log x \cdot x\right), \left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)\right), z\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log x, x\right), \left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)\right), z\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)\right), z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\log y\right)\right), x\right)\right), z\right) \]

      8. neg-log N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\log \left(\frac{1}{y}\right), x\right)\right), z\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{y}\right)\right), x\right)\right), z\right) \]

      10. /-lowering-/.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(1, y\right)\right), x\right)\right), z\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\log x \cdot x + \log \left(\frac{1}{y}\right) \cdot x\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- 0.0 z)
     (if (<= t_1 2e+270)
       (fma t_0 x (- 0.0 z))
       (- (* x (log x)) (* x (log y)))))))

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 = 0.0 - z;
	} else if (t_1 <= 2e+270) {
		tmp = fma(t_0, x, (0.0 - z));
	} else {
		tmp = (x * log(x)) - (x * log(y));
	}
	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(0.0 - z);
	elseif (t_1 <= 2e+270)
		tmp = fma(t_0, x, Float64(0.0 - z));
	else
		tmp = Float64(Float64(x * log(x)) - Float64(x * log(y)));
	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[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$1, 2e+270], N[(t$95$0 * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.9%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 60.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 60.3%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 60.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.0000000000000001e270

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. fmm-def N/A

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

      3. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\log \left(\frac{x}{y}\right), \color{blue}{x}, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{y}\right)\right), x, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(0 - z\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(\log 1 - z\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \mathsf{\_.f64}\left(\log 1, z\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \mathsf{\_.f64}\left(0, z\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)} \]
   5. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-lowering-neg.f64 99.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \mathsf{neg.f64}\left(z\right)\right) \]

   6. Applied egg-rr 99.7%

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

   if 2.0000000000000001e270 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 15.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + \log \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{\log y}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \color{blue}{\log y}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \log \color{blue}{y}\right)\right) \]

      15. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \log y\right)\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log x, \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \log \color{blue}{y}\right)\right) \]

      18. log-lowering-log.f64 57.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 57.2%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\log x \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\log x, x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)} \cdot x\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\log y}\right)\right) \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\log y\right)\right), \color{blue}{x}\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\left(0 - \log y\right), x\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \log y\right), x\right)\right) \]

      9. log-lowering-log.f64 57.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(y\right)\right), x\right)\right) \]

   7. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\log x \cdot x + \left(0 - \log y\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_1 \leq 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- 0.0 z)
     (if (<= t_1 1e+277) (fma t_0 x (- 0.0 z)) (* x (- (log x) (log y)))))))

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 = 0.0 - z;
	} else if (t_1 <= 1e+277) {
		tmp = fma(t_0, x, (0.0 - z));
	} else {
		tmp = x * (log(x) - log(y));
	}
	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(0.0 - z);
	elseif (t_1 <= 1e+277)
		tmp = fma(t_0, x, Float64(0.0 - z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	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[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$1, 1e+277], N[(t$95$0 * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.9%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 60.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 60.3%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 60.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e277

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. fmm-def N/A

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

      3. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\log \left(\frac{x}{y}\right), \color{blue}{x}, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{y}\right)\right), x, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(0 - z\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(\log 1 - z\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \mathsf{\_.f64}\left(\log 1, z\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \mathsf{\_.f64}\left(0, z\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)} \]
   5. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-lowering-neg.f64 99.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), x, \mathsf{neg.f64}\left(z\right)\right) \]

   6. Applied egg-rr 99.7%

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

   if 1e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + \log \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{\log y}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \color{blue}{\log y}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \log \color{blue}{y}\right)\right) \]

      15. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \log y\right)\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log x, \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \log \color{blue}{y}\right)\right) \]

      18. log-lowering-log.f64 55.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 55.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_0 \leq 10^{+277}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- 0.0 z)
     (if (<= t_0 1e+277) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0 - z;
	} else if (t_0 <= 1e+277) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - z;
	} else if (t_0 <= 1e+277) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.0 - z
	elif t_0 <= 1e+277:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0 - z);
	elseif (t_0 <= 1e+277)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.0 - z;
	elseif (t_0 <= 1e+277)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$0, 1e+277], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.9%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 60.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 60.3%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 60.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e277

   1. Initial program 99.7%

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

   if 1e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + \log \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{\log y}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \color{blue}{\log y}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \log \color{blue}{y}\right)\right) \]

      15. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \log y\right)\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log x, \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \log \color{blue}{y}\right)\right) \]

      18. log-lowering-log.f64 55.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 55.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+277}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- 0.0 z)
     (if (<= t_0 5e+306) (- t_0 z) (- 0.0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0 - z;
	} else if (t_0 <= 5e+306) {
		tmp = t_0 - z;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - z;
	} else if (t_0 <= 5e+306) {
		tmp = t_0 - z;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.0 - z
	elif t_0 <= 5e+306:
		tmp = t_0 - z
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0 - z);
	elseif (t_0 <= 5e+306)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.0 - z;
	elseif (t_0 <= 5e+306)
		tmp = t_0 - z;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(t$95$0 - z), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.99999999999999993e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.3%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 52.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 52.5%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 52.5%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 52.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.99999999999999993e306

   1. Initial program 99.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-31}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+166}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e-241)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (if (<= x 1.85e-31)
     (- 0.0 z)
     (if (<= x 2.05e+166)
       (- (* (- 0.0 x) (log (/ y x))) z)
       (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-241) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	} else if (x <= 1.85e-31) {
		tmp = 0.0 - z;
	} else if (x <= 2.05e+166) {
		tmp = ((0.0 - x) * log((y / x))) - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7.5d-241)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - y)))) - z
    else if (x <= 1.85d-31) then
        tmp = 0.0d0 - z
    else if (x <= 2.05d+166) then
        tmp = ((0.0d0 - x) * log((y / x))) - z
    else
        tmp = x * (log(x) - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-241) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else if (x <= 1.85e-31) {
		tmp = 0.0 - z;
	} else if (x <= 2.05e+166) {
		tmp = ((0.0 - x) * Math.log((y / x))) - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.5e-241:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - y)))) - z
	elif x <= 1.85e-31:
		tmp = 0.0 - z
	elif x <= 2.05e+166:
		tmp = ((0.0 - x) * math.log((y / x))) - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.5e-241)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y)))) - z);
	elseif (x <= 1.85e-31)
		tmp = Float64(0.0 - z);
	elseif (x <= 2.05e+166)
		tmp = Float64(Float64(Float64(0.0 - x) * log(Float64(y / x))) - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.5e-241)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	elseif (x <= 1.85e-31)
		tmp = 0.0 - z;
	elseif (x <= 2.05e+166)
		tmp = ((0.0 - x) * log((y / x))) - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.5e-241], N[(N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.85e-31], N[(0.0 - z), $MachinePrecision], If[LessEqual[x, 2.05e+166], N[(N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.49999999999999977e-241

   1. Initial program 79.5%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log \left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(y\right)}\right)\right), z\right) \]

      2. log-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\log \left(\mathsf{neg}\left(x\right)\right) - \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log \left(\mathsf{neg}\left(x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\log 1 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), z\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(0 - y\right)\right)\right)\right), z\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\log 1 - y\right)\right)\right)\right), z\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, y\right)\right)\right)\right), z\right) \]

      13. metadata-eval 99.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(0 - x\right) - \log \left(0 - y\right)\right)} - z \]
   5. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

      2. neg-lowering-neg.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto x \cdot \left(\log \color{blue}{\left(-x\right)} - \log \left(0 - y\right)\right) - z \]

   if -7.49999999999999977e-241 < x < 1.8499999999999999e-31

   1. Initial program 69.8%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 89.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 89.2%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 89.2%

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

   if 1.8499999999999999e-31 < x < 2.0500000000000001e166

   1. Initial program 90.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log \left(\frac{1}{\frac{y}{x}}\right)\right), z\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\log \left(\frac{y}{x}\right)\right)\right)\right), z\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\log \left(\frac{y}{x}\right)\right)\right), z\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x}\right)\right)\right)\right), z\right) \]

      5. /-lowering-/.f64 95.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right), z\right) \]

   4. Applied egg-rr 95.2%

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

   if 2.0500000000000001e166 < x

   1. Initial program 70.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + \log \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{\log y}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \color{blue}{\log y}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \log \color{blue}{y}\right)\right) \]

      15. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \log y\right)\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log x, \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \log \color{blue}{y}\right)\right) \]

      18. log-lowering-log.f64 93.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 93.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-31}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+166}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-33}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-15}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e-33)
   (* (- 0.0 x) (log (/ y x)))
   (if (<= x 7e-15) (- 0.0 z) (* x (log (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e-33) {
		tmp = (0.0 - x) * log((y / x));
	} else if (x <= 7e-15) {
		tmp = 0.0 - z;
	} else {
		tmp = x * log((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.8d-33)) then
        tmp = (0.0d0 - x) * log((y / x))
    else if (x <= 7d-15) then
        tmp = 0.0d0 - z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e-33) {
		tmp = (0.0 - x) * Math.log((y / x));
	} else if (x <= 7e-15) {
		tmp = 0.0 - z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.8e-33:
		tmp = (0.0 - x) * math.log((y / x))
	elif x <= 7e-15:
		tmp = 0.0 - z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e-33)
		tmp = Float64(Float64(0.0 - x) * log(Float64(y / x)));
	elseif (x <= 7e-15)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.8e-33)
		tmp = (0.0 - x) * log((y / x));
	elseif (x <= 7e-15)
		tmp = 0.0 - z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e-33], N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-15], N[(0.0 - z), $MachinePrecision], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e-33

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + \log \left(\frac{\color{blue}{1}}{y}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{\log y}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \color{blue}{\log y}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \log \color{blue}{y}\right)\right) \]

      15. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \log y\right)\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log x, \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \log \color{blue}{y}\right)\right) \]

      18. log-lowering-log.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 0.0%

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

      1. diff-log N/A

         \[\leadsto \mathsf{*.f64}\left(x, \log \left(\frac{x}{y}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(x, \log \left(\frac{1}{\frac{y}{x}}\right)\right) \]

      3. neg-log N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\log \left(\frac{y}{x}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\log \left(\frac{y}{x}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right) \]

   7. Applied egg-rr 64.6%

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

   if -4.8e-33 < x < 7.0000000000000001e-15

   1. Initial program 74.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 83.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.3%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 83.3%

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

   if 7.0000000000000001e-15 < x

   1. Initial program 80.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log \left(\frac{x}{y}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(\left(\frac{x}{y}\right)\right)\right) \]

      3. /-lowering-/.f64 58.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right) \]

   5. Simplified 58.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-33}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-15}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= x -5.6e-52) t_0 (if (<= x 2.5e-16) (- 0.0 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (x <= -5.6e-52) {
		tmp = t_0;
	} else if (x <= 2.5e-16) {
		tmp = 0.0 - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * log((x / y))
    if (x <= (-5.6d-52)) then
        tmp = t_0
    else if (x <= 2.5d-16) then
        tmp = 0.0d0 - z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (x <= -5.6e-52) {
		tmp = t_0;
	} else if (x <= 2.5e-16) {
		tmp = 0.0 - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if x <= -5.6e-52:
		tmp = t_0
	elif x <= 2.5e-16:
		tmp = 0.0 - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (x <= -5.6e-52)
		tmp = t_0;
	elseif (x <= 2.5e-16)
		tmp = Float64(0.0 - z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (x <= -5.6e-52)
		tmp = t_0;
	elseif (x <= 2.5e-16)
		tmp = 0.0 - z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e-52], t$95$0, If[LessEqual[x, 2.5e-16], N[(0.0 - z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999989e-52 or 2.5000000000000002e-16 < x

   1. Initial program 80.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log \left(\frac{x}{y}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(\left(\frac{x}{y}\right)\right)\right) \]

      3. /-lowering-/.f64 60.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right) \]

   5. Simplified 60.5%

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

   if -5.59999999999999989e-52 < x < 2.5000000000000002e-16

   1. Initial program 74.2%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 83.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 83.9%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.9%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 83.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.7% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 0 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 0.0 z))

C

double code(double x, double y, double z) {
	return 0.0 - 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 = 0.0d0 - z
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 - z;
}

Python

def code(x, y, z):
	return 0.0 - z

Julia

function code(x, y, z)
	return Float64(0.0 - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 - z;
end

Wolfram

code[x_, y_, z_] := N[(0.0 - z), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 50.7%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

5. Simplified 50.7%

   \[\leadsto \color{blue}{0 - z} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 50.7%

      \[\leadsto \mathsf{neg.f64}\left(z\right) \]

7. Applied egg-rr 50.7%

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

8. Final simplification 50.7%

   \[\leadsto 0 - z \]
9. Add Preprocessing

Alternative 10: 2.2% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 77.3%

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

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 50.7%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

5. Simplified 50.7%

   \[\leadsto \color{blue}{0 - z} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 50.7%

      \[\leadsto \mathsf{neg.f64}\left(z\right) \]

7. Applied egg-rr 50.7%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. metadata-eval N/A

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

   4. cube-unmult N/A

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

   5. sub0-neg N/A

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

   6. cube-unmult N/A

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

   7. cube-neg N/A

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

   8. neg-sub0 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. neg-sub0 N/A

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

   12. neg-sub0 N/A

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

   13. sqr-neg N/A

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

   14. pow-prod-down N/A

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

   15. sqr-pow N/A

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

   16. metadata-eval N/A

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

   17. +-lft-identity N/A

       \[\leadsto \frac{{z}^{3}}{z \cdot z + \color{blue}{0 \cdot z}} \]

   18. distribute-rgt-out N/A

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

   19. +-commutative N/A

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

   20. +-lft-identity N/A

       \[\leadsto \frac{{z}^{3}}{z \cdot z} \]

   21. pow2 N/A

       \[\leadsto \frac{{z}^{3}}{{z}^{\color{blue}{2}}} \]

   22. pow-div N/A

       \[\leadsto {z}^{\color{blue}{\left(3 - 2\right)}} \]

   23. metadata-eval N/A

       \[\leadsto {z}^{1} \]

   24. unpow1 2.1%

       \[\leadsto z \]

9. Applied egg-rr 2.1%

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

Developer Target 1: 87.9% accurate, 0.5× 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 2024158 
(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))