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

Percentage Accurate: 77.2% → 99.4%

Time: 13.1s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (- (fma (log x) x (* x (log (/ 1.0 y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	} else {
		tmp = fma(log(x), x, (x * log((1.0 / y)))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-309)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y)))) - z);
	else
		tmp = Float64(fma(log(x), x, Float64(x * log(Float64(1.0 / y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-309], 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[Log[x], $MachinePrecision] * x + N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 77.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -1.000000000000002e-309 < y

   1. Initial program 73.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 72.7%

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

   4. Applied egg-rr 72.7%

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

      1. neg-log N/A

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

      2. clear-num N/A

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

      3. diff-log N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. fma-define N/A

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

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

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

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

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

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

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

      10. neg-log N/A

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

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

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

      12. /-lowering-/.f64 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.3× speedup

Math

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

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 4.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 49.4%

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

   5. Simplified 49.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 49.4%

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

   7. Applied egg-rr 49.4%

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

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

   1. Initial program 99.9%

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

4. Final simplification 86.9%

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

Alternative 3: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.35 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-126}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.35e+170)
   (* x (+ (log (- 0.0 x)) (log (/ -1.0 y))))
   (if (<= x -6.6e-126)
     (- (* (- 0.0 x) (log (/ y x))) z)
     (if (<= x -2e-308) (- 0.0 z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.35e+170) {
		tmp = x * (log((0.0 - x)) + log((-1.0 / y)));
	} else if (x <= -6.6e-126) {
		tmp = ((0.0 - x) * log((y / x))) - z;
	} else if (x <= -2e-308) {
		tmp = 0.0 - 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 (x <= (-4.35d+170)) then
        tmp = x * (log((0.0d0 - x)) + log(((-1.0d0) / y)))
    else if (x <= (-6.6d-126)) then
        tmp = ((0.0d0 - x) * log((y / x))) - z
    else if (x <= (-2d-308)) then
        tmp = 0.0d0 - 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 (x <= -4.35e+170) {
		tmp = x * (Math.log((0.0 - x)) + Math.log((-1.0 / y)));
	} else if (x <= -6.6e-126) {
		tmp = ((0.0 - x) * Math.log((y / x))) - z;
	} else if (x <= -2e-308) {
		tmp = 0.0 - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.35e+170:
		tmp = x * (math.log((0.0 - x)) + math.log((-1.0 / y)))
	elif x <= -6.6e-126:
		tmp = ((0.0 - x) * math.log((y / x))) - z
	elif x <= -2e-308:
		tmp = 0.0 - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.35e+170)
		tmp = Float64(x * Float64(log(Float64(0.0 - x)) + log(Float64(-1.0 / y))));
	elseif (x <= -6.6e-126)
		tmp = Float64(Float64(Float64(0.0 - x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-308)
		tmp = Float64(0.0 - 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 (x <= -4.35e+170)
		tmp = x * (log((0.0 - x)) + log((-1.0 / y)));
	elseif (x <= -6.6e-126)
		tmp = ((0.0 - x) * log((y / x))) - z;
	elseif (x <= -2e-308)
		tmp = 0.0 - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.35e+170], N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-126], N[(N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], N[(0.0 - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.35000000000000009e170

   1. Initial program 58.5%

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. remove-double-div N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 86.0%

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

   5. Simplified 86.0%

      \[\leadsto \color{blue}{x \cdot \left(\log \left(\frac{-1}{y}\right) + \log \left(0 - x\right)\right)} \]

   if -4.35000000000000009e170 < x < -6.6000000000000001e-126

   1. Initial program 90.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 92.4%

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

   4. Applied egg-rr 92.4%

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

   if -6.6000000000000001e-126 < x < -1.9999999999999998e-308

   1. Initial program 72.6%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 86.3%

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

   5. Simplified 86.3%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 86.3%

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

   7. Applied egg-rr 86.3%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 73.4%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.35 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-126}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{-124}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.75e-124)
   (- (* (- 0.0 x) (log (/ y x))) z)
   (if (<= x -2e-308) (- 0.0 z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.75e-124) {
		tmp = ((0.0 - x) * log((y / x))) - z;
	} else if (x <= -2e-308) {
		tmp = 0.0 - 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 (x <= (-3.75d-124)) then
        tmp = ((0.0d0 - x) * log((y / x))) - z
    else if (x <= (-2d-308)) then
        tmp = 0.0d0 - 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 (x <= -3.75e-124) {
		tmp = ((0.0 - x) * Math.log((y / x))) - z;
	} else if (x <= -2e-308) {
		tmp = 0.0 - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.75e-124:
		tmp = ((0.0 - x) * math.log((y / x))) - z
	elif x <= -2e-308:
		tmp = 0.0 - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.75e-124)
		tmp = Float64(Float64(Float64(0.0 - x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-308)
		tmp = Float64(0.0 - 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 (x <= -3.75e-124)
		tmp = ((0.0 - x) * log((y / x))) - z;
	elseif (x <= -2e-308)
		tmp = 0.0 - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.75e-124], N[(N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], N[(0.0 - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7499999999999998e-124

   1. Initial program 79.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 81.7%

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

   4. Applied egg-rr 81.7%

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

   if -3.7499999999999998e-124 < x < -1.9999999999999998e-308

   1. Initial program 72.6%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 86.3%

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

   5. Simplified 86.3%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 86.3%

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

   7. Applied egg-rr 86.3%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 73.4%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{-124}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log \left(\frac{1}{y}\right) + x \cdot \log x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (- (+ (* x (log (/ 1.0 y))) (* x (log x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	} else {
		tmp = ((x * log((1.0 / y))) + (x * log(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) :: tmp
    if (y <= (-1d-309)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - y)))) - z
    else
        tmp = ((x * log((1.0d0 / y))) + (x * log(x))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else {
		tmp = ((x * Math.log((1.0 / y))) + (x * Math.log(x))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-309:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - y)))) - z
	else:
		tmp = ((x * math.log((1.0 / y))) + (x * math.log(x))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-309)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(Float64(1.0 / y))) + Float64(x * log(x))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-309)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	else
		tmp = ((x * log((1.0 / y))) + (x * log(x))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-309], 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[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 77.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -1.000000000000002e-309 < y

   1. Initial program 73.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log \left(\frac{1}{y}\right) + x \cdot \log x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - 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 <= (-1d-309)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - 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 <= -1e-309) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-309:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - 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 <= -1e-309)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - 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 <= -1e-309)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-309], 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[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 77.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.6%

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

   4. Applied egg-rr 99.6%

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

   if -1.000000000000002e-309 < y

   1. Initial program 73.4%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

Alternative 7: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-153}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-83}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e-153)
   (- 0.0 z)
   (if (<= z 8.5e-83) (* (- 0.0 x) (log (/ y x))) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e-153) {
		tmp = 0.0 - z;
	} else if (z <= 8.5e-83) {
		tmp = (0.0 - x) * log((y / x));
	} else {
		tmp = 0.0 - 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 (z <= (-9d-153)) then
        tmp = 0.0d0 - z
    else if (z <= 8.5d-83) then
        tmp = (0.0d0 - x) * log((y / x))
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e-153) {
		tmp = 0.0 - z;
	} else if (z <= 8.5e-83) {
		tmp = (0.0 - x) * Math.log((y / x));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9e-153:
		tmp = 0.0 - z
	elif z <= 8.5e-83:
		tmp = (0.0 - x) * math.log((y / x))
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e-153)
		tmp = Float64(0.0 - z);
	elseif (z <= 8.5e-83)
		tmp = Float64(Float64(0.0 - x) * log(Float64(y / x)));
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9e-153)
		tmp = 0.0 - z;
	elseif (z <= 8.5e-83)
		tmp = (0.0 - x) * log((y / x));
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e-153], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 8.5e-83], N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e-153 or 8.49999999999999938e-83 < z

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 74.7%

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

   5. Simplified 74.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.7%

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

   7. Applied egg-rr 74.7%

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

   if -9e-153 < z < 8.49999999999999938e-83

   1. Initial program 72.0%

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

      1. unpow1 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 N/A

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

      11. unpow-1 N/A

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

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

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

      13. /-lowering-/.f64 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in x around -inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 42.5%

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

   7. Simplified 42.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. log-rec N/A

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

      6. sub-neg N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. remove-double-div N/A

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

      11. log-div N/A

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

      12. frac-2neg N/A

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

      13. clear-num N/A

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

      14. neg-log N/A

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

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

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

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

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

      17. /-lowering-/.f64 63.2%

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

   9. Applied egg-rr 63.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-153}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-83}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-153}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e-153)
   (- 0.0 z)
   (if (<= z 3.1e-84) (* x (log (/ x y))) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e-153) {
		tmp = 0.0 - z;
	} else if (z <= 3.1e-84) {
		tmp = x * log((x / y));
	} else {
		tmp = 0.0 - 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 (z <= (-9d-153)) then
        tmp = 0.0d0 - z
    else if (z <= 3.1d-84) then
        tmp = x * log((x / y))
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e-153) {
		tmp = 0.0 - z;
	} else if (z <= 3.1e-84) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9e-153:
		tmp = 0.0 - z
	elif z <= 3.1e-84:
		tmp = x * math.log((x / y))
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e-153)
		tmp = Float64(0.0 - z);
	elseif (z <= 3.1e-84)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9e-153)
		tmp = 0.0 - z;
	elseif (z <= 3.1e-84)
		tmp = x * log((x / y));
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e-153], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 3.1e-84], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e-153 or 3.10000000000000002e-84 < z

   1. Initial program 76.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 74.4%

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

   5. Simplified 74.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.4%

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

   7. Applied egg-rr 74.4%

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

   if -9e-153 < z < 3.10000000000000002e-84

   1. Initial program 73.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 62.7%

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

   5. Simplified 62.7%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-153}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.6% 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 75.2%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 57.0%

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

5. Simplified 57.0%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 57.0%

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

7. Applied egg-rr 57.0%

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

8. Final simplification 57.0%

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

Developer Target 1: 88.3% 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 2024160 
(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))