Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 72.2% → 99.8%

Time: 10.5s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.2:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{1}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y} \cdot e^{\frac{\mathsf{fma}\left(0.5, \frac{2 - {\left(\frac{1 - x}{x + -1}\right)}^{2}}{y}, 1\right)}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.2)
   (- 1.0 (log1p (* (- x y) (/ 1.0 (+ y -1.0)))))
   (-
    1.0
    (log
     (*
      (/ (+ x -1.0) y)
      (exp
       (/
        (fma 0.5 (/ (- 2.0 (pow (/ (- 1.0 x) (+ x -1.0)) 2.0)) y) 1.0)
        y)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.2) {
		tmp = 1.0 - log1p(((x - y) * (1.0 / (y + -1.0))));
	} else {
		tmp = 1.0 - log((((x + -1.0) / y) * exp((fma(0.5, ((2.0 - pow(((1.0 - x) / (x + -1.0)), 2.0)) / y), 1.0) / y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.2)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) * Float64(1.0 / Float64(y + -1.0)))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x + -1.0) / y) * exp(Float64(fma(0.5, Float64(Float64(2.0 - (Float64(Float64(1.0 - x) / Float64(x + -1.0)) ^ 2.0)) / y), 1.0) / y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.2], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] * N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[Exp[N[(N[(0.5 * N[(N[(2.0 - N[Power[N[(N[(1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.20000000000000001

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{y + -1}{x - y}}}\right) \]

      2. associate-/r/ 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{1}{y + -1} \cdot \left(x - y\right)}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{1}{y + -1} \cdot \left(x - y\right)}\right) \]

   if 0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 7.2%

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

      2. log1p-define 7.2%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 7.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 7.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 7.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 7.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 7.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 7.2%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 80.2%

      \[\leadsto 1 - \color{blue}{\left(\log \left(-1 \cdot \left(x - 1\right)\right) + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\left(-0.5 \cdot \frac{2 + -1 \cdot \frac{{\left(1 - x\right)}^{2}}{{\left(x - 1\right)}^{2}}}{y} + \frac{1}{x - 1}\right) - \frac{x}{x - 1}}{y}\right)\right)} \]

   7. Simplified 80.2%

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{-1}{y}\right) + \left(\mathsf{log1p}\left(-x\right) + \frac{-1 \cdot \frac{-0.5 \cdot \left(2 - \frac{{\left(1 - x\right)}^{2}}{{\left(x + -1\right)}^{2}}\right)}{y} + 1}{y}\right)\right)} \]

   9. Applied egg-rr 100.0%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{-1}{y} \cdot \left(\left(1 - x\right) \cdot e^{\frac{\mathsf{fma}\left(-1, \frac{-0.5 \cdot \left(2 - {\left(\frac{1 - x}{x + -1}\right)}^{2}\right)}{y}, 1\right)}{y}}\right)\right)} \]

   11. Simplified 100.0%

       \[\leadsto 1 - \color{blue}{\log \left(\frac{x + -1}{y} \cdot e^{\frac{\mathsf{fma}\left(0.5, \frac{2 - {\left(\frac{1 - x}{x + -1}\right)}^{2}}{y}, 1\right)}{y}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.2:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{1}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y} \cdot e^{\frac{\mathsf{fma}\left(0.5, \frac{2 - {\left(\frac{1 - x}{x + -1}\right)}^{2}}{y}, 1\right)}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (/ E (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((((double) M_E) / ((x + -1.0) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((Math.E / ((x + -1.0) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((math.e / ((x + -1.0) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999998999999999971

   1. Initial program 99.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. log1p-define 99.8%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 99.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 99.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 99.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 99.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 99.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   if 0.999998999999999971 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 5.5%

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

      2. log1p-define 5.5%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 5.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 5.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 5.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 5.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 5.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 5.5%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 18.5%

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

      1. log-rec 18.5%

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

      2. unsub-neg 18.5%

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

      3. sub-neg 18.5%

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

      4. metadata-eval 18.5%

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

   7. Simplified 18.5%

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

      1. add-log-exp 18.5%

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

      2. exp-diff 18.5%

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

      3. diff-log 99.6%

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

      4. add-exp-log 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. exp-1-e 99.6%

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

       2. +-commutative 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7) (not (<= y 1.0)))
   (log (/ E (/ (+ x -1.0) y)))
   (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7) || !(y <= 1.0)) {
		tmp = log((((double) M_E) / ((x + -1.0) / y)));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.7) || !(y <= 1.0)) {
		tmp = Math.log((Math.E / ((x + -1.0) / y)));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.7) or not (y <= 1.0):
		tmp = math.log((math.e / ((x + -1.0) / y)))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.7) || !(y <= 1.0))
		tmp = log(Float64(exp(1) / Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[Log[N[(E / N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.69999999999999996 or 1 < y

   1. Initial program 32.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 32.2%

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

      2. log1p-define 32.2%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 32.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 32.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 32.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 32.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 32.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 32.2%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 29.8%

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

      1. log-rec 29.8%

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

      2. unsub-neg 29.8%

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

      3. sub-neg 29.8%

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

      4. metadata-eval 29.8%

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

   7. Simplified 29.8%

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

      1. add-log-exp 29.8%

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

      2. exp-diff 29.8%

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

      3. diff-log 98.9%

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

      4. add-exp-log 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. exp-1-e 98.9%

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

       2. +-commutative 98.9%

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

   11. Simplified 98.9%

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

   if -1.69999999999999996 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. div-sub 98.2%

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

      3. mul-1-neg 98.2%

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

      4. sub-neg 98.2%

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

      5. *-inverses 98.2%

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

      6. *-rgt-identity 98.2%

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

      7. log1p-define 98.2%

         \[\leadsto 1 - \left(y + \color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)}\right) \]

      8. mul-1-neg 98.2%

         \[\leadsto 1 - \left(y + \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \]

   7. Simplified 98.2%

      \[\leadsto 1 - \color{blue}{\left(y + \mathsf{log1p}\left(-x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12.6:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -12.6)
   (log (* y (- E)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -12.6) {
		tmp = log((y * -((double) M_E)));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -12.6) {
		tmp = Math.log((y * -Math.E));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -12.6:
		tmp = math.log((y * -math.e))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -12.6)
		tmp = log(Float64(y * Float64(-exp(1))));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -12.6], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -12.5999999999999996

   1. Initial program 20.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 20.8%

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

      2. log1p-define 20.8%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 98.4%

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

      4. add-exp-log 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. exp-1-e 98.5%

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

       2. +-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around 0 68.4%

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

       1. associate-*r* 68.4%

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

       2. neg-mul-1 68.4%

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

   14. Simplified 68.4%

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

   if -12.5999999999999996 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. div-sub 98.2%

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

      3. mul-1-neg 98.2%

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

      4. sub-neg 98.2%

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

      5. *-inverses 98.2%

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

      6. *-rgt-identity 98.2%

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

      7. log1p-define 98.2%

         \[\leadsto 1 - \left(y + \color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)}\right) \]

      8. mul-1-neg 98.2%

         \[\leadsto 1 - \left(y + \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \]

   7. Simplified 98.2%

      \[\leadsto 1 - \color{blue}{\left(y + \mathsf{log1p}\left(-x\right)\right)} \]

   if 1 < y

   1. Initial program 58.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 58.3%

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

      2. log1p-define 58.3%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.5%

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

      1. log-rec 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. exp-diff 98.5%

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

      3. diff-log 99.9%

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

      4. add-exp-log 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. exp-1-e 99.9%

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

       2. +-commutative 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around inf 99.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12.6:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -240:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -240.0)
   (log (* y (- E)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -240.0) {
		tmp = log((y * -((double) M_E)));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -240.0) {
		tmp = Math.log((y * -Math.E));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -240.0:
		tmp = math.log((y * -math.e))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -240.0)
		tmp = log(Float64(y * Float64(-exp(1))));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -240.0], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -240

   1. Initial program 20.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 20.8%

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

      2. log1p-define 20.8%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 98.4%

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

      4. add-exp-log 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. exp-1-e 98.5%

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

       2. +-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around 0 68.4%

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

       1. associate-*r* 68.4%

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

       2. neg-mul-1 68.4%

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

   14. Simplified 68.4%

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

   if -240 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 97.2%

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

      1. log1p-define 97.2%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} \]

      2. mul-1-neg 97.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-x}\right) \]

   7. Simplified 97.2%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]

   if 1 < y

   1. Initial program 58.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 58.3%

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

      2. log1p-define 58.3%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 58.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.5%

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

      1. log-rec 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. exp-diff 98.5%

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

      3. diff-log 99.9%

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

      4. add-exp-log 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. exp-1-e 99.9%

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

       2. +-commutative 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around inf 99.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -240:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -330.0) (log (* y (- E))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -330.0) {
		tmp = log((y * -((double) M_E)));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -330.0) {
		tmp = Math.log((y * -Math.E));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -330.0:
		tmp = math.log((y * -math.e))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -330.0)
		tmp = log(Float64(y * Float64(-exp(1))));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -330.0], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -330

   1. Initial program 20.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 20.8%

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

      2. log1p-define 20.8%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 98.4%

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

      4. add-exp-log 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. exp-1-e 98.5%

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

       2. +-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around 0 68.4%

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

       1. associate-*r* 68.4%

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

       2. neg-mul-1 68.4%

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

   14. Simplified 68.4%

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

   if -330 < y

   1. Initial program 91.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 91.3%

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

      2. log1p-define 91.3%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 91.3%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.0%

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

      1. log1p-define 77.0%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} \]

      2. mul-1-neg 77.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-x}\right) \]

   7. Simplified 77.0%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -330:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-1 - y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + y \cdot 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45)
   (log (* y (- E)))
   (+
    1.0
    (* y (- -1.0 (* y (+ 0.5 (* y (+ 0.3333333333333333 (* y 0.25))))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45) {
		tmp = log((y * -((double) M_E)));
	} else {
		tmp = 1.0 + (y * (-1.0 - (y * (0.5 + (y * (0.3333333333333333 + (y * 0.25)))))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45) {
		tmp = Math.log((y * -Math.E));
	} else {
		tmp = 1.0 + (y * (-1.0 - (y * (0.5 + (y * (0.3333333333333333 + (y * 0.25)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45:
		tmp = math.log((y * -math.e))
	else:
		tmp = 1.0 + (y * (-1.0 - (y * (0.5 + (y * (0.3333333333333333 + (y * 0.25)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45)
		tmp = log(Float64(y * Float64(-exp(1))));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-1.0 - Float64(y * Float64(0.5 + Float64(y * Float64(0.3333333333333333 + Float64(y * 0.25))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45)
		tmp = log((y * -2.71828182845904523536));
	else
		tmp = 1.0 + (y * (-1.0 - (y * (0.5 + (y * (0.3333333333333333 + (y * 0.25)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(y * N[(-1.0 - N[(y * N[(0.5 + N[(y * N[(0.3333333333333333 + N[(y * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996

   1. Initial program 20.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 20.8%

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

      2. log1p-define 20.8%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 20.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 98.4%

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

      4. add-exp-log 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. exp-1-e 98.5%

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

       2. +-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around 0 68.4%

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

       1. associate-*r* 68.4%

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

       2. neg-mul-1 68.4%

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

   14. Simplified 68.4%

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

   if -1.44999999999999996 < y

   1. Initial program 91.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 91.3%

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

      2. log1p-define 91.3%

         \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

      3. distribute-neg-frac2 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

      4. neg-sub0 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

      7. +-commutative 91.3%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

   3. Simplified 91.3%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 50.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{y}{y - 1}}\right) \]
   6. Step-by-step derivation

      1. sub-neg 50.5%

         \[\leadsto 1 - \mathsf{log1p}\left(-1 \cdot \frac{y}{\color{blue}{y + \left(-1\right)}}\right) \]

      2. metadata-eval 50.5%

         \[\leadsto 1 - \mathsf{log1p}\left(-1 \cdot \frac{y}{y + \color{blue}{-1}}\right) \]

      3. neg-mul-1 50.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-\frac{y}{y + -1}}\right) \]

      4. distribute-neg-frac 50.5%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{-y}{y + -1}}\right) \]

   7. Simplified 50.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{-y}{y + -1}}\right) \]

   8. Taylor expanded in y around 0 50.3%

      \[\leadsto 1 - \color{blue}{y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 50.3%

         \[\leadsto 1 - y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + \color{blue}{y \cdot 0.25}\right)\right)\right) \]

   10. Simplified 50.3%

       \[\leadsto 1 - \color{blue}{y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + y \cdot 0.25\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-1 - y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + y \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.2% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 68.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Step-by-step derivation

   1. sub-neg 68.4%

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

   2. log1p-define 68.5%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]

   3. distribute-neg-frac2 68.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - y}{-\left(1 - y\right)}}\right) \]

   4. neg-sub0 68.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

   5. associate--r- 68.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

   6. metadata-eval 68.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{-1} + y}\right) \]

   7. +-commutative 68.5%

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{y + -1}}\right) \]

3. Simplified 68.5%

   \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 68.9%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x}{y - 1}}\right) \]

6. Taylor expanded in x around 0 37.5%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024135 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))