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

Percentage Accurate: 72.5% → 99.7%

Time: 13.7s

Alternatives: 13

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: 72.5% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380000:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -380000.0)
   (+ 1.0 (- (/ -1.0 y) (+ (log1p (- x)) (log (/ -1.0 y)))))
   (if (<= y 1.2e+16)
     (- 1.0 (log1p (* (/ 1.0 (+ y -1.0)) (- x y))))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -380000.0) {
		tmp = 1.0 + ((-1.0 / y) - (log1p(-x) + log((-1.0 / y))));
	} else if (y <= 1.2e+16) {
		tmp = 1.0 - log1p(((1.0 / (y + -1.0)) * (x - y)));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -380000.0) {
		tmp = 1.0 + ((-1.0 / y) - (Math.log1p(-x) + Math.log((-1.0 / y))));
	} else if (y <= 1.2e+16) {
		tmp = 1.0 - Math.log1p(((1.0 / (y + -1.0)) * (x - y)));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -380000.0:
		tmp = 1.0 + ((-1.0 / y) - (math.log1p(-x) + math.log((-1.0 / y))))
	elif y <= 1.2e+16:
		tmp = 1.0 - math.log1p(((1.0 / (y + -1.0)) * (x - y)))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -380000.0)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y)))));
	elseif (y <= 1.2e+16)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(y + -1.0)) * Float64(x - y))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -380000.0], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+16], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e5

   1. Initial program 25.2%

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

      1. sub-neg 25.2%

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

      2. log1p-define 25.2%

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

      3. distribute-neg-frac2 25.2%

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

      4. neg-sub0 25.2%

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

      5. associate--r- 25.2%

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

      6. metadata-eval 25.2%

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

      7. +-commutative 25.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in y around -inf 99.3%

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

   7. Simplified 99.3%

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

   if -3.8e5 < y < 1.2e16

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 1.2e16 < y

   1. Initial program 41.0%

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

      1. sub-neg 41.0%

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

      2. log1p-define 41.0%

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

      3. distribute-neg-frac2 41.0%

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

      4. neg-sub0 41.0%

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

      5. associate--r- 41.0%

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

      6. metadata-eval 41.0%

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

      7. +-commutative 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in y around inf 41.0%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. log-rec 98.5%

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

      2. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

      1. sub-neg 98.5%

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

      2. diff-log 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. sub-neg 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380000:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1650000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1650000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 5e+15)
     (- 1.0 (log1p (* (/ 1.0 (+ y -1.0)) (- x y))))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1650000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 5e+15) {
		tmp = 1.0 - log1p(((1.0 / (y + -1.0)) * (x - y)));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1650000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 5e+15) {
		tmp = 1.0 - Math.log1p(((1.0 / (y + -1.0)) * (x - y)));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1650000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 5e+15:
		tmp = 1.0 - math.log1p(((1.0 / (y + -1.0)) * (x - y)))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1650000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 5e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(y + -1.0)) * Float64(x - y))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1650000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+15], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.65e9

   1. Initial program 25.2%

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

      1. sub-neg 25.2%

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

      2. log1p-define 25.2%

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

      3. distribute-neg-frac2 25.2%

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

      4. neg-sub0 25.2%

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

      5. associate--r- 25.2%

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

      6. metadata-eval 25.2%

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

      7. +-commutative 25.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in y around -inf 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-lft-in 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. log1p-define 99.1%

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

      7. mul-1-neg 99.1%

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

   7. Simplified 99.1%

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

   if -1.65e9 < y < 5e15

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 5e15 < y

   1. Initial program 41.0%

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

      1. sub-neg 41.0%

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

      2. log1p-define 41.0%

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

      3. distribute-neg-frac2 41.0%

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

      4. neg-sub0 41.0%

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

      5. associate--r- 41.0%

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

      6. metadata-eval 41.0%

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

      7. +-commutative 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in y around inf 41.0%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. log-rec 98.5%

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

      2. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

      1. sub-neg 98.5%

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

      2. diff-log 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. sub-neg 99.9%

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

   12. Simplified 99.9%

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

Alternative 3: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\frac{1}{y + -1} - \frac{\frac{y}{x}}{y + -1}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(y \cdot \left(-1 - x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.6e-6)
   (- 1.0 (log1p (* x (- (/ 1.0 (+ y -1.0)) (/ (/ y x) (+ y -1.0))))))
   (if (<= x 1.0)
     (+ 1.0 (log1p (- (* y (- -1.0 x)) x)))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.6e-6) {
		tmp = 1.0 - log1p((x * ((1.0 / (y + -1.0)) - ((y / x) / (y + -1.0)))));
	} else if (x <= 1.0) {
		tmp = 1.0 + log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.6e-6) {
		tmp = 1.0 - Math.log1p((x * ((1.0 / (y + -1.0)) - ((y / x) / (y + -1.0)))));
	} else if (x <= 1.0) {
		tmp = 1.0 + Math.log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.6e-6:
		tmp = 1.0 - math.log1p((x * ((1.0 / (y + -1.0)) - ((y / x) / (y + -1.0)))))
	elif x <= 1.0:
		tmp = 1.0 + math.log1p(((y * (-1.0 - x)) - x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.6e-6)
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(1.0 / Float64(y + -1.0)) - Float64(Float64(y / x) / Float64(y + -1.0))))));
	elseif (x <= 1.0)
		tmp = Float64(1.0 + log1p(Float64(Float64(y * Float64(-1.0 - x)) - x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.6e-6], N[(1.0 - N[Log[1 + N[(x * N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / x), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[Log[1 + N[(N[(y * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.60000000000000034e-6

   1. Initial program 88.3%

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

      1. sub-neg 88.3%

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

      2. log1p-define 88.3%

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

      3. distribute-neg-frac2 88.3%

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

      4. neg-sub0 88.3%

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

      5. associate--r- 88.3%

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

      6. metadata-eval 88.3%

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

      7. +-commutative 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around -inf 86.8%

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

      1. associate-*r* 86.8%

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

      2. mul-1-neg 86.8%

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

      3. sub-neg 86.8%

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

      4. metadata-eval 86.8%

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

      5. associate-/r* 89.1%

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

      6. sub-neg 89.1%

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

      7. metadata-eval 89.1%

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

   7. Simplified 89.1%

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

   if -6.60000000000000034e-6 < x < 1

   1. Initial program 70.3%

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

      1. sub-neg 70.3%

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

      2. log1p-define 70.4%

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

      3. distribute-neg-frac2 70.4%

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

      4. neg-sub0 70.4%

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

      5. associate--r- 70.4%

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

      6. metadata-eval 70.4%

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

      7. +-commutative 70.4%

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

   3. Simplified 70.4%

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

      1. add-exp-log 70.4%

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

      2. sub-neg 70.4%

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

      3. log1p-define 70.4%

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

   6. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}} \]
   7. Step-by-step derivation

      1. log1p-undefine 70.4%

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

      2. rem-exp-log 70.4%

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

      3. add-sqr-sqrt 38.9%

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

      4. sqrt-unprod 70.2%

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

      5. sqr-neg 70.2%

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

      6. sqrt-unprod 31.4%

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

      7. add-sqr-sqrt 67.1%

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

      8. sub-neg 67.1%

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

      9. add-sqr-sqrt 37.3%

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

      10. sqrt-unprod 67.0%

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

      11. sqr-neg 67.0%

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

      12. sqrt-unprod 29.8%

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

      13. add-sqr-sqrt 71.6%

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

   8. Applied egg-rr 71.6%

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

   9. Taylor expanded in y around 0 98.5%

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

       1. +-commutative 98.5%

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

       2. mul-1-neg 98.5%

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

       3. unsub-neg 98.5%

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

       4. mul-1-neg 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

       6. distribute-neg-in 98.5%

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

       7. metadata-eval 98.5%

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

       8. unsub-neg 98.5%

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

   11. Simplified 98.5%

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

   if 1 < x

   1. Initial program 45.8%

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

      1. sub-neg 45.8%

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

      2. log1p-define 45.8%

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

      3. distribute-neg-frac2 45.8%

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

      4. neg-sub0 45.8%

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

      5. associate--r- 45.8%

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

      6. metadata-eval 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in y around inf 41.4%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. log-rec 94.3%

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

      2. unsub-neg 94.3%

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

   8. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. diff-log 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. sub-neg 95.6%

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

   12. Simplified 95.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\frac{1}{y + -1} - \frac{\frac{y}{x}}{y + -1}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(y \cdot \left(-1 - x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\frac{1}{y + -1} - \frac{\frac{y}{y + -1}}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(y \cdot \left(-1 - x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e-5)
   (- 1.0 (log1p (* x (- (/ 1.0 (+ y -1.0)) (/ (/ y (+ y -1.0)) x)))))
   (if (<= x 1.0)
     (+ 1.0 (log1p (- (* y (- -1.0 x)) x)))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-5) {
		tmp = 1.0 - log1p((x * ((1.0 / (y + -1.0)) - ((y / (y + -1.0)) / x))));
	} else if (x <= 1.0) {
		tmp = 1.0 + log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-5) {
		tmp = 1.0 - Math.log1p((x * ((1.0 / (y + -1.0)) - ((y / (y + -1.0)) / x))));
	} else if (x <= 1.0) {
		tmp = 1.0 + Math.log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.25e-5:
		tmp = 1.0 - math.log1p((x * ((1.0 / (y + -1.0)) - ((y / (y + -1.0)) / x))))
	elif x <= 1.0:
		tmp = 1.0 + math.log1p(((y * (-1.0 - x)) - x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.25e-5)
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(1.0 / Float64(y + -1.0)) - Float64(Float64(y / Float64(y + -1.0)) / x)))));
	elseif (x <= 1.0)
		tmp = Float64(1.0 + log1p(Float64(Float64(y * Float64(-1.0 - x)) - x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.25e-5], N[(1.0 - N[Log[1 + N[(x * N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[Log[1 + N[(N[(y * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.25000000000000014e-5

   1. Initial program 88.3%

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

      1. sub-neg 88.3%

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

      2. log1p-define 88.3%

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

      3. distribute-neg-frac2 88.3%

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

      4. neg-sub0 88.3%

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

      5. associate--r- 88.3%

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

      6. metadata-eval 88.3%

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

      7. +-commutative 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf 86.8%

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

      1. +-commutative 86.8%

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

      2. mul-1-neg 86.8%

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

      3. unsub-neg 86.8%

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

      4. sub-neg 86.8%

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

      5. metadata-eval 86.8%

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

      6. sub-neg 86.8%

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

      7. metadata-eval 86.8%

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

      8. *-commutative 86.8%

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

      9. associate-/r* 88.5%

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

   7. Simplified 88.5%

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

   if -2.25000000000000014e-5 < x < 1

   1. Initial program 70.3%

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

      1. sub-neg 70.3%

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

      2. log1p-define 70.4%

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

      3. distribute-neg-frac2 70.4%

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

      4. neg-sub0 70.4%

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

      5. associate--r- 70.4%

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

      6. metadata-eval 70.4%

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

      7. +-commutative 70.4%

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

   3. Simplified 70.4%

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

      1. add-exp-log 70.4%

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

      2. sub-neg 70.4%

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

      3. log1p-define 70.4%

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

   6. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}} \]
   7. Step-by-step derivation

      1. log1p-undefine 70.4%

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

      2. rem-exp-log 70.4%

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

      3. add-sqr-sqrt 38.9%

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

      4. sqrt-unprod 70.2%

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

      5. sqr-neg 70.2%

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

      6. sqrt-unprod 31.4%

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

      7. add-sqr-sqrt 67.1%

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

      8. sub-neg 67.1%

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

      9. add-sqr-sqrt 37.3%

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

      10. sqrt-unprod 67.0%

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

      11. sqr-neg 67.0%

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

      12. sqrt-unprod 29.8%

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

      13. add-sqr-sqrt 71.6%

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

   8. Applied egg-rr 71.6%

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

   9. Taylor expanded in y around 0 98.5%

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

       1. +-commutative 98.5%

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

       2. mul-1-neg 98.5%

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

       3. unsub-neg 98.5%

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

       4. mul-1-neg 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

       6. distribute-neg-in 98.5%

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

       7. metadata-eval 98.5%

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

       8. unsub-neg 98.5%

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

   11. Simplified 98.5%

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

   if 1 < x

   1. Initial program 45.8%

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

      1. sub-neg 45.8%

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

      2. log1p-define 45.8%

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

      3. distribute-neg-frac2 45.8%

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

      4. neg-sub0 45.8%

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

      5. associate--r- 45.8%

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

      6. metadata-eval 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in y around inf 41.4%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. log-rec 94.3%

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

      2. unsub-neg 94.3%

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

   8. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. diff-log 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. sub-neg 95.6%

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

   12. Simplified 95.6%

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

Alternative 5: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(y \cdot \left(-1 - x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e-5)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (if (<= x 1.0)
     (+ 1.0 (log1p (- (* y (- -1.0 x)) x)))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-5) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else if (x <= 1.0) {
		tmp = 1.0 + log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-5) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else if (x <= 1.0) {
		tmp = 1.0 + Math.log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e-5:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	elif x <= 1.0:
		tmp = 1.0 + math.log1p(((y * (-1.0 - x)) - x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e-5)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	elseif (x <= 1.0)
		tmp = Float64(1.0 + log1p(Float64(Float64(y * Float64(-1.0 - x)) - x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e-5], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[Log[1 + N[(N[(y * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-5

   1. Initial program 88.3%

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

      1. sub-neg 88.3%

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

      2. log1p-define 88.3%

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

      3. distribute-neg-frac2 88.3%

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

      4. neg-sub0 88.3%

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

      5. associate--r- 88.3%

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

      6. metadata-eval 88.3%

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

      7. +-commutative 88.3%

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

   3. Simplified 88.3%

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

   if -2.9e-5 < x < 1

   1. Initial program 70.3%

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

      1. sub-neg 70.3%

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

      2. log1p-define 70.4%

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

      3. distribute-neg-frac2 70.4%

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

      4. neg-sub0 70.4%

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

      5. associate--r- 70.4%

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

      6. metadata-eval 70.4%

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

      7. +-commutative 70.4%

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

   3. Simplified 70.4%

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

      1. add-exp-log 70.4%

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

      2. sub-neg 70.4%

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

      3. log1p-define 70.4%

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

   6. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}} \]
   7. Step-by-step derivation

      1. log1p-undefine 70.4%

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

      2. rem-exp-log 70.4%

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

      3. add-sqr-sqrt 38.9%

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

      4. sqrt-unprod 70.2%

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

      5. sqr-neg 70.2%

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

      6. sqrt-unprod 31.4%

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

      7. add-sqr-sqrt 67.1%

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

      8. sub-neg 67.1%

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

      9. add-sqr-sqrt 37.3%

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

      10. sqrt-unprod 67.0%

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

      11. sqr-neg 67.0%

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

      12. sqrt-unprod 29.8%

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

      13. add-sqr-sqrt 71.6%

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

   8. Applied egg-rr 71.6%

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

   9. Taylor expanded in y around 0 98.5%

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

       1. +-commutative 98.5%

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

       2. mul-1-neg 98.5%

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

       3. unsub-neg 98.5%

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

       4. mul-1-neg 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

       6. distribute-neg-in 98.5%

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

       7. metadata-eval 98.5%

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

       8. unsub-neg 98.5%

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

   11. Simplified 98.5%

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

   if 1 < x

   1. Initial program 45.8%

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

      1. sub-neg 45.8%

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

      2. log1p-define 45.8%

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

      3. distribute-neg-frac2 45.8%

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

      4. neg-sub0 45.8%

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

      5. associate--r- 45.8%

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

      6. metadata-eval 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in y around inf 41.4%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. log-rec 94.3%

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

      2. unsub-neg 94.3%

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

   8. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. diff-log 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. sub-neg 95.6%

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

   12. Simplified 95.6%

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

Alternative 6: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(y \cdot \left(-1 - x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.6e-6)
   (- 1.0 (log1p (/ x (+ y -1.0))))
   (if (<= x 1.0)
     (+ 1.0 (log1p (- (* y (- -1.0 x)) x)))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.6e-6) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else if (x <= 1.0) {
		tmp = 1.0 + log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.6e-6) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else if (x <= 1.0) {
		tmp = 1.0 + Math.log1p(((y * (-1.0 - x)) - x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.6e-6:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	elif x <= 1.0:
		tmp = 1.0 + math.log1p(((y * (-1.0 - x)) - x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.6e-6)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	elseif (x <= 1.0)
		tmp = Float64(1.0 + log1p(Float64(Float64(y * Float64(-1.0 - x)) - x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.6e-6], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[Log[1 + N[(N[(y * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.60000000000000034e-6

   1. Initial program 88.3%

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

      1. sub-neg 88.3%

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

      2. log1p-define 88.3%

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

      3. distribute-neg-frac2 88.3%

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

      4. neg-sub0 88.3%

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

      5. associate--r- 88.3%

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

      6. metadata-eval 88.3%

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

      7. +-commutative 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf 86.6%

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

   if -6.60000000000000034e-6 < x < 1

   1. Initial program 70.3%

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

      1. sub-neg 70.3%

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

      2. log1p-define 70.4%

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

      3. distribute-neg-frac2 70.4%

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

      4. neg-sub0 70.4%

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

      5. associate--r- 70.4%

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

      6. metadata-eval 70.4%

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

      7. +-commutative 70.4%

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

   3. Simplified 70.4%

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

      1. add-exp-log 70.4%

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

      2. sub-neg 70.4%

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

      3. log1p-define 70.4%

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

   6. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}} \]
   7. Step-by-step derivation

      1. log1p-undefine 70.4%

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

      2. rem-exp-log 70.4%

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

      3. add-sqr-sqrt 38.9%

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

      4. sqrt-unprod 70.2%

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

      5. sqr-neg 70.2%

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

      6. sqrt-unprod 31.4%

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

      7. add-sqr-sqrt 67.1%

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

      8. sub-neg 67.1%

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

      9. add-sqr-sqrt 37.3%

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

      10. sqrt-unprod 67.0%

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

      11. sqr-neg 67.0%

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

      12. sqrt-unprod 29.8%

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

      13. add-sqr-sqrt 71.6%

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

   8. Applied egg-rr 71.6%

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

   9. Taylor expanded in y around 0 98.5%

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

       1. +-commutative 98.5%

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

       2. mul-1-neg 98.5%

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

       3. unsub-neg 98.5%

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

       4. mul-1-neg 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

       6. distribute-neg-in 98.5%

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

       7. metadata-eval 98.5%

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

       8. unsub-neg 98.5%

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

   11. Simplified 98.5%

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

   if 1 < x

   1. Initial program 45.8%

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

      1. sub-neg 45.8%

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

      2. log1p-define 45.8%

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

      3. distribute-neg-frac2 45.8%

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

      4. neg-sub0 45.8%

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

      5. associate--r- 45.8%

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

      6. metadata-eval 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in y around inf 41.4%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. log-rec 94.3%

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

      2. unsub-neg 94.3%

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

   8. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. diff-log 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. sub-neg 95.6%

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

   12. Simplified 95.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(y \cdot \left(-1 - x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 40000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -65.0)
   (+ 1.0 (log (- y)))
   (if (<= y 40000.0) (- 1.0 (log1p (/ x (+ y -1.0)))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -65.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 40000.0) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -65.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 40000.0) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -65.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 40000.0:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -65.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 40000.0)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -65.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 40000.0], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -65

   1. Initial program 27.1%

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

      1. sub-neg 27.1%

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

      2. log1p-define 27.1%

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

      3. distribute-neg-frac2 27.1%

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

      4. neg-sub0 27.1%

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

      5. associate--r- 27.1%

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

      6. metadata-eval 27.1%

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

      7. +-commutative 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in x around 0 5.7%

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

      1. sub-neg 5.7%

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

      2. metadata-eval 5.7%

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

      3. neg-mul-1 5.7%

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

      4. distribute-neg-frac 5.7%

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

   7. Simplified 5.7%

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

   8. Taylor expanded in y around -inf 63.2%

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

      1. sub-neg 63.2%

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

      2. neg-log 63.2%

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

      3. clear-num 63.2%

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

      4. div-inv 63.2%

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

      5. metadata-eval 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-commutative 63.2%

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

       2. neg-mul-1 63.2%

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

   12. Simplified 63.2%

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

   if -65 < y < 4e4

   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 x around inf 99.9%

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

   if 4e4 < y

   1. Initial program 41.0%

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

      1. sub-neg 41.0%

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

      2. log1p-define 41.0%

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

      3. distribute-neg-frac2 41.0%

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

      4. neg-sub0 41.0%

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

      5. associate--r- 41.0%

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

      6. metadata-eval 41.0%

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

      7. +-commutative 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in y around inf 41.0%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. log-rec 98.5%

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

      2. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

      1. sub-neg 98.5%

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

      2. diff-log 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. sub-neg 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 40000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -27.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -27.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -27.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -27.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -27.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -27.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -27

   1. Initial program 27.1%

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

      1. sub-neg 27.1%

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

      2. log1p-define 27.1%

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

      3. distribute-neg-frac2 27.1%

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

      4. neg-sub0 27.1%

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

      5. associate--r- 27.1%

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

      6. metadata-eval 27.1%

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

      7. +-commutative 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in x around 0 5.7%

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

      1. sub-neg 5.7%

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

      2. metadata-eval 5.7%

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

      3. neg-mul-1 5.7%

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

      4. distribute-neg-frac 5.7%

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

   7. Simplified 5.7%

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

   8. Taylor expanded in y around -inf 63.2%

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

      1. sub-neg 63.2%

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

      2. neg-log 63.2%

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

      3. clear-num 63.2%

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

      4. div-inv 63.2%

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

      5. metadata-eval 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-commutative 63.2%

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

       2. neg-mul-1 63.2%

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

   12. Simplified 63.2%

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

   if -27 < 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 0 100.0%

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

      1. +-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. log1p-define 100.0%

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

      8. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if 1 < y

   1. Initial program 45.8%

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

      1. sub-neg 45.8%

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

      2. log1p-define 45.8%

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

      3. distribute-neg-frac2 45.8%

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

      4. neg-sub0 45.8%

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

      5. associate--r- 45.8%

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

      6. metadata-eval 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in y around inf 41.4%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. log-rec 94.3%

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

      2. unsub-neg 94.3%

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

   8. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. diff-log 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. sub-neg 95.6%

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

   12. Simplified 95.6%

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

Alternative 9: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -55:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -55.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -55.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -55.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -55.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -55.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -55.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -55

   1. Initial program 27.1%

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

      1. sub-neg 27.1%

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

      2. log1p-define 27.1%

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

      3. distribute-neg-frac2 27.1%

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

      4. neg-sub0 27.1%

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

      5. associate--r- 27.1%

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

      6. metadata-eval 27.1%

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

      7. +-commutative 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in x around 0 5.7%

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

      1. sub-neg 5.7%

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

      2. metadata-eval 5.7%

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

      3. neg-mul-1 5.7%

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

      4. distribute-neg-frac 5.7%

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

   7. Simplified 5.7%

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

   8. Taylor expanded in y around -inf 63.2%

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

      1. sub-neg 63.2%

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

      2. neg-log 63.2%

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

      3. clear-num 63.2%

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

      4. div-inv 63.2%

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

      5. metadata-eval 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-commutative 63.2%

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

       2. neg-mul-1 63.2%

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

   12. Simplified 63.2%

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

   if -55 < 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 99.9%

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

      1. log1p-define 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   if 1 < y

   1. Initial program 45.8%

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

      1. sub-neg 45.8%

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

      2. log1p-define 45.8%

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

      3. distribute-neg-frac2 45.8%

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

      4. neg-sub0 45.8%

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

      5. associate--r- 45.8%

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

      6. metadata-eval 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in y around inf 41.4%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. log-rec 94.3%

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

      2. unsub-neg 94.3%

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

   8. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. diff-log 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. sub-neg 95.6%

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

   12. Simplified 95.6%

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

Alternative 10: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -31.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -31.5) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -31.5) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -31.5) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -31.5:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -31.5)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -31.5], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -31.5

   1. Initial program 27.1%

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

      1. sub-neg 27.1%

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

      2. log1p-define 27.1%

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

      3. distribute-neg-frac2 27.1%

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

      4. neg-sub0 27.1%

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

      5. associate--r- 27.1%

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

      6. metadata-eval 27.1%

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

      7. +-commutative 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in x around 0 5.7%

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

      1. sub-neg 5.7%

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

      2. metadata-eval 5.7%

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

      3. neg-mul-1 5.7%

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

      4. distribute-neg-frac 5.7%

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

   7. Simplified 5.7%

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

   8. Taylor expanded in y around -inf 63.2%

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

      1. sub-neg 63.2%

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

      2. neg-log 63.2%

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

      3. clear-num 63.2%

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

      4. div-inv 63.2%

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

      5. metadata-eval 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. *-commutative 63.2%

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

       2. neg-mul-1 63.2%

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

   12. Simplified 63.2%

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

   if -31.5 < y

   1. Initial program 92.6%

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

      1. sub-neg 92.6%

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

      2. log1p-define 92.6%

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

      3. distribute-neg-frac2 92.6%

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

      4. neg-sub0 92.6%

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

      5. associate--r- 92.6%

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

      6. metadata-eval 92.6%

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

      7. +-commutative 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in y around 0 86.2%

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

      1. log1p-define 86.3%

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

      2. mul-1-neg 86.3%

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

   7. Simplified 86.3%

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

Alternative 11: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.85e-13) (+ 1.0 (log (- y))) (- 1.0 (/ x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.85e-13) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - (x / (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.85d-13)) then
        tmp = 1.0d0 + log(-y)
    else
        tmp = 1.0d0 - (x / (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.85e-13) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - (x / (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.85e-13:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - (x / (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.85e-13)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.85e-13)
		tmp = 1.0 + log(-y);
	else
		tmp = 1.0 - (x / (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.85e-13], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999994e-13

   1. Initial program 28.1%

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

      1. sub-neg 28.1%

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

      2. log1p-define 28.1%

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

      3. distribute-neg-frac2 28.1%

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

      4. neg-sub0 28.1%

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

      5. associate--r- 28.1%

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

      6. metadata-eval 28.1%

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

      7. +-commutative 28.1%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in x around 0 5.6%

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

      1. sub-neg 5.6%

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

      2. metadata-eval 5.6%

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

      3. neg-mul-1 5.6%

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

      4. distribute-neg-frac 5.6%

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

   7. Simplified 5.6%

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

   8. Taylor expanded in y around -inf 62.6%

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

      1. sub-neg 62.6%

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

      2. neg-log 62.6%

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

      3. clear-num 62.6%

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

      4. div-inv 62.6%

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

      5. metadata-eval 62.6%

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

   10. Applied egg-rr 62.6%

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

       1. *-commutative 62.6%

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

       2. neg-mul-1 62.6%

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

   12. Simplified 62.6%

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

   if -1.84999999999999994e-13 < y

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. log1p-define 92.5%

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

      3. distribute-neg-frac2 92.5%

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

      4. neg-sub0 92.5%

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

      5. associate--r- 92.5%

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

      6. metadata-eval 92.5%

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

      7. +-commutative 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around inf 93.3%

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

   6. Taylor expanded in x around 0 57.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{y - 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{y + -1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - (x / (y + -1.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. sub-neg 73.9%

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

   2. log1p-define 73.9%

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

   3. distribute-neg-frac2 73.9%

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

   4. neg-sub0 73.9%

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

   5. associate--r- 73.9%

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

   6. metadata-eval 73.9%

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

   7. +-commutative 73.9%

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

3. Simplified 73.9%

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

5. Taylor expanded in x around inf 75.1%

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

6. Taylor expanded in x around 0 44.4%

   \[\leadsto 1 - \color{blue}{\frac{x}{y - 1}} \]

7. Final simplification 44.4%

   \[\leadsto 1 - \frac{x}{y + -1} \]
8. Add Preprocessing

Alternative 13: 43.5% 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 73.9%

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

   1. sub-neg 73.9%

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

   2. log1p-define 73.9%

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

   3. distribute-neg-frac2 73.9%

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

   4. neg-sub0 73.9%

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

   5. associate--r- 73.9%

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

   6. metadata-eval 73.9%

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

   7. +-commutative 73.9%

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

3. Simplified 73.9%

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

5. Taylor expanded in x around inf 75.1%

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

6. Taylor expanded in x around 0 42.8%

   \[\leadsto \color{blue}{1} \]
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 2024116 
(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))))))