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

Percentage Accurate: 72.2% → 99.8%

Time: 11.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999996)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (* y (/ E (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999996) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((y * (((double) M_E) / (x + -1.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999996) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((y * (Math.E / (x + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999996:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((y * (math.e / (x + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999996)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(y * Float64(exp(1) / Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999996], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[(E / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999995999999999996

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      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

   if 0.999995999999999996 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 3.5%

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

      1. sub-neg 3.5%

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

      2. log1p-define 3.5%

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

      3. distribute-neg-frac2 3.5%

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

      4. neg-sub0 3.5%

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

      5. associate--r- 3.5%

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

      6. metadata-eval 3.5%

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

      7. +-commutative 3.5%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in y around inf 16.9%

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

      1. log-rec 16.9%

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

      2. unsub-neg 16.9%

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

      3. sub-neg 16.9%

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

      4. metadata-eval 16.9%

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

      5. +-commutative 16.9%

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

   7. Simplified 16.9%

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

      1. add-log-exp 16.9%

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

      2. exp-diff 16.9%

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

      3. diff-log 100.0%

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

      4. add-exp-log 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

       2. exp-1-e 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7600000 \lor \neg \left(y \leq 3800000000000\right):\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7600000.0) (not (<= y 3800000000000.0)))
   (log (* y (/ E (+ x -1.0))))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7600000.0) || !(y <= 3800000000000.0)) {
		tmp = log((y * (((double) M_E) / (x + -1.0))));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7600000.0) || !(y <= 3800000000000.0)) {
		tmp = Math.log((y * (Math.E / (x + -1.0))));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7600000.0) or not (y <= 3800000000000.0):
		tmp = math.log((y * (math.e / (x + -1.0))))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7600000.0) || !(y <= 3800000000000.0))
		tmp = log(Float64(y * Float64(exp(1) / Float64(x + -1.0))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7600000.0], N[Not[LessEqual[y, 3800000000000.0]], $MachinePrecision]], N[Log[N[(y * N[(E / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.6e6 or 3.8e12 < y

   1. Initial program 29.3%

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

      1. sub-neg 29.3%

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

      2. log1p-define 29.3%

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

      3. distribute-neg-frac2 29.3%

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

      4. neg-sub0 29.3%

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

      5. associate--r- 29.3%

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

      6. metadata-eval 29.3%

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

      7. +-commutative 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in y around inf 25.4%

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

      1. log-rec 25.4%

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

      2. unsub-neg 25.4%

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

      3. sub-neg 25.4%

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

      4. metadata-eval 25.4%

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

      5. +-commutative 25.4%

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

   7. Simplified 25.4%

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

      1. add-log-exp 25.4%

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

      2. exp-diff 25.4%

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

      3. diff-log 100.0%

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

      4. add-exp-log 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

       2. exp-1-e 100.0%

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

   11. Simplified 100.0%

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

   if -7.6e6 < y < 3.8e12

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      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. Taylor expanded in x around inf 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7600000 \lor \neg \left(y \leq 3800000000000\right):\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.69999999999999996 or 1 < y

   1. Initial program 32.2%

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

      1. sub-neg 32.2%

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

      2. log1p-define 32.2%

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

      3. distribute-neg-frac2 32.2%

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

      4. neg-sub0 32.2%

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

      5. associate--r- 32.2%

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

      6. metadata-eval 32.2%

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

      7. +-commutative 32.2%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in y around inf 24.3%

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

      1. log-rec 24.3%

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

      2. unsub-neg 24.3%

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

      3. sub-neg 24.3%

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

      4. metadata-eval 24.3%

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

      5. +-commutative 24.3%

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

   7. Simplified 24.3%

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

      1. add-log-exp 24.3%

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

      2. exp-diff 24.3%

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

      3. diff-log 97.8%

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

      4. add-exp-log 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. associate-/r/ 97.9%

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

       2. exp-1-e 97.9%

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

   11. Simplified 97.9%

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

   if -1.69999999999999996 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. div-sub 98.6%

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

      3. mul-1-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. *-inverses 98.6%

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

      6. *-rgt-identity 98.6%

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

      7. log1p-define 98.7%

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

      8. mul-1-neg 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 98.3%

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

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -145:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -145.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -145.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -145.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -145.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -145.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -145.0], 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[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -145

   1. Initial program 23.6%

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

      1. sub-neg 23.6%

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

      2. log1p-define 23.6%

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

      3. distribute-neg-frac2 23.6%

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

      4. neg-sub0 23.6%

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

      5. associate--r- 23.6%

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

      6. metadata-eval 23.6%

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

      7. +-commutative 23.6%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in x around 0 3.6%

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

      1. mul-1-neg 3.6%

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

      2. sub-neg 3.6%

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

      3. metadata-eval 3.6%

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

      4. distribute-neg-frac 3.6%

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

   7. Simplified 3.6%

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

   8. Taylor expanded in y around -inf 67.1%

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

      1. sub-neg 67.1%

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

      2. +-commutative 67.1%

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

      3. neg-log 67.1%

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

      4. clear-num 67.1%

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

      5. div-inv 67.1%

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

      6. metadata-eval 67.1%

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

      7. *-commutative 67.1%

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

      8. neg-mul-1 67.1%

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

   10. Applied egg-rr 67.1%

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

   if -145 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.5%

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

      1. +-commutative 97.5%

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

      2. div-sub 97.5%

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

      3. mul-1-neg 97.5%

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

      4. sub-neg 97.5%

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

      5. *-inverses 97.5%

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

      6. *-rgt-identity 97.5%

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

      7. log1p-define 97.5%

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

      8. mul-1-neg 97.5%

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

   7. Simplified 97.5%

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

   if 1 < y

   1. Initial program 53.1%

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

      1. sub-neg 53.1%

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

      2. log1p-define 53.1%

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

      3. distribute-neg-frac2 53.1%

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

      4. neg-sub0 53.1%

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

      5. associate--r- 53.1%

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

      6. metadata-eval 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. log-rec 98.1%

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

      2. unsub-neg 98.1%

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

      3. sub-neg 98.1%

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

      4. metadata-eval 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

      1. add-log-exp 98.1%

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

      2. exp-diff 98.1%

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

      3. diff-log 99.9%

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

      4. add-exp-log 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

       2. exp-1-e 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 99.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -145:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7600000:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7600000.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7600000.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7600000.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7600000.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7600000.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7600000.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.6e6

   1. Initial program 21.0%

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

      1. sub-neg 21.0%

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

      2. log1p-define 21.0%

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

      3. distribute-neg-frac2 21.0%

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

      4. neg-sub0 21.0%

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

      5. associate--r- 21.0%

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

      6. metadata-eval 21.0%

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

      7. +-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in x around 0 2.7%

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

      1. mul-1-neg 2.7%

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

      2. sub-neg 2.7%

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

      3. metadata-eval 2.7%

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

      4. distribute-neg-frac 2.7%

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

   7. Simplified 2.7%

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

   8. Taylor expanded in y around -inf 68.9%

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

      1. sub-neg 68.9%

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

      2. +-commutative 68.9%

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

      3. neg-log 68.9%

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

      4. clear-num 68.9%

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

      5. div-inv 68.9%

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

      6. metadata-eval 68.9%

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

      7. *-commutative 68.9%

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

      8. neg-mul-1 68.9%

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

   10. Applied egg-rr 68.9%

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

   if -7.6e6 < y < 1

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      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. Taylor expanded in y around 0 95.4%

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

      1. log1p-define 95.4%

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

      2. mul-1-neg 95.4%

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

   7. Simplified 95.4%

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

   if 1 < y

   1. Initial program 53.1%

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

      1. sub-neg 53.1%

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

      2. log1p-define 53.1%

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

      3. distribute-neg-frac2 53.1%

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

      4. neg-sub0 53.1%

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

      5. associate--r- 53.1%

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

      6. metadata-eval 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. log-rec 98.1%

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

      2. unsub-neg 98.1%

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

      3. sub-neg 98.1%

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

      4. metadata-eval 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

      1. add-log-exp 98.1%

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

      2. exp-diff 98.1%

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

      3. diff-log 99.9%

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

      4. add-exp-log 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

       2. exp-1-e 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 99.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7600000:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7600000:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7600000.0) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7600000.0) {
		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 <= -7600000.0) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7600000.0:
		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 <= -7600000.0)
		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, -7600000.0], 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 < -7.6e6

   1. Initial program 21.0%

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

      1. sub-neg 21.0%

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

      2. log1p-define 21.0%

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

      3. distribute-neg-frac2 21.0%

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

      4. neg-sub0 21.0%

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

      5. associate--r- 21.0%

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

      6. metadata-eval 21.0%

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

      7. +-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in x around 0 2.7%

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

      1. mul-1-neg 2.7%

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

      2. sub-neg 2.7%

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

      3. metadata-eval 2.7%

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

      4. distribute-neg-frac 2.7%

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

   7. Simplified 2.7%

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

   8. Taylor expanded in y around -inf 68.9%

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

      1. sub-neg 68.9%

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

      2. +-commutative 68.9%

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

      3. neg-log 68.9%

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

      4. clear-num 68.9%

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

      5. div-inv 68.9%

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

      6. metadata-eval 68.9%

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

      7. *-commutative 68.9%

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

      8. neg-mul-1 68.9%

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

   10. Applied egg-rr 68.9%

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

   if -7.6e6 < y

   1. Initial program 92.0%

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

      1. sub-neg 92.0%

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

      2. log1p-define 92.0%

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

      3. distribute-neg-frac2 92.0%

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

      4. neg-sub0 92.0%

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

      5. associate--r- 92.0%

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

      6. metadata-eval 92.0%

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

      7. +-commutative 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. log1p-define 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7600000:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00033:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.00033) (+ 1.0 (log (- y))) (- 1.0 (/ x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.00033) {
		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 <= (-0.00033d0)) 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 <= -0.00033) {
		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 <= -0.00033:
		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 <= -0.00033)
		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 <= -0.00033)
		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, -0.00033], 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 < -3.3e-4

   1. Initial program 26.2%

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

      1. sub-neg 26.2%

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

      2. log1p-define 26.2%

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

      3. distribute-neg-frac2 26.2%

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

      4. neg-sub0 26.2%

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

      5. associate--r- 26.2%

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

      6. metadata-eval 26.2%

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

      7. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in x around 0 4.6%

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

      1. mul-1-neg 4.6%

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

      2. sub-neg 4.6%

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

      3. metadata-eval 4.6%

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

      4. distribute-neg-frac 4.6%

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

   7. Simplified 4.6%

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

   8. Taylor expanded in y around -inf 65.3%

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

      1. sub-neg 65.3%

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

      2. +-commutative 65.3%

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

      3. neg-log 65.3%

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

      4. clear-num 65.3%

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

      5. div-inv 65.3%

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

      6. metadata-eval 65.3%

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

      7. *-commutative 65.3%

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

      8. neg-mul-1 65.3%

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

   10. Applied egg-rr 65.3%

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

   if -3.3e-4 < y

   1. Initial program 91.8%

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

      1. sub-neg 91.8%

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

      2. log1p-define 91.9%

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

      3. distribute-neg-frac2 91.9%

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

      4. neg-sub0 91.9%

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

      5. associate--r- 91.9%

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

      6. metadata-eval 91.9%

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

      7. +-commutative 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in x around inf 89.9%

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

   6. Taylor expanded in x around 0 58.1%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{x}{y - 1}} \]
   7. Step-by-step derivation

      1. mul-1-neg 58.1%

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

      2. sub-neg 58.1%

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

      3. metadata-eval 58.1%

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

      4. unsub-neg 58.1%

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

      5. +-commutative 58.1%

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

   8. Simplified 58.1%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00033:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.2% 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 69.0%

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

   1. sub-neg 69.0%

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

   2. log1p-define 69.0%

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

   3. distribute-neg-frac2 69.0%

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

   4. neg-sub0 69.0%

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

   5. associate--r- 69.0%

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

   6. metadata-eval 69.0%

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

   7. +-commutative 69.0%

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

3. Simplified 69.0%

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

5. Taylor expanded in x around inf 69.6%

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

6. Taylor expanded in x around 0 42.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{x}{y - 1}} \]
7. Step-by-step derivation

   1. mul-1-neg 42.3%

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

   2. sub-neg 42.3%

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

   3. metadata-eval 42.3%

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

   4. unsub-neg 42.3%

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

   5. +-commutative 42.3%

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

8. Simplified 42.3%

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

9. Final simplification 42.3%

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

Alternative 9: 42.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. sub-neg 69.0%

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

   2. log1p-define 69.0%

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

   3. distribute-neg-frac2 69.0%

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

   4. neg-sub0 69.0%

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

   5. associate--r- 69.0%

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

   6. metadata-eval 69.0%

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

   7. +-commutative 69.0%

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

3. Simplified 69.0%

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

5. Taylor expanded in y around 0 57.8%

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

   1. log1p-define 57.8%

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

   2. mul-1-neg 57.8%

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

7. Simplified 57.8%

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

8. Taylor expanded in x around 0 40.6%

   \[\leadsto \color{blue}{1 + x} \]

9. Final simplification 40.6%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 10: 42.5% 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 69.0%

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

   1. sub-neg 69.0%

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

   2. log1p-define 69.0%

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

   3. distribute-neg-frac2 69.0%

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

   4. neg-sub0 69.0%

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

   5. associate--r- 69.0%

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

   6. metadata-eval 69.0%

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

   7. +-commutative 69.0%

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

3. Simplified 69.0%

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

5. Taylor expanded in x around inf 69.6%

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

6. Taylor expanded in x around 0 40.5%

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

Developer Target 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024157 
(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))))))