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

Percentage Accurate: 72.1% → 99.8%

Time: 10.3s

Alternatives: 6

Speedup: 0.9×

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: 72.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99999)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99999:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99999)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99999], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999990000000000046

   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-frac 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   if 0.999990000000000046 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.9%

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

      1. sub-neg 6.9%

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

      2. log1p-define 6.9%

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

      3. distribute-neg-frac 6.9%

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

      4. sub-neg 6.9%

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

      5. distribute-neg-in 6.9%

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

      6. remove-double-neg 6.9%

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

      7. +-commutative 6.9%

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

      8. sub-neg 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in y around -inf 79.1%

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

   6. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-+r+ 0.0%

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

      7. log-rec 0.0%

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

      8. sub-neg 0.0%

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

      9. log-div 81.5%

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

      10. +-commutative 81.5%

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

      11. log-prod 99.7%

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

      12. distribute-lft-neg-out 99.7%

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

      13. associate-*r/ 99.7%

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

      14. *-commutative 99.7%

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

      15. neg-mul-1 99.7%

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

      16. distribute-neg-in 99.7%

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

      17. metadata-eval 99.7%

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

      18. sub-neg 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5200 \lor \neg \left(y \leq 69000000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5200.0) (not (<= y 69000000000000.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5200.0) || !(y <= 69000000000000.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5200.0) || !(y <= 69000000000000.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - Math.log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5200.0) or not (y <= 69000000000000.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - math.log1p((-x / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5200.0) || !(y <= 69000000000000.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(-x) / Float64(1.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5200.0], N[Not[LessEqual[y, 69000000000000.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5200 or 6.9e13 < y

   1. Initial program 35.7%

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

      1. sub-neg 35.7%

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

      2. log1p-define 35.7%

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

      3. distribute-neg-frac 35.7%

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

      4. sub-neg 35.7%

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

      5. distribute-neg-in 35.7%

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

      6. remove-double-neg 35.7%

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

      7. +-commutative 35.7%

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

      8. sub-neg 35.7%

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

   3. Simplified 35.7%

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

   5. Taylor expanded in y around -inf 60.1%

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

   6. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-+r+ 0.0%

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

      7. log-rec 0.0%

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

      8. sub-neg 0.0%

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

      9. log-div 72.2%

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

      10. +-commutative 72.2%

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

      11. log-prod 99.0%

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

      12. distribute-lft-neg-out 99.0%

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

      13. associate-*r/ 99.0%

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

      14. *-commutative 99.0%

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

      15. neg-mul-1 99.0%

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

      16. distribute-neg-in 99.0%

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

      17. metadata-eval 99.0%

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

      18. sub-neg 99.0%

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

   8. Simplified 99.0%

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

   if -5200 < y < 6.9e13

   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-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-neg-frac 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5200 \lor \neg \left(y \leq 69000000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.72 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.72) (not (<= y 1.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.72) || !(y <= 1.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.72) || !(y <= 1.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.72) or not (y <= 1.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.72) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.72], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.71999999999999997 or 1 < y

   1. Initial program 36.9%

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

      1. sub-neg 36.9%

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

      2. log1p-define 36.9%

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

      3. distribute-neg-frac 36.9%

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

      4. sub-neg 36.9%

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

      5. distribute-neg-in 36.9%

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

      6. remove-double-neg 36.9%

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

      7. +-commutative 36.9%

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

      8. sub-neg 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in y around -inf 59.4%

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

   6. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-+r+ 0.0%

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

      7. log-rec 0.0%

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

      8. sub-neg 0.0%

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

      9. log-div 71.2%

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

      10. +-commutative 71.2%

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

      11. log-prod 98.0%

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

      12. distribute-lft-neg-out 98.0%

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

      13. associate-*r/ 98.0%

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

      14. *-commutative 98.0%

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

      15. neg-mul-1 98.0%

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

      16. distribute-neg-in 98.0%

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

      17. metadata-eval 98.0%

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

      18. sub-neg 98.0%

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

   8. Simplified 98.0%

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

   if -1.71999999999999997 < 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-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. mul-1-neg 99.4%

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

      3. log1p-define 99.5%

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

      4. mul-1-neg 99.5%

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

      5. +-commutative 99.5%

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

      6. mul-1-neg 99.5%

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

      7. unsub-neg 99.5%

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

      8. div-sub 99.5%

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

      9. *-inverses 99.5%

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

      10. *-rgt-identity 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.72 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(-x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p (- x))))

C

double code(double x, double y) {
	return 1.0 - log1p(-x);
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p(-x);
}

Python

def code(x, y):
	return 1.0 - math.log1p(-x)

Julia

function code(x, y)
	return Float64(1.0 - log1p(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sub-neg 73.3%

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

   2. log1p-define 73.4%

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

   3. distribute-neg-frac 73.4%

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

   4. sub-neg 73.4%

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

   5. distribute-neg-in 73.4%

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

   6. remove-double-neg 73.4%

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

   7. +-commutative 73.4%

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

   8. sub-neg 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in y around 0 60.9%

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

   1. sub-neg 60.9%

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

   2. mul-1-neg 60.9%

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

   3. log1p-define 60.9%

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

   4. mul-1-neg 60.9%

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

7. Simplified 60.9%

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

8. Final simplification 60.9%

   \[\leadsto 1 - \mathsf{log1p}\left(-x\right) \]
9. Add Preprocessing

Alternative 5: 6.4% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sub-neg 73.3%

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

   2. log1p-define 73.4%

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

   3. distribute-neg-frac 73.4%

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

   4. sub-neg 73.4%

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

   5. distribute-neg-in 73.4%

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

   6. remove-double-neg 73.4%

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

   7. +-commutative 73.4%

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

   8. sub-neg 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in y around -inf 31.5%

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

   1. sub-neg 31.5%

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

   2. metadata-eval 31.5%

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

   3. distribute-lft-in 31.5%

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

   4. metadata-eval 31.5%

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

   5. +-commutative 31.5%

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

   6. log1p-define 31.5%

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

   7. mul-1-neg 31.5%

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

   8. div-sub 31.5%

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

   9. associate-*r/ 31.5%

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

7. Simplified 31.5%

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

8. Taylor expanded in y around 0 6.2%

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

9. Final simplification 6.2%

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

Alternative 6: 3.8% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sub-neg 73.3%

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

   2. log1p-define 73.4%

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

   3. distribute-neg-frac 73.4%

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

   4. sub-neg 73.4%

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

   5. distribute-neg-in 73.4%

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

   6. remove-double-neg 73.4%

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

   7. +-commutative 73.4%

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

   8. sub-neg 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in y around -inf 31.5%

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

   1. sub-neg 31.5%

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

   2. metadata-eval 31.5%

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

   3. distribute-lft-in 31.5%

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

   4. metadata-eval 31.5%

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

   5. +-commutative 31.5%

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

   6. log1p-define 31.5%

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

   7. mul-1-neg 31.5%

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

   8. div-sub 31.5%

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

   9. associate-*r/ 31.5%

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

7. Simplified 31.5%

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

8. Taylor expanded in y around 0 3.6%

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

9. Final simplification 3.6%

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

Developer target: 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 2024034 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

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