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

Percentage Accurate: 72.0% → 99.8%

Time: 12.5s

Alternatives: 9

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: 72.0% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 + \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.02)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (+
    (+ 1.0 (/ (+ -1.0 (/ (+ -0.5 (/ -0.3333333333333333 y)) y)) y))
    (log (/ y (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.02) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 + ((-1.0 + ((-0.5 + (-0.3333333333333333 / y)) / y)) / y)) + log((y / (x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.02) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 + ((-1.0 + ((-0.5 + (-0.3333333333333333 / y)) / y)) / y)) + Math.log((y / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.02:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = (1.0 + ((-1.0 + ((-0.5 + (-0.3333333333333333 / y)) / y)) / y)) + math.log((y / (x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.02)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-1.0 + Float64(Float64(-0.5 + Float64(-0.3333333333333333 / y)) / y)) / y)) + log(Float64(y / 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], 0.02], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-1.0 + N[(N[(-0.5 + N[(-0.3333333333333333 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y / 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)) < 0.0200000000000000004

   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 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

   if 0.0200000000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) 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-frac2 6.9%

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

      4. neg-sub0 6.9%

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

      5. associate--r- 6.9%

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

      6. metadata-eval 6.9%

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

      7. +-commutative 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in y around -inf 72.4%

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{\left(-1 \cdot \frac{0.16666666666666666 \cdot \frac{-6 \cdot \frac{1 - x}{x - 1} + \left(2 \cdot \frac{{\left(1 - x\right)}^{3}}{{\left(x - 1\right)}^{3}} + 6 \cdot \frac{1 - x}{x - 1}\right)}{y} - 0.5 \cdot \left(2 + -1 \cdot \frac{{\left(1 - x\right)}^{2}}{{\left(x - 1\right)}^{2}}\right)}{y} + \frac{x}{x - 1}\right) - \frac{1}{x - 1}}{y}\right) - \left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right)} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{\left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 82.4%

         \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \left(\color{blue}{\log \left(1 + \left(-x\right)\right)} + \log \left(\frac{-1}{y}\right)\right) \]

      2. sum-log 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. +-commutative 100.0%

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

      6. frac-2neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. div-inv 100.0%

         \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \log \color{blue}{\left(\frac{-\left(x + -1\right)}{-y}\right)} \]

      9. frac-2neg 100.0%

         \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \log \color{blue}{\left(\frac{x + -1}{y}\right)} \]

      10. clear-num 100.0%

          \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \log \color{blue}{\left(\frac{1}{\frac{y}{x + -1}}\right)} \]

      11. log-div 100.0%

          \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \color{blue}{\left(\log 1 - \log \left(\frac{y}{x + -1}\right)\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \left(\color{blue}{0} - \log \left(\frac{y}{x + -1}\right)\right) \]

      13. +-commutative 100.0%

          \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \left(0 - \log \left(\frac{y}{\color{blue}{-1 + x}}\right)\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \color{blue}{\left(0 - \log \left(\frac{y}{-1 + x}\right)\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 100.0%

          \[\leadsto \left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \color{blue}{\left(-\log \left(\frac{y}{-1 + x}\right)\right)} \]

   11. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1 + \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99995)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99995) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} 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)) <= 0.99995) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} 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)) <= 0.99995:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	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)) <= 0.99995)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	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], 0.99995], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $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 #s(literal 1 binary64) y)) < 0.999950000000000006

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.999950000000000006 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.8%

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

      1. sub-neg 5.8%

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

      2. log1p-define 5.8%

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

      3. distribute-neg-frac2 5.8%

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

      4. neg-sub0 5.8%

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

      5. associate--r- 5.8%

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

      6. metadata-eval 5.8%

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

      7. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in y around inf 5.8%

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

      1. +-commutative 5.8%

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

      2. associate--r+ 5.8%

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

      3. sub-neg 5.8%

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

      4. div-sub 5.8%

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

      5. sub-neg 5.8%

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

      6. metadata-eval 5.8%

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

      7. metadata-eval 5.8%

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

   7. Simplified 5.8%

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

      1. *-un-lft-identity 5.8%

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

      2. +-commutative 5.8%

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

      3. +-commutative 5.8%

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

   9. Applied egg-rr 5.8%

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

       1. *-lft-identity 5.8%

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

       2. log1p-undefine 5.8%

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

       3. associate-+r+ 99.3%

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

       4. metadata-eval 99.3%

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

       5. +-lft-identity 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.76 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.76) (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 <= -1.76) || !(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 <= -1.76) || !(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 <= -1.76) 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 <= -1.76) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.76], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.76000000000000001 or 1 < y

   1. Initial program 30.2%

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

      1. sub-neg 30.2%

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

      2. log1p-define 30.2%

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

      3. distribute-neg-frac2 30.2%

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

      4. neg-sub0 30.2%

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

      5. associate--r- 30.2%

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

      6. metadata-eval 30.2%

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

      7. +-commutative 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in y around inf 28.8%

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

      1. +-commutative 28.8%

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

      2. associate--r+ 28.8%

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

      3. sub-neg 28.8%

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

      4. div-sub 28.8%

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

      5. sub-neg 28.8%

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

      6. metadata-eval 28.8%

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

      7. metadata-eval 28.8%

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

   7. Simplified 28.8%

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

      1. *-un-lft-identity 28.8%

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

      2. +-commutative 28.8%

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

      3. +-commutative 28.8%

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

   9. Applied egg-rr 28.8%

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

       1. *-lft-identity 28.8%

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

       2. log1p-undefine 28.8%

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

       3. associate-+r+ 97.9%

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

       4. metadata-eval 97.9%

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

       5. +-lft-identity 97.9%

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

   11. Simplified 97.9%

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

   if -1.76000000000000001 < y < 1

   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 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.76 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8500 \lor \neg \left(y \leq 4800000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8500.0) (not (<= y 4800000000000.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8500.0) || !(y <= 4800000000000.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8500.0) || !(y <= 4800000000000.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8500.0) or not (y <= 4800000000000.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8500.0) || !(y <= 4800000000000.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8500.0], N[Not[LessEqual[y, 4800000000000.0]], $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[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8500 or 4.8e12 < y

   1. Initial program 29.4%

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

      1. sub-neg 29.4%

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

      2. log1p-define 29.4%

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

      3. distribute-neg-frac2 29.4%

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

      4. neg-sub0 29.4%

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

      5. associate--r- 29.4%

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

      6. metadata-eval 29.4%

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

      7. +-commutative 29.4%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in y around inf 28.5%

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

      1. +-commutative 28.5%

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

      2. associate--r+ 28.5%

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

      3. sub-neg 28.5%

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

      4. div-sub 28.5%

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

      5. sub-neg 28.5%

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

      6. metadata-eval 28.5%

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

      7. metadata-eval 28.5%

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

   7. Simplified 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. +-commutative 28.5%

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

      3. +-commutative 28.5%

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

   9. Applied egg-rr 28.5%

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

       1. *-lft-identity 28.5%

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

       2. log1p-undefine 28.5%

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

       3. associate-+r+ 98.4%

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

       4. metadata-eval 98.4%

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

       5. +-lft-identity 98.4%

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

   11. Simplified 98.4%

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

   if -8500 < y < 4.8e12

   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 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8500 \lor \neg \left(y \leq 4800000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -12.5999999999999996

   1. Initial program 25.1%

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

      1. sub-neg 25.1%

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

      2. log1p-define 25.1%

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

      3. distribute-neg-frac2 25.1%

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

      4. neg-sub0 25.1%

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

      5. associate--r- 25.1%

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

      6. metadata-eval 25.1%

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

      7. +-commutative 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in y around inf 23.9%

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

      1. +-commutative 23.9%

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

      2. associate--r+ 23.9%

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

      3. sub-neg 23.9%

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

      4. div-sub 23.9%

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

      5. sub-neg 23.9%

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

      6. metadata-eval 23.9%

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

      7. metadata-eval 23.9%

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

   7. Simplified 23.9%

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

   8. Taylor expanded in x around 0 65.7%

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

      1. distribute-neg-frac 65.7%

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

      2. metadata-eval 65.7%

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

   10. Simplified 65.7%

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

       1. sub-neg 65.7%

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

       2. neg-log 65.7%

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

       3. clear-num 65.7%

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

       4. div-inv 65.7%

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

       5. metadata-eval 65.7%

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

   12. Applied egg-rr 65.7%

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

       1. *-commutative 65.7%

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

       2. neg-mul-1 65.7%

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

   14. Simplified 65.7%

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

   if -12.5999999999999996 < y

   1. Initial program 94.0%

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

      1. sub-neg 94.0%

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

      2. log1p-define 94.0%

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

      3. distribute-neg-frac2 94.0%

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

      4. neg-sub0 94.0%

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

      5. associate--r- 94.0%

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

      6. metadata-eval 94.0%

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

      7. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around 0 88.0%

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

   7. Simplified 88.1%

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

4. Final simplification 81.9%

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

Alternative 6: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8) (+ 1.0 (log (- y))) (+ x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d0)) then
        tmp = 1.0d0 + log(-y)
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8)
		tmp = 1.0 + log(-y);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000004

   1. Initial program 25.1%

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

      1. sub-neg 25.1%

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

      2. log1p-define 25.1%

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

      3. distribute-neg-frac2 25.1%

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

      4. neg-sub0 25.1%

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

      5. associate--r- 25.1%

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

      6. metadata-eval 25.1%

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

      7. +-commutative 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in y around inf 23.9%

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

      1. +-commutative 23.9%

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

      2. associate--r+ 23.9%

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

      3. sub-neg 23.9%

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

      4. div-sub 23.9%

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

      5. sub-neg 23.9%

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

      6. metadata-eval 23.9%

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

      7. metadata-eval 23.9%

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

   7. Simplified 23.9%

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

   8. Taylor expanded in x around 0 65.7%

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

      1. distribute-neg-frac 65.7%

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

      2. metadata-eval 65.7%

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

   10. Simplified 65.7%

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

       1. sub-neg 65.7%

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

       2. neg-log 65.7%

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

       3. clear-num 65.7%

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

       4. div-inv 65.7%

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

       5. metadata-eval 65.7%

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

   12. Applied egg-rr 65.7%

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

       1. *-commutative 65.7%

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

       2. neg-mul-1 65.7%

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

   14. Simplified 65.7%

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

   if -1.80000000000000004 < y

   1. Initial program 94.0%

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

      1. sub-neg 94.0%

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

      2. log1p-define 94.0%

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

      3. distribute-neg-frac2 94.0%

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

      4. neg-sub0 94.0%

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

      5. associate--r- 94.0%

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

      6. metadata-eval 94.0%

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

      7. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around 0 87.6%

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

      1. log1p-define 87.6%

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

      2. mul-1-neg 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in x around 0 58.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.4) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.4) {
		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 <= -5.4) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.4:
		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 <= -5.4)
		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, -5.4], 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 < -5.4000000000000004

   1. Initial program 25.1%

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

      1. sub-neg 25.1%

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

      2. log1p-define 25.1%

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

      3. distribute-neg-frac2 25.1%

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

      4. neg-sub0 25.1%

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

      5. associate--r- 25.1%

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

      6. metadata-eval 25.1%

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

      7. +-commutative 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in y around inf 23.9%

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

      1. +-commutative 23.9%

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

      2. associate--r+ 23.9%

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

      3. sub-neg 23.9%

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

      4. div-sub 23.9%

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

      5. sub-neg 23.9%

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

      6. metadata-eval 23.9%

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

      7. metadata-eval 23.9%

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

   7. Simplified 23.9%

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

   8. Taylor expanded in x around 0 65.7%

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

      1. distribute-neg-frac 65.7%

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

      2. metadata-eval 65.7%

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

   10. Simplified 65.7%

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

       1. sub-neg 65.7%

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

       2. neg-log 65.7%

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

       3. clear-num 65.7%

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

       4. div-inv 65.7%

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

       5. metadata-eval 65.7%

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

   12. Applied egg-rr 65.7%

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

       1. *-commutative 65.7%

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

       2. neg-mul-1 65.7%

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

   14. Simplified 65.7%

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

   if -5.4000000000000004 < y

   1. Initial program 94.0%

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

      1. sub-neg 94.0%

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

      2. log1p-define 94.0%

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

      3. distribute-neg-frac2 94.0%

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

      4. neg-sub0 94.0%

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

      5. associate--r- 94.0%

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

      6. metadata-eval 94.0%

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

      7. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around 0 87.6%

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

      1. log1p-define 87.6%

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

      2. mul-1-neg 87.6%

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

   7. Simplified 87.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% 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 74.9%

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

   1. sub-neg 74.9%

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

   2. log1p-define 74.9%

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

   3. distribute-neg-frac2 74.9%

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

   4. neg-sub0 74.9%

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

   5. associate--r- 74.9%

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

   6. metadata-eval 74.9%

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

   7. +-commutative 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in y around 0 66.9%

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

   1. log1p-define 66.9%

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

   2. mul-1-neg 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in x around 0 44.8%

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

9. Final simplification 44.8%

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

Alternative 9: 42.9% 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 74.9%

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

   1. sub-neg 74.9%

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

   2. log1p-define 74.9%

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

   3. distribute-neg-frac2 74.9%

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

   4. neg-sub0 74.9%

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

   5. associate--r- 74.9%

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

   6. metadata-eval 74.9%

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

   7. +-commutative 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in y around 0 66.9%

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

   1. log1p-define 66.9%

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

   2. mul-1-neg 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in x around 0 44.0%

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

9. Final simplification 44.0%

   \[\leadsto 1 \]
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 2024130 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (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))))))