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

Percentage Accurate: 73.2% → 99.8%

Time: 10.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

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 (/ y (+ x -1.0))))))

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((y / (x + -1.0)));
	}
	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((y / (x + -1.0)));
	}
	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((y / (x + -1.0)))
	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(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], 0.99995], 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[(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)) < 0.999950000000000006

   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 6.3%

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

      1. sub-neg 6.3%

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

      2. log1p-define 6.3%

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

      3. distribute-neg-frac2 6.3%

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

      4. neg-sub0 6.3%

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

      5. associate--r- 6.3%

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

      6. metadata-eval 6.3%

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

      7. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in y around inf 6.4%

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

      1. +-commutative 6.4%

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

      2. associate--r+ 6.4%

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

      3. div-sub 6.4%

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

      4. sub-neg 6.4%

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

      5. metadata-eval 6.4%

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

   7. Simplified 6.4%

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

      1. add-exp-log 6.4%

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

      2. expm1-define 6.4%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{x + -1}{y}\right)\right)}\right) \]

      3. log1p-expm1-u 99.9%

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

      4. clear-num 99.9%

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

      5. log-rec 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. sub-neg 99.9%

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

       2. remove-double-neg 99.9%

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

       3. +-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10 \lor \neg \left(y \leq 68000000000\right):\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -10.0) (not (<= y 68000000000.0)))
   (+ 1.0 (log (/ y (+ x -1.0))))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -10.0) || !(y <= 68000000000.0)) {
		tmp = 1.0 + log((y / (x + -1.0)));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -10.0) || !(y <= 68000000000.0)) {
		tmp = 1.0 + Math.log((y / (x + -1.0)));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -10.0) or not (y <= 68000000000.0):
		tmp = 1.0 + math.log((y / (x + -1.0)))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -10.0) || !(y <= 68000000000.0))
		tmp = Float64(1.0 + log(Float64(y / Float64(x + -1.0))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -10.0], N[Not[LessEqual[y, 68000000000.0]], $MachinePrecision]], N[(1.0 + N[Log[N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -10 or 6.8e10 < y

   1. Initial program 30.9%

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

      1. sub-neg 30.9%

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

      2. log1p-define 30.9%

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

      3. distribute-neg-frac2 30.9%

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

      4. neg-sub0 30.9%

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

      5. associate--r- 30.9%

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

      6. metadata-eval 30.9%

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

      7. +-commutative 30.9%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in y around inf 29.4%

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

      1. +-commutative 29.4%

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

      2. associate--r+ 29.4%

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

      3. div-sub 29.4%

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

      4. sub-neg 29.4%

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

      5. metadata-eval 29.4%

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

   7. Simplified 29.4%

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

      1. add-exp-log 29.4%

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

      2. expm1-define 29.4%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{x + -1}{y}\right)\right)}\right) \]

      3. log1p-expm1-u 98.3%

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

      4. clear-num 98.3%

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

      5. log-rec 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. sub-neg 98.3%

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

       2. remove-double-neg 98.3%

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

       3. +-commutative 98.3%

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

   11. Applied egg-rr 98.3%

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

   if -10 < y < 6.8e10

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10 \lor \neg \left(y \leq 68000000000\right):\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.8) (not (<= y 1.0)))
   (+ 1.0 (log (/ y (+ x -1.0))))
   (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.8) || !(y <= 1.0)) {
		tmp = 1.0 + log((y / (x + -1.0)));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.8) || !(y <= 1.0)) {
		tmp = 1.0 + Math.log((y / (x + -1.0)));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.8) or not (y <= 1.0):
		tmp = 1.0 + math.log((y / (x + -1.0)))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.8) || !(y <= 1.0))
		tmp = Float64(1.0 + log(Float64(y / Float64(x + -1.0))));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.8], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 + N[Log[N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000004 or 1 < y

   1. Initial program 32.4%

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

      1. sub-neg 32.4%

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

      2. log1p-define 32.4%

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

      3. distribute-neg-frac2 32.4%

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

      4. neg-sub0 32.4%

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

      5. associate--r- 32.4%

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

      6. metadata-eval 32.4%

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

      7. +-commutative 32.4%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in y around inf 29.8%

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

      1. +-commutative 29.8%

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

      2. associate--r+ 29.8%

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

      3. div-sub 29.8%

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

      4. sub-neg 29.8%

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

      5. metadata-eval 29.8%

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

   7. Simplified 29.8%

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

      1. add-exp-log 29.8%

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

      2. expm1-define 29.8%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{x + -1}{y}\right)\right)}\right) \]

      3. log1p-expm1-u 97.2%

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

      4. clear-num 97.2%

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

      5. log-rec 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. sub-neg 97.2%

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

       2. remove-double-neg 97.2%

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

       3. +-commutative 97.2%

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

   11. Applied egg-rr 97.2%

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

   if -1.80000000000000004 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. div-sub 99.0%

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

      3. *-commutative 99.0%

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

      4. mul-1-neg 99.0%

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

      5. sub-neg 99.0%

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

      6. *-inverses 99.0%

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

      7. metadata-eval 99.0%

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

      8. distribute-lft-neg-in 99.0%

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

      9. neg-mul-1 99.0%

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

      10. remove-double-neg 99.0%

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

      11. log1p-define 99.0%

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

      12. mul-1-neg 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.8)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (+ 1.0 (log (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.8) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 + log((y / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.8) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 + Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.8:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 + math.log((y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.8)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 + log(Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.8], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.79999999999999982

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. log1p-define 24.7%

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

      3. distribute-neg-frac2 24.7%

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

      4. neg-sub0 24.7%

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

      5. associate--r- 24.7%

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

      6. metadata-eval 24.7%

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

      7. +-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

      6. log1p-define 97.3%

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

      7. mul-1-neg 97.3%

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

      8. metadata-eval 97.3%

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

      9. distribute-neg-frac 97.3%

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

      10. distribute-neg-frac2 97.3%

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

      11. log-rec 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0 70.5%

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

   if -5.79999999999999982 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. div-sub 99.0%

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

      3. *-commutative 99.0%

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

      4. mul-1-neg 99.0%

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

      5. sub-neg 99.0%

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

      6. *-inverses 99.0%

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

      7. metadata-eval 99.0%

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

      8. distribute-lft-neg-in 99.0%

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

      9. neg-mul-1 99.0%

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

      10. remove-double-neg 99.0%

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

      11. log1p-define 99.0%

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

      12. mul-1-neg 99.0%

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

   7. Simplified 99.0%

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

   if 1 < y

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. log1p-define 55.8%

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

      3. distribute-neg-frac2 55.8%

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

      4. neg-sub0 55.8%

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

      5. associate--r- 55.8%

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

      6. metadata-eval 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in y around inf 51.2%

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

      1. +-commutative 51.2%

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

      2. associate--r+ 51.2%

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

      3. div-sub 51.2%

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

      4. sub-neg 51.2%

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

      5. metadata-eval 51.2%

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

   7. Simplified 51.2%

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

      1. add-exp-log 51.2%

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

      2. expm1-define 51.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{x + -1}{y}\right)\right)}\right) \]

      3. log1p-expm1-u 95.4%

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

      4. clear-num 95.4%

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

      5. log-rec 95.4%

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

   9. Applied egg-rr 95.4%

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

   10. Taylor expanded in x around inf 92.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (+ 1.0 (log (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 + log((y / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 + Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 + math.log((y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 + log(Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. log1p-define 24.7%

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

      3. distribute-neg-frac2 24.7%

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

      4. neg-sub0 24.7%

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

      5. associate--r- 24.7%

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

      6. metadata-eval 24.7%

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

      7. +-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

      6. log1p-define 97.3%

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

      7. mul-1-neg 97.3%

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

      8. metadata-eval 97.3%

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

      9. distribute-neg-frac 97.3%

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

      10. distribute-neg-frac2 97.3%

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

      11. log-rec 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0 70.5%

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

   if -9.5 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. log1p-define 98.6%

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

      2. mul-1-neg 98.6%

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

   7. Simplified 98.6%

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

   if 1 < y

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. log1p-define 55.8%

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

      3. distribute-neg-frac2 55.8%

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

      4. neg-sub0 55.8%

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

      5. associate--r- 55.8%

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

      6. metadata-eval 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in y around inf 51.2%

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

      1. +-commutative 51.2%

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

      2. associate--r+ 51.2%

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

      3. div-sub 51.2%

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

      4. sub-neg 51.2%

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

      5. metadata-eval 51.2%

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

   7. Simplified 51.2%

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

      1. add-exp-log 51.2%

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

      2. expm1-define 51.2%

         \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{x + -1}{y}\right)\right)}\right) \]

      3. log1p-expm1-u 95.4%

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

      4. clear-num 95.4%

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

      5. log-rec 95.4%

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

   9. Applied egg-rr 95.4%

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

   10. Taylor expanded in x around inf 92.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.5) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. log1p-define 24.7%

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

      3. distribute-neg-frac2 24.7%

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

      4. neg-sub0 24.7%

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

      5. associate--r- 24.7%

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

      6. metadata-eval 24.7%

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

      7. +-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

      6. log1p-define 97.3%

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

      7. mul-1-neg 97.3%

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

      8. metadata-eval 97.3%

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

      9. distribute-neg-frac 97.3%

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

      10. distribute-neg-frac2 97.3%

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

      11. log-rec 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0 70.5%

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

   if -8.5 < y

   1. Initial program 94.5%

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

      1. sub-neg 94.5%

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

      2. log1p-define 94.5%

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

      3. distribute-neg-frac2 94.5%

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

      4. neg-sub0 94.5%

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

      5. associate--r- 94.5%

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

      6. metadata-eval 94.5%

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

      7. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 86.4%

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

      1. log1p-define 86.4%

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

      2. mul-1-neg 86.4%

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

   7. Simplified 86.4%

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

Alternative 7: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000004

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. log1p-define 24.7%

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

      3. distribute-neg-frac2 24.7%

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

      4. neg-sub0 24.7%

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

      5. associate--r- 24.7%

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

      6. metadata-eval 24.7%

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

      7. +-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

      6. log1p-define 97.3%

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

      7. mul-1-neg 97.3%

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

      8. metadata-eval 97.3%

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

      9. distribute-neg-frac 97.3%

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

      10. distribute-neg-frac2 97.3%

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

      11. log-rec 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0 70.5%

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

   if -1.80000000000000004 < y

   1. Initial program 94.5%

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

      1. sub-neg 94.5%

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

      2. log1p-define 94.5%

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

      3. distribute-neg-frac2 94.5%

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

      4. neg-sub0 94.5%

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

      5. associate--r- 94.5%

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

      6. metadata-eval 94.5%

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

      7. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around inf 93.4%

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

   6. Taylor expanded in x around 0 59.8%

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

4. Final simplification 62.7%

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

Alternative 8: 45.1% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. sub-neg 75.4%

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

   2. log1p-define 75.4%

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

   3. distribute-neg-frac2 75.4%

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

   4. neg-sub0 75.4%

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

   5. associate--r- 75.4%

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

   6. metadata-eval 75.4%

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

   7. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in x around inf 75.7%

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

6. Taylor expanded in x around 0 47.0%

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

7. Final simplification 47.0%

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

Alternative 9: 43.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. sub-neg 75.4%

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

   2. log1p-define 75.4%

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

   3. distribute-neg-frac2 75.4%

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

   4. neg-sub0 75.4%

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

   5. associate--r- 75.4%

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

   6. metadata-eval 75.4%

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

   7. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in y around 0 66.5%

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

   1. log1p-define 66.5%

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

   2. mul-1-neg 66.5%

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

7. Simplified 66.5%

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

8. Taylor expanded in x around 0 45.9%

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

9. Final simplification 45.9%

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

Alternative 10: 43.4% 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 75.4%

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

   1. sub-neg 75.4%

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

   2. log1p-define 75.4%

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

   3. distribute-neg-frac2 75.4%

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

   4. neg-sub0 75.4%

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

   5. associate--r- 75.4%

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

   6. metadata-eval 75.4%

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

   7. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in y around 0 66.5%

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

   1. log1p-define 66.5%

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

   2. mul-1-neg 66.5%

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

7. Simplified 66.5%

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

8. Taylor expanded in x around 0 45.4%

   \[\leadsto \color{blue}{1} \]
9. 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 2024096 
(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))))))