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

Percentage Accurate: 71.9% → 98.6%

Time: 17.1s

Alternatives: 14

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: 71.9% 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: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 + \frac{x + \left(-1 + \frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 2e-10)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (-
    1.0
    (log
     (/
      (+ x (+ -1.0 (/ (+ x (+ -1.0 (/ (+ (+ x -1.0) (/ (+ x -1.0) y)) y))) y)))
      y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-10) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + (-1.0 + ((x + (-1.0 + (((x + -1.0) + ((x + -1.0) / y)) / y))) / y))) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-10) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + (-1.0 + ((x + (-1.0 + (((x + -1.0) + ((x + -1.0) / y)) / y))) / y))) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 2e-10:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + (-1.0 + ((x + (-1.0 + (((x + -1.0) + ((x + -1.0) / y)) / y))) / y))) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 2e-10)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + Float64(-1.0 + Float64(Float64(x + Float64(-1.0 + Float64(Float64(Float64(x + -1.0) + Float64(Float64(x + -1.0) / y)) / y))) / y))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2e-10], 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 + N[(-1.0 + N[(N[(x + N[(-1.0 + N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2.00000000000000007e-10

   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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 11.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 + \frac{x + \left(-1 + \frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{-1 + \left(x + \frac{x + \left(-1 - \frac{1 + \left(\frac{1 - x}{y} - x\right)}{y}\right)}{y}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 2e-10)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log
    (*
     y
     (/
      E
      (+
       -1.0
       (+ x (/ (+ x (- -1.0 (/ (+ 1.0 (- (/ (- 1.0 x) y) x)) y))) y))))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-10) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((y * (((double) M_E) / (-1.0 + (x + ((x + (-1.0 - ((1.0 + (((1.0 - x) / y) - x)) / y))) / y))))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-10) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((y * (Math.E / (-1.0 + (x + ((x + (-1.0 - ((1.0 + (((1.0 - x) / y) - x)) / y))) / y))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 2e-10:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((y * (math.e / (-1.0 + (x + ((x + (-1.0 - ((1.0 + (((1.0 - x) / y) - x)) / y))) / y))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 2e-10)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(y * Float64(exp(1) / Float64(-1.0 + Float64(x + Float64(Float64(x + Float64(-1.0 - Float64(Float64(1.0 + Float64(Float64(Float64(1.0 - x) / y) - x)) / y))) / y))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2e-10], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[(E / N[(-1.0 + N[(x + N[(N[(x + N[(-1.0 - N[(N[(1.0 + N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2.00000000000000007e-10

   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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 11.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.9%

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

   5. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. exp-diff 99.9%

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

      3. add-exp-log 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.9%

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

      2. exp-1-e 99.9%

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

      3. associate--l- 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{-1 + \left(x + \frac{x + \left(-1 - \frac{1 + \left(\frac{1 - x}{y} - x\right)}{y}\right)}{y}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 - \frac{1 + \left(\frac{1 - x}{y} - x\right)}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9996)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x (- -1.0 (/ (+ 1.0 (- (/ (- 1.0 x) y) x)) y))) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9996) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + (-1.0 - ((1.0 + (((1.0 - x) / y) - x)) / y))) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9996) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + (-1.0 - ((1.0 + (((1.0 - x) / y) - x)) / y))) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9996:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + (-1.0 - ((1.0 + (((1.0 - x) / y) - x)) / y))) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9996)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + Float64(-1.0 - Float64(Float64(1.0 + Float64(Float64(Float64(1.0 - x) / y) - x)) / y))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.9996], 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 + N[(-1.0 - N[(N[(1.0 + N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99960000000000004

   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 0.99960000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 10.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 - \frac{1 + \left(\frac{1 - x}{y} - x\right)}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -80:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log (/ x y)))))
   (if (<= y -9e+122)
     t_0
     (if (<= y -3e+68)
       t_1
       (if (<= y -80.0)
         t_0
         (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double t_1 = 1.0 - log((x / y));
	double tmp;
	if (y <= -9e+122) {
		tmp = t_0;
	} else if (y <= -3e+68) {
		tmp = t_1;
	} else if (y <= -80.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -9e+122) {
		tmp = t_0;
	} else if (y <= -3e+68) {
		tmp = t_1;
	} else if (y <= -80.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	t_1 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -9e+122:
		tmp = t_0
	elif y <= -3e+68:
		tmp = t_1
	elif y <= -80.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	t_1 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -9e+122)
		tmp = t_0;
	elseif (y <= -3e+68)
		tmp = t_1;
	elseif (y <= -80.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+122], t$95$0, If[LessEqual[y, -3e+68], t$95$1, If[LessEqual[y, -80.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999995e122 or -3.0000000000000002e68 < y < -80

   1. Initial program 25.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 6.9%

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

      1. log1p-define 6.9%

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

   5. Simplified 6.9%

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

   6. Taylor expanded in y around -inf 58.9%

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

   if -8.99999999999999995e122 < y < -3.0000000000000002e68 or 1 < y

   1. Initial program 45.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. mul-1-neg 98.7%

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

      6. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf 90.2%

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

   if -80 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.4%

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

   5. Simplified 99.4%

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

Alternative 5: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -24.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log (/ x y)))))
   (if (<= y -9e+122)
     t_0
     (if (<= y -1.4e+69)
       t_1
       (if (<= y -24.5) t_0 (if (<= y 1.0) (- 1.0 (log1p (- x))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double t_1 = 1.0 - log((x / y));
	double tmp;
	if (y <= -9e+122) {
		tmp = t_0;
	} else if (y <= -1.4e+69) {
		tmp = t_1;
	} else if (y <= -24.5) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -9e+122) {
		tmp = t_0;
	} else if (y <= -1.4e+69) {
		tmp = t_1;
	} else if (y <= -24.5) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	t_1 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -9e+122:
		tmp = t_0
	elif y <= -1.4e+69:
		tmp = t_1
	elif y <= -24.5:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	t_1 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -9e+122)
		tmp = t_0;
	elseif (y <= -1.4e+69)
		tmp = t_1;
	elseif (y <= -24.5)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+122], t$95$0, If[LessEqual[y, -1.4e+69], t$95$1, If[LessEqual[y, -24.5], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999995e122 or -1.39999999999999991e69 < y < -24.5

   1. Initial program 25.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 6.9%

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

      1. log1p-define 6.9%

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

   5. Simplified 6.9%

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

   6. Taylor expanded in y around -inf 58.9%

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

   if -8.99999999999999995e122 < y < -1.39999999999999991e69 or 1 < y

   1. Initial program 45.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. mul-1-neg 98.7%

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

      6. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf 90.2%

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

   if -24.5 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.8%

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

      1. sub-neg 98.8%

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

      2. mul-1-neg 98.8%

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

      3. log1p-define 98.8%

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

      4. mul-1-neg 98.8%

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

   5. Simplified 98.8%

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

Alternative 6: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99998)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ (+ x -1.0) (/ (+ x -1.0) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99998) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((((x + -1.0) + ((x + -1.0) / y)) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99998) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((((x + -1.0) + ((x + -1.0) / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99998:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((((x + -1.0) + ((x + -1.0) / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99998)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x + -1.0) + Float64(Float64(x + -1.0) / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99998], 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[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99997999999999998

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-define 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 0.99997999999999998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.8%

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

      1. associate-*r/ 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9999995)
   (- 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)) <= 0.9999995) {
		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)) <= 0.9999995) {
		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)) <= 0.9999995:
		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)) <= 0.9999995)
		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], 0.9999995], 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)) < 0.999999500000000041

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 0.999999500000000041 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\log \left(e \cdot \frac{y + -1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -500000.0) (not (<= x 1.0)))
   (log (* E (/ (+ y -1.0) x)))
   (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -500000.0) || !(x <= 1.0)) {
		tmp = log((((double) M_E) * ((y + -1.0) / x)));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -500000.0) || !(x <= 1.0)) {
		tmp = Math.log((Math.E * ((y + -1.0) / x)));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -500000.0) or not (x <= 1.0):
		tmp = math.log((math.e * ((y + -1.0) / x)))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -500000.0) || !(x <= 1.0))
		tmp = log(Float64(exp(1) * Float64(Float64(y + -1.0) / x)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -500000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[Log[N[(E * N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e5 or 1 < x

   1. Initial program 69.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-neg-frac2 99.1%

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

      3. sub-neg 99.1%

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

      4. distribute-neg-in 99.1%

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

      5. metadata-eval 99.1%

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

      6. remove-double-neg 99.1%

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

      7. +-commutative 99.1%

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

   5. Simplified 99.1%

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

      1. add-log-exp 99.1%

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

      2. sub-neg 99.1%

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

      3. exp-sum 99.1%

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

      4. neg-log 99.1%

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

      5. clear-num 99.1%

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

      6. add-exp-log 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. exp-1-e 99.1%

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

   9. Simplified 99.1%

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

   if -5e5 < x < 1

   1. Initial program 74.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.3%

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

      1. sub-neg 75.3%

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

      2. mul-1-neg 75.3%

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

      3. log1p-define 75.3%

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

      4. mul-1-neg 75.3%

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

   5. Simplified 75.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\log \left(e \cdot \frac{y + -1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 0.028:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y + -1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65)
   (- 1.0 (log (/ (+ x -1.0) y)))
   (if (<= y 0.028) (- (- 1.0 y) (log1p (- x))) (log (* E (/ (+ y -1.0) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (y <= 0.028) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log((((double) M_E) * ((y + -1.0) / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (y <= 0.028) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log((Math.E * ((y + -1.0) / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif y <= 0.028:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log((math.e * ((y + -1.0) / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (y <= 0.028)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(exp(1) * Float64(Float64(y + -1.0) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.028], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(E * N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999

   1. Initial program 28.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. neg-mul-1 97.7%

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

      3. distribute-neg-in 97.7%

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

      4. metadata-eval 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

   5. Simplified 97.7%

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

   if -1.6499999999999999 < y < 0.0280000000000000006

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.4%

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

   5. Simplified 99.4%

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

   if 0.0280000000000000006 < y

   1. Initial program 45.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-neg-frac2 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-in 98.7%

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

      5. metadata-eval 98.7%

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

      6. remove-double-neg 98.7%

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

      7. +-commutative 98.7%

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

   5. Simplified 98.7%

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

      1. add-log-exp 98.7%

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

      2. sub-neg 98.7%

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

      3. exp-sum 98.7%

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

      4. neg-log 98.7%

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

      5. clear-num 98.7%

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

      6. add-exp-log 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. exp-1-e 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 0.028:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y + -1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1050000:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y + -1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1050000.0)
   (- 1.0 (log (/ x (+ y -1.0))))
   (if (<= x 1.0) (- 1.0 (log1p (- x))) (log (* E (/ (+ y -1.0) x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1050000.0) {
		tmp = 1.0 - log((x / (y + -1.0)));
	} else if (x <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = log((((double) M_E) * ((y + -1.0) / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1050000.0) {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	} else if (x <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = Math.log((Math.E * ((y + -1.0) / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1050000.0:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	elif x <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = math.log((math.e * ((y + -1.0) / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1050000.0)
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	elseif (x <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = log(Float64(exp(1) * Float64(Float64(y + -1.0) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1050000.0], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(E * N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e6

   1. Initial program 76.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-neg-frac2 99.3%

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

      3. sub-neg 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. metadata-eval 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   if -1.05e6 < x < 1

   1. Initial program 74.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.3%

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

      1. sub-neg 75.3%

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

      2. mul-1-neg 75.3%

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

      3. log1p-define 75.3%

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

      4. mul-1-neg 75.3%

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

   5. Simplified 75.3%

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

   if 1 < x

   1. Initial program 45.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-neg-frac2 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-in 98.7%

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

      5. metadata-eval 98.7%

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

      6. remove-double-neg 98.7%

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

      7. +-commutative 98.7%

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

   5. Simplified 98.7%

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

      1. add-log-exp 98.7%

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

      2. sub-neg 98.7%

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

      3. exp-sum 98.7%

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

      4. neg-log 98.7%

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

      5. clear-num 98.7%

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

      6. add-exp-log 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. exp-1-e 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1050000:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y + -1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -15.5) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -15.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -15.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -15.5:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -15.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -15.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -15.5

   1. Initial program 28.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 6.2%

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

      1. log1p-define 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in y around -inf 55.0%

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

   if -15.5 < y

   1. Initial program 91.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 84.3%

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

      1. sub-neg 84.3%

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

      2. mul-1-neg 84.3%

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

      3. log1p-define 84.3%

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

      4. mul-1-neg 84.3%

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

   5. Simplified 84.3%

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

Alternative 12: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8600000000:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8600000000.0) (- 1.0 (log (- x))) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8600000000.0) {
		tmp = 1.0 - log(-x);
	} else {
		tmp = 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 (x <= (-8600000000.0d0)) then
        tmp = 1.0d0 - log(-x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8600000000.0) {
		tmp = 1.0 - Math.log(-x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8600000000.0:
		tmp = 1.0 - math.log(-x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8600000000.0)
		tmp = Float64(1.0 - log(Float64(-x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8600000000.0)
		tmp = 1.0 - log(-x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8600000000.0], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -8.6e9

   1. Initial program 78.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 58.8%

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

      1. neg-mul-1 58.8%

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

   8. Simplified 58.8%

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

   if -8.6e9 < x

   1. Initial program 69.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.0%

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

      1. log1p-define 62.1%

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

   5. Simplified 62.1%

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

   6. Taylor expanded in y around 0 61.0%

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

   7. Taylor expanded in y around 0 63.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 61.8%

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

   1. sub-neg 61.8%

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

   2. mul-1-neg 61.8%

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

   3. log1p-define 61.8%

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

   4. mul-1-neg 61.8%

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

5. Simplified 61.8%

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

Alternative 14: 43.6% 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 72.3%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 42.7%

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

   1. log1p-define 42.8%

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

5. Simplified 42.8%

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

6. Taylor expanded in y around 0 42.1%

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

7. Taylor expanded in y around 0 44.9%

   \[\leadsto \color{blue}{1} \]
8. 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 2024110 
(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))))))