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

Percentage Accurate: 72.4% → 99.6%

Time: 9.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.95:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.95)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.95) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.95) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.95:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.95)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.95], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.94999999999999996

   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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.4%

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

      1. sub-neg 6.4%

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

      2. log1p-def 6.4%

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

      3. neg-sub0 6.4%

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

      4. div-sub 6.5%

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

      5. associate--r- 6.5%

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

      6. neg-sub0 6.5%

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

      7. +-commutative 6.5%

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

      8. sub-neg 6.5%

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

      9. div-sub 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in y around inf 6.5%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate--r+ 0.0%

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

      4. sub-neg 0.0%

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

      5. sub-div 0.0%

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

      6. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 6.5%

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

      3. log1p-def 6.5%

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

      4. +-commutative 6.5%

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

      5. associate-+l+ 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. metadata-eval 99.7%

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

   8. Simplified 99.7%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{-1 + x}{y} + 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.95:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternative 2: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.15 \cdot 10^{+22}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -10.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.41:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))))
   (if (<= y -1.12e+36)
     t_0
     (if (<= y -3.15e+22)
       (- 1.0 (log1p (/ x y)))
       (if (<= y -10.5)
         t_0
         (if (<= y 0.41)
           (- 1.0 (+ y (log1p (- x))))
           (- 1.0 (log (/ x (+ y -1.0))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double tmp;
	if (y <= -1.12e+36) {
		tmp = t_0;
	} else if (y <= -3.15e+22) {
		tmp = 1.0 - log1p((x / y));
	} else if (y <= -10.5) {
		tmp = t_0;
	} else if (y <= 0.41) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double tmp;
	if (y <= -1.12e+36) {
		tmp = t_0;
	} else if (y <= -3.15e+22) {
		tmp = 1.0 - Math.log1p((x / y));
	} else if (y <= -10.5) {
		tmp = t_0;
	} else if (y <= 0.41) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	tmp = 0
	if y <= -1.12e+36:
		tmp = t_0
	elif y <= -3.15e+22:
		tmp = 1.0 - math.log1p((x / y))
	elif y <= -10.5:
		tmp = t_0
	elif y <= 0.41:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	tmp = 0.0
	if (y <= -1.12e+36)
		tmp = t_0;
	elseif (y <= -3.15e+22)
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	elseif (y <= -10.5)
		tmp = t_0;
	elseif (y <= 0.41)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.12e+36], t$95$0, If[LessEqual[y, -3.15e+22], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -10.5], t$95$0, If[LessEqual[y, 0.41], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.11999999999999999e36 or -3.1500000000000001e22 < y < -10.5

   1. Initial program 20.9%

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

      1. sub-neg 20.9%

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

      2. log1p-def 20.9%

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

      3. neg-sub0 20.9%

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

      4. div-sub 20.9%

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

      5. associate--r- 20.9%

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

      6. neg-sub0 20.9%

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

      7. +-commutative 20.9%

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

      8. sub-neg 20.9%

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

      9. div-sub 20.9%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in y around inf 19.7%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. distribute-neg-frac 73.5%

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

      2. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   if -1.11999999999999999e36 < y < -3.1500000000000001e22

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. log1p-def 83.9%

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

      3. neg-sub0 83.9%

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

      4. div-sub 83.9%

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

      5. associate--r- 83.9%

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

      6. neg-sub0 83.9%

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

      7. +-commutative 83.9%

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

      8. sub-neg 83.9%

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

      9. div-sub 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in y around inf 83.9%

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

   5. Taylor expanded in x around inf 85.8%

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

   if -10.5 < y < 0.409999999999999976

   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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.3%

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

      1. div-sub 98.3%

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

      2. mul-1-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. *-inverses 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. log1p-def 98.3%

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

      7. mul-1-neg 98.3%

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

   6. Simplified 98.3%

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

   if 0.409999999999999976 < y

   1. Initial program 63.3%

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

      1. sub-neg 63.3%

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

      2. log1p-def 63.3%

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

      3. neg-sub0 63.3%

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

      4. div-sub 63.3%

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

      5. associate--r- 63.3%

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

      6. neg-sub0 63.3%

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

      7. +-commutative 63.3%

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

      8. sub-neg 63.3%

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

      9. div-sub 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around inf 60.3%

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

      1. neg-mul-1 60.3%

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

      2. distribute-neg-frac 60.3%

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

   6. Simplified 60.3%

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

      1. frac-2neg 60.3%

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

      2. div-inv 60.3%

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

      3. remove-double-neg 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. associate-*r/ 60.3%

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

      2. *-rgt-identity 60.3%

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

      3. neg-sub0 60.3%

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

      4. associate--r- 60.3%

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

      5. metadata-eval 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in x around inf 97.1%

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

       1. log-rec 97.1%

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

       2. unsub-neg 97.1%

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

       3. mul-1-neg 97.1%

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

       4. log-rec 97.1%

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

       5. remove-double-neg 97.1%

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

       6. sub-neg 97.1%

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

       7. metadata-eval 97.1%

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

       8. +-commutative 97.1%

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

       9. log-div 98.3%

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

       10. +-commutative 98.3%

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

   13. Simplified 98.3%

       \[\leadsto 1 - \color{blue}{\log \left(\frac{x}{y + -1}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -3.15 \cdot 10^{+22}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -10.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.41:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -12:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log1p (/ x y)))))
   (if (<= y -8e+38)
     t_0
     (if (<= y -1.02e+20)
       t_1
       (if (<= y -12.0)
         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 - log1p((x / y));
	double tmp;
	if (y <= -8e+38) {
		tmp = t_0;
	} else if (y <= -1.02e+20) {
		tmp = t_1;
	} else if (y <= -12.0) {
		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.log1p((x / y));
	double tmp;
	if (y <= -8e+38) {
		tmp = t_0;
	} else if (y <= -1.02e+20) {
		tmp = t_1;
	} else if (y <= -12.0) {
		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.log1p((x / y))
	tmp = 0
	if y <= -8e+38:
		tmp = t_0
	elif y <= -1.02e+20:
		tmp = t_1
	elif y <= -12.0:
		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 - log1p(Float64(x / y)))
	tmp = 0.0
	if (y <= -8e+38)
		tmp = t_0;
	elseif (y <= -1.02e+20)
		tmp = t_1;
	elseif (y <= -12.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(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[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+38], t$95$0, If[LessEqual[y, -1.02e+20], t$95$1, If[LessEqual[y, -12.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.99999999999999982e38 or -1.02e20 < y < -12

   1. Initial program 20.9%

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

      1. sub-neg 20.9%

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

      2. log1p-def 20.9%

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

      3. neg-sub0 20.9%

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

      4. div-sub 20.9%

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

      5. associate--r- 20.9%

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

      6. neg-sub0 20.9%

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

      7. +-commutative 20.9%

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

      8. sub-neg 20.9%

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

      9. div-sub 20.9%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in y around inf 19.7%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. distribute-neg-frac 73.5%

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

      2. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   if -7.99999999999999982e38 < y < -1.02e20 or 1 < y

   1. Initial program 66.8%

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

      1. sub-neg 66.8%

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

      2. log1p-def 66.8%

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

      3. neg-sub0 66.8%

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

      4. div-sub 66.8%

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

      5. associate--r- 66.8%

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

      6. neg-sub0 66.8%

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

      7. +-commutative 66.8%

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

      8. sub-neg 66.8%

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

      9. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in y around inf 65.5%

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

   5. Taylor expanded in x around inf 63.3%

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

   if -12 < 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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.3%

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

      1. div-sub 98.3%

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

      2. mul-1-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. *-inverses 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. log1p-def 98.3%

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

      7. mul-1-neg 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+38}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -12:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -13500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log1p (/ x y)))))
   (if (<= y -1.12e+36)
     t_0
     (if (<= y -6e+20)
       t_1
       (if (<= y -13500000.0)
         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 - log1p((x / y));
	double tmp;
	if (y <= -1.12e+36) {
		tmp = t_0;
	} else if (y <= -6e+20) {
		tmp = t_1;
	} else if (y <= -13500000.0) {
		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.log1p((x / y));
	double tmp;
	if (y <= -1.12e+36) {
		tmp = t_0;
	} else if (y <= -6e+20) {
		tmp = t_1;
	} else if (y <= -13500000.0) {
		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.log1p((x / y))
	tmp = 0
	if y <= -1.12e+36:
		tmp = t_0
	elif y <= -6e+20:
		tmp = t_1
	elif y <= -13500000.0:
		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 - log1p(Float64(x / y)))
	tmp = 0.0
	if (y <= -1.12e+36)
		tmp = t_0;
	elseif (y <= -6e+20)
		tmp = t_1;
	elseif (y <= -13500000.0)
		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[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.12e+36], t$95$0, If[LessEqual[y, -6e+20], t$95$1, If[LessEqual[y, -13500000.0], 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 < -1.11999999999999999e36 or -6e20 < y < -1.35e7

   1. Initial program 19.0%

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

      1. sub-neg 19.0%

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

      2. log1p-def 19.0%

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

      3. neg-sub0 19.0%

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

      4. div-sub 19.0%

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

      5. associate--r- 19.0%

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

      6. neg-sub0 19.0%

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

      7. +-commutative 19.0%

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

      8. sub-neg 19.0%

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

      9. div-sub 19.0%

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

   3. Simplified 19.0%

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

   4. Taylor expanded in y around inf 19.0%

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

   5. Taylor expanded in x around 0 74.9%

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

      1. distribute-neg-frac 74.9%

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

      2. metadata-eval 74.9%

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

   7. Simplified 74.9%

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

   if -1.11999999999999999e36 < y < -6e20 or 1 < y

   1. Initial program 66.8%

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

      1. sub-neg 66.8%

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

      2. log1p-def 66.8%

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

      3. neg-sub0 66.8%

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

      4. div-sub 66.8%

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

      5. associate--r- 66.8%

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

      6. neg-sub0 66.8%

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

      7. +-commutative 66.8%

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

      8. sub-neg 66.8%

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

      9. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in y around inf 65.5%

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

   5. Taylor expanded in x around inf 63.3%

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

   if -1.35e7 < 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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.3%

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

      1. log1p-def 96.3%

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

      2. mul-1-neg 96.3%

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

   6. Simplified 96.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -13500000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13500000 \lor \neg \left(y \leq 3700000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -13500000.0) (not (<= y 3700000000000.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -13500000.0) || !(y <= 3700000000000.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -13500000.0) || !(y <= 3700000000000.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -13500000.0) or not (y <= 3700000000000.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -13500000.0) || !(y <= 3700000000000.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -13500000.0], N[Not[LessEqual[y, 3700000000000.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e7 or 3.7e12 < 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-def 32.4%

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

      3. neg-sub0 32.4%

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

      4. div-sub 32.4%

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

      5. associate--r- 32.4%

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

      6. neg-sub0 32.4%

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

      7. +-commutative 32.4%

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

      8. sub-neg 32.4%

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

      9. div-sub 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in y around inf 32.4%

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

      1. expm1-log1p-u 26.5%

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

      2. expm1-udef 26.5%

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

      3. associate--r+ 26.5%

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

      4. sub-neg 26.5%

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

      5. sub-div 26.5%

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

      6. metadata-eval 26.5%

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

   6. Applied egg-rr 26.5%

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

      1. expm1-def 26.5%

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

      2. expm1-log1p 32.4%

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

      3. log1p-def 32.4%

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

      4. +-commutative 32.4%

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

      5. associate-+l+ 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   if -1.35e7 < y < 3.7e12

   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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.3%

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

      1. neg-mul-1 97.3%

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

      2. distribute-neg-frac 97.3%

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

   6. Simplified 97.3%

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

      1. frac-2neg 97.3%

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

      2. div-inv 97.3%

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

      3. remove-double-neg 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. associate-*r/ 97.3%

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

      2. *-rgt-identity 97.3%

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

      3. neg-sub0 97.3%

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

      4. associate--r- 97.3%

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

      5. metadata-eval 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13500000 \lor \neg \left(y \leq 3700000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 6: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + -1}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(t_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y -1.0))))
   (if (<= y -1.12e+36)
     (- 1.0 (log (/ -1.0 y)))
     (if (<= y 1.0) (- 1.0 (log1p t_0)) (- 1.0 (log t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + -1.0);
	double tmp;
	if (y <= -1.12e+36) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(t_0);
	} else {
		tmp = 1.0 - log(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = x / (y + -1.0);
	double tmp;
	if (y <= -1.12e+36) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(t_0);
	} else {
		tmp = 1.0 - Math.log(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + -1.0)
	tmp = 0
	if y <= -1.12e+36:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(t_0)
	else:
		tmp = 1.0 - math.log(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + -1.0))
	tmp = 0.0
	if (y <= -1.12e+36)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(t_0));
	else
		tmp = Float64(1.0 - log(t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.12e+36], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.11999999999999999e36

   1. Initial program 18.2%

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

      1. sub-neg 18.2%

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

      2. log1p-def 18.2%

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

      3. neg-sub0 18.2%

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

      4. div-sub 18.2%

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

      5. associate--r- 18.2%

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

      6. neg-sub0 18.2%

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

      7. +-commutative 18.2%

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

      8. sub-neg 18.2%

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

      9. div-sub 18.2%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in y around inf 18.2%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. distribute-neg-frac 74.3%

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

      2. metadata-eval 74.3%

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

   7. Simplified 74.3%

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

   if -1.11999999999999999e36 < y < 1

   1. Initial program 98.1%

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

      1. sub-neg 98.1%

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

      2. log1p-def 98.1%

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

      3. neg-sub0 98.1%

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

      4. div-sub 98.1%

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

      5. associate--r- 98.1%

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

      6. neg-sub0 98.1%

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

      7. +-commutative 98.1%

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

      8. sub-neg 98.1%

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

      9. div-sub 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 95.1%

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

      1. neg-mul-1 95.1%

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

      2. distribute-neg-frac 95.1%

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

   6. Simplified 95.1%

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

      1. frac-2neg 95.1%

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

      2. div-inv 95.1%

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

      3. remove-double-neg 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. associate-*r/ 95.1%

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

      2. *-rgt-identity 95.1%

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

      3. neg-sub0 95.1%

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

      4. associate--r- 95.1%

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

      5. metadata-eval 95.1%

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

   10. Simplified 95.1%

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

   if 1 < y

   1. Initial program 63.3%

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

      1. sub-neg 63.3%

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

      2. log1p-def 63.3%

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

      3. neg-sub0 63.3%

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

      4. div-sub 63.3%

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

      5. associate--r- 63.3%

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

      6. neg-sub0 63.3%

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

      7. +-commutative 63.3%

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

      8. sub-neg 63.3%

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

      9. div-sub 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around inf 60.3%

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

      1. neg-mul-1 60.3%

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

      2. distribute-neg-frac 60.3%

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

   6. Simplified 60.3%

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

      1. frac-2neg 60.3%

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

      2. div-inv 60.3%

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

      3. remove-double-neg 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. associate-*r/ 60.3%

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

      2. *-rgt-identity 60.3%

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

      3. neg-sub0 60.3%

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

      4. associate--r- 60.3%

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

      5. metadata-eval 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in x around inf 97.1%

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

       1. log-rec 97.1%

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

       2. unsub-neg 97.1%

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

       3. mul-1-neg 97.1%

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

       4. log-rec 97.1%

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

       5. remove-double-neg 97.1%

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

       6. sub-neg 97.1%

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

       7. metadata-eval 97.1%

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

       8. +-commutative 97.1%

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

       9. log-div 98.3%

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

       10. +-commutative 98.3%

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

   13. Simplified 98.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+36}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13500000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13500000.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13500000.0) {
		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 <= -13500000.0) {
		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 <= -13500000.0:
		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 <= -13500000.0)
		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, -13500000.0], 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 < -1.35e7

   1. Initial program 23.3%

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

      1. sub-neg 23.3%

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

      2. log1p-def 23.3%

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

      3. neg-sub0 23.3%

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

      4. div-sub 23.3%

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

      5. associate--r- 23.3%

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

      6. neg-sub0 23.3%

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

      7. +-commutative 23.3%

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

      8. sub-neg 23.3%

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

      9. div-sub 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in y around inf 23.3%

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

   5. Taylor expanded in x around 0 71.1%

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

      1. distribute-neg-frac 71.1%

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

      2. metadata-eval 71.1%

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

   7. Simplified 71.1%

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

   if -1.35e7 < y

   1. Initial program 93.5%

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

      1. sub-neg 93.5%

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

      2. log1p-def 93.5%

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

      3. neg-sub0 93.5%

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

      4. div-sub 93.5%

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

      5. associate--r- 93.5%

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

      6. neg-sub0 93.5%

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

      7. +-commutative 93.5%

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

      8. sub-neg 93.5%

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

      9. div-sub 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 79.4%

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

      1. log1p-def 79.4%

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

      2. mul-1-neg 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13500000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 8: 63.0% 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 68.6%

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

   1. sub-neg 68.6%

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

   2. log1p-def 68.6%

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

   3. neg-sub0 68.6%

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

   4. div-sub 68.6%

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

   5. associate--r- 68.6%

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

   6. neg-sub0 68.6%

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

   7. +-commutative 68.6%

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

   8. sub-neg 68.6%

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

   9. div-sub 68.6%

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

3. Simplified 68.6%

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

4. Taylor expanded in y around 0 55.9%

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

   1. log1p-def 55.9%

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

   2. mul-1-neg 55.9%

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

6. Simplified 55.9%

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

7. Final simplification 55.9%

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

Alternative 9: 44.5% 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 68.6%

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

   1. sub-neg 68.6%

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

   2. log1p-def 68.6%

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

   3. neg-sub0 68.6%

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

   4. div-sub 68.6%

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

   5. associate--r- 68.6%

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

   6. neg-sub0 68.6%

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

   7. +-commutative 68.6%

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

   8. sub-neg 68.6%

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

   9. div-sub 68.6%

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

3. Simplified 68.6%

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

4. Taylor expanded in x around inf 68.9%

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

   1. neg-mul-1 68.9%

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

   2. distribute-neg-frac 68.9%

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

6. Simplified 68.9%

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

   1. frac-2neg 68.9%

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

   2. div-inv 68.9%

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

   3. remove-double-neg 68.9%

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

8. Applied egg-rr 68.9%

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

   1. associate-*r/ 68.9%

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

   2. *-rgt-identity 68.9%

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

   3. neg-sub0 68.9%

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

   4. associate--r- 68.9%

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

   5. metadata-eval 68.9%

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

10. Simplified 68.9%

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

11. Taylor expanded in x around 0 40.0%

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

12. Final simplification 40.0%

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

Alternative 10: 43.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. sub-neg 68.6%

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

   2. log1p-def 68.6%

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

   3. neg-sub0 68.6%

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

   4. div-sub 68.6%

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

   5. associate--r- 68.6%

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

   6. neg-sub0 68.6%

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

   7. +-commutative 68.6%

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

   8. sub-neg 68.6%

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

   9. div-sub 68.6%

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

3. Simplified 68.6%

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

4. Taylor expanded in y around 0 55.9%

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

5. Taylor expanded in x around 0 37.8%

   \[\leadsto 1 - \color{blue}{\left(-0.5 \cdot {x}^{2} + -1 \cdot x\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 37.8%

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

   2. unsub-neg 37.8%

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

   3. *-commutative 37.8%

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

   4. unpow2 37.8%

      \[\leadsto 1 - \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - x\right) \]

   5. associate-*l* 37.8%

      \[\leadsto 1 - \left(\color{blue}{x \cdot \left(x \cdot -0.5\right)} - x\right) \]

7. Simplified 37.8%

   \[\leadsto 1 - \color{blue}{\left(x \cdot \left(x \cdot -0.5\right) - x\right)} \]

8. Taylor expanded in x around 0 38.6%

   \[\leadsto \color{blue}{1 + x} \]
9. Step-by-step derivation

   1. +-commutative 38.6%

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

10. Simplified 38.6%

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

11. Final simplification 38.6%

    \[\leadsto x + 1 \]

Alternative 11: 42.8% 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 68.6%

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

   1. sub-neg 68.6%

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

   2. log1p-def 68.6%

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

   3. neg-sub0 68.6%

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

   4. div-sub 68.6%

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

   5. associate--r- 68.6%

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

   6. neg-sub0 68.6%

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

   7. +-commutative 68.6%

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

   8. sub-neg 68.6%

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

   9. div-sub 68.6%

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

3. Simplified 68.6%

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

4. Taylor expanded in y around 0 55.9%

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

5. Taylor expanded in x around 0 37.8%

   \[\leadsto 1 - \color{blue}{\left(-0.5 \cdot {x}^{2} + -1 \cdot x\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 37.8%

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

   2. unsub-neg 37.8%

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

   3. *-commutative 37.8%

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

   4. unpow2 37.8%

      \[\leadsto 1 - \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - x\right) \]

   5. associate-*l* 37.8%

      \[\leadsto 1 - \left(\color{blue}{x \cdot \left(x \cdot -0.5\right)} - x\right) \]

7. Simplified 37.8%

   \[\leadsto 1 - \color{blue}{\left(x \cdot \left(x \cdot -0.5\right) - x\right)} \]

8. Taylor expanded in x around 0 38.3%

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

9. Final simplification 38.3%

   \[\leadsto 1 \]

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 2023171 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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