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

Percentage Accurate: 73.0% → 98.1%

Time: 10.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{\frac{y \cdot e}{x + -1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 5e-11)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (* 2.0 (log (sqrt (/ (* y E) (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 5e-11) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 2.0 * log(sqrt(((y * ((double) M_E)) / (x + -1.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 5e-11) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 2.0 * Math.log(Math.sqrt(((y * Math.E) / (x + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 5e-11:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 2.0 * math.log(math.sqrt(((y * math.e) / (x + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 5e-11)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(2.0 * log(sqrt(Float64(Float64(y * exp(1)) / Float64(x + -1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 5e-11], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Log[N[Sqrt[N[(N[(y * E), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 5.00000000000000018e-11

   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-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 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.8%

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

      1. sub-neg 6.8%

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

      2. log1p-def 6.8%

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

      3. neg-sub0 6.8%

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

      4. div-sub 6.8%

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

      5. associate--r- 6.8%

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

      6. neg-sub0 6.8%

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

      7. +-commutative 6.8%

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

      8. sub-neg 6.8%

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

      9. div-sub 6.8%

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

   3. Simplified 6.8%

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

      1. add-log-exp 6.8%

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

      2. add-sqr-sqrt 6.8%

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

      3. log-prod 6.8%

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

      4. exp-diff 6.8%

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

      5. exp-1-e 6.8%

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

      6. log1p-udef 6.8%

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

      7. add-exp-log 6.8%

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

      8. exp-diff 6.8%

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

      9. exp-1-e 6.8%

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

      10. log1p-udef 6.8%

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

      11. add-exp-log 6.8%

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

   5. Applied egg-rr 6.8%

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

      1. count-2 6.8%

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

      2. remove-double-neg 6.8%

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

      3. unsub-neg 6.8%

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

      4. *-rgt-identity 6.8%

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

      5. associate-*r/ 8.4%

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

      6. distribute-rgt-neg-in 8.4%

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

      7. distribute-neg-frac 8.4%

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

      8. metadata-eval 8.4%

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

      9. metadata-eval 8.4%

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

      10. associate-/r* 8.4%

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

      11. neg-mul-1 8.4%

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

      12. associate-*r/ 6.8%

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

      13. *-rgt-identity 6.8%

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

      14. neg-sub0 6.8%

          \[\leadsto 2 \cdot \log \left(\sqrt{\frac{e}{1 - \frac{y - x}{\color{blue}{0 - \left(1 - y\right)}}}}\right) \]

      15. associate--r- 6.8%

          \[\leadsto 2 \cdot \log \left(\sqrt{\frac{e}{1 - \frac{y - x}{\color{blue}{\left(0 - 1\right) + y}}}}\right) \]

      16. metadata-eval 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in y around inf 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{\frac{y \cdot e}{x + -1}}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -470000:\\ \;\;\;\;\left(1 - \frac{\frac{x}{x + -1} + \frac{-1}{x + -1}}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -470000.0)
   (-
    (- 1.0 (/ (+ (/ x (+ x -1.0)) (/ -1.0 (+ x -1.0))) y))
    (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 1.7e+16)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -470000.0) {
		tmp = (1.0 - (((x / (x + -1.0)) + (-1.0 / (x + -1.0))) / y)) - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 1.7e+16) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -470000.0) {
		tmp = (1.0 - (((x / (x + -1.0)) + (-1.0 / (x + -1.0))) / y)) - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 1.7e+16) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -470000.0:
		tmp = (1.0 - (((x / (x + -1.0)) + (-1.0 / (x + -1.0))) / y)) - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 1.7e+16:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -470000.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(x / Float64(x + -1.0)) + Float64(-1.0 / Float64(x + -1.0))) / y)) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 1.7e+16)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -470000.0], N[(N[(1.0 - N[(N[(N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+16], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7e5

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in y around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

      9. sub-neg 99.5%

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

      10. metadata-eval 99.5%

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

      11. +-commutative 99.5%

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

   6. Simplified 99.5%

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

   if -4.7e5 < y < 1.7e16

   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 1.7e16 < y

   1. Initial program 56.9%

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

      1. sub-neg 56.9%

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

      2. log1p-def 56.9%

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

      3. neg-sub0 56.9%

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

      4. div-sub 56.9%

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

      5. associate--r- 56.9%

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

      6. neg-sub0 56.9%

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

      7. +-commutative 56.9%

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

      8. sub-neg 56.9%

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

      9. div-sub 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in y around inf 98.1%

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

      1. +-commutative 98.1%

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

      2. log-rec 98.1%

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

      3. unsub-neg 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. +-commutative 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -470000:\\ \;\;\;\;\left(1 - \frac{\frac{x}{x + -1} + \frac{-1}{x + -1}}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -500000:\\ \;\;\;\;1 + \left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -500000.0)
   (+
    1.0
    (- (- (/ (- 1.0 x) (* y (+ x -1.0))) (log (/ -1.0 y))) (log1p (- x))))
   (if (<= y 5e+14)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -500000.0) {
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - log((-1.0 / y))) - log1p(-x));
	} else if (y <= 5e+14) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -500000.0) {
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - Math.log((-1.0 / y))) - Math.log1p(-x));
	} else if (y <= 5e+14) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -500000.0:
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - math.log((-1.0 / y))) - math.log1p(-x))
	elif y <= 5e+14:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -500000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - x) / Float64(y * Float64(x + -1.0))) - log(Float64(-1.0 / y))) - log1p(Float64(-x))));
	elseif (y <= 5e+14)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -500000.0], N[(1.0 + N[(N[(N[(N[(1.0 - x), $MachinePrecision] / N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+14], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5e5

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in y around -inf 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

      6. log1p-def 99.4%

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

      7. mul-1-neg 99.4%

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

      8. mul-1-neg 99.4%

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

      9. unsub-neg 99.4%

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

      10. div-sub 99.4%

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

      11. associate-/l/ 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if -5e5 < y < 5e14

   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 5e14 < y

   1. Initial program 56.9%

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

      1. sub-neg 56.9%

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

      2. log1p-def 56.9%

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

      3. neg-sub0 56.9%

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

      4. div-sub 56.9%

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

      5. associate--r- 56.9%

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

      6. neg-sub0 56.9%

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

      7. +-commutative 56.9%

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

      8. sub-neg 56.9%

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

      9. div-sub 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in y around inf 98.1%

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

      1. +-commutative 98.1%

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

      2. log-rec 98.1%

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

      3. unsub-neg 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. +-commutative 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -500000:\\ \;\;\;\;1 + \left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5400000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 7600000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5400000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 7600000000000.0)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5400000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 7600000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5400000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 7600000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5400000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 7600000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5400000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 7600000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5400000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7600000000000.0], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e9

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in y around -inf 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-lft-in 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. log1p-def 99.1%

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

      7. mul-1-neg 99.1%

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

   6. Simplified 99.1%

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

   if -5.4e9 < y < 7.6e12

   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 7.6e12 < y

   1. Initial program 56.9%

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

      1. sub-neg 56.9%

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

      2. log1p-def 56.9%

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

      3. neg-sub0 56.9%

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

      4. div-sub 56.9%

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

      5. associate--r- 56.9%

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

      6. neg-sub0 56.9%

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

      7. +-commutative 56.9%

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

      8. sub-neg 56.9%

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

      9. div-sub 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in y around inf 98.1%

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

      1. +-commutative 98.1%

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

      2. log-rec 98.1%

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

      3. unsub-neg 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. +-commutative 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5400000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 7600000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]

Alternative 5: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999995)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (+ 1.0 (- (/ -1.0 y) (log (/ -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999995) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - log((-1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999995) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - Math.log((-1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999995:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + ((-1.0 / y) - math.log((-1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999995)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999995], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.99999999500000003

   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-def 99.8%

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

      3. neg-sub0 99.8%

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

      4. div-sub 99.8%

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

      5. associate--r- 99.8%

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

      6. neg-sub0 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

   if 0.99999999500000003 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.0%

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

      1. sub-neg 6.0%

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

      2. log1p-def 6.0%

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

      3. neg-sub0 6.0%

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

      4. div-sub 6.0%

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

      5. associate--r- 6.0%

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

      6. neg-sub0 6.0%

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

      7. +-commutative 6.0%

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

      8. sub-neg 6.0%

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

      9. div-sub 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in y around -inf 86.8%

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

      1. sub-neg 86.8%

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

      2. metadata-eval 86.8%

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

      3. distribute-lft-in 86.8%

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

      4. metadata-eval 86.8%

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

      5. +-commutative 86.8%

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

      6. log1p-def 86.8%

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

      7. mul-1-neg 86.8%

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

      8. mul-1-neg 86.8%

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

      9. unsub-neg 86.8%

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

      10. div-sub 86.8%

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

      11. associate-/l/ 86.8%

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

      12. sub-neg 86.8%

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

      13. metadata-eval 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in x around 0 70.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \]

Alternative 6: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -270000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -270000000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -270000000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -270000000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -270000000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -270000000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -270000000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e11

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in x around 0 5.6%

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

      1. log1p-def 5.6%

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

   6. Simplified 5.6%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 69.5%

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

   9. Simplified 69.5%

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

   if -2.7e11 < y

   1. Initial program 95.1%

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

      1. sub-neg 95.1%

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

      2. log1p-def 95.1%

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

      3. neg-sub0 95.1%

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

      4. div-sub 95.1%

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

      5. associate--r- 95.1%

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

      6. neg-sub0 95.1%

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

      7. +-commutative 95.1%

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

      8. sub-neg 95.1%

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

      9. div-sub 95.1%

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

   3. Simplified 95.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \end{array} \]

Alternative 7: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -17.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -17.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -17.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -17

   1. Initial program 20.7%

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

      1. sub-neg 20.7%

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

      2. log1p-def 20.7%

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

      3. neg-sub0 20.7%

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

      4. div-sub 20.7%

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

      5. associate--r- 20.7%

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

      6. neg-sub0 20.7%

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

      7. +-commutative 20.7%

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

      8. sub-neg 20.7%

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

      9. div-sub 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in x around 0 5.5%

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

      1. log1p-def 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 68.5%

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

   9. Simplified 68.5%

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

   if -17 < 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 0 100.0%

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

      1. div-sub 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. *-inverses 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. log1p-def 100.0%

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

      7. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if 1 < y

   1. Initial program 56.9%

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

      1. sub-neg 56.9%

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

      2. log1p-def 56.9%

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

      3. neg-sub0 56.9%

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

      4. div-sub 56.9%

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

      5. associate--r- 56.9%

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

      6. neg-sub0 56.9%

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

      7. +-commutative 56.9%

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

      8. sub-neg 56.9%

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

      9. div-sub 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in x around inf 52.6%

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

      1. neg-mul-1 52.6%

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

      2. distribute-neg-frac 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around inf 52.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 8: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -780000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -780000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -780000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -780000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -780000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((-x / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -780000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(-x) / Float64(1.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -780000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e8

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in x around 0 5.6%

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

      1. log1p-def 5.6%

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

   6. Simplified 5.6%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 69.5%

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

   9. Simplified 69.5%

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

   if -7.8e8 < y

   1. Initial program 95.1%

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

      1. sub-neg 95.1%

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

      2. log1p-def 95.1%

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

      3. neg-sub0 95.1%

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

      4. div-sub 95.1%

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

      5. associate--r- 95.1%

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

      6. neg-sub0 95.1%

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

      7. +-commutative 95.1%

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

      8. sub-neg 95.1%

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

      9. div-sub 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around inf 94.4%

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

      1. neg-mul-1 94.4%

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

      2. distribute-neg-frac 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -780000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternative 9: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -25000.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -25000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -25000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -25000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -25000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -25000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -25000

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in x around 0 5.6%

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

      1. log1p-def 5.6%

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

   6. Simplified 5.6%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 69.5%

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

   9. Simplified 69.5%

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

   if -25000 < 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 99.2%

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

      1. log1p-def 99.3%

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

      2. mul-1-neg 99.3%

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

   6. Simplified 99.3%

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

   if 1 < y

   1. Initial program 56.9%

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

      1. sub-neg 56.9%

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

      2. log1p-def 56.9%

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

      3. neg-sub0 56.9%

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

      4. div-sub 56.9%

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

      5. associate--r- 56.9%

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

      6. neg-sub0 56.9%

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

      7. +-commutative 56.9%

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

      8. sub-neg 56.9%

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

      9. div-sub 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in x around inf 52.6%

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

      1. neg-mul-1 52.6%

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

      2. distribute-neg-frac 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around inf 52.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 10: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -25000.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -25000.0) {
		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 <= -25000.0) {
		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 <= -25000.0:
		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 <= -25000.0)
		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, -25000.0], 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 < -25000

   1. Initial program 19.6%

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

      1. sub-neg 19.6%

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

      2. log1p-def 19.6%

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

      3. neg-sub0 19.6%

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

      4. div-sub 19.6%

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

      5. associate--r- 19.6%

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

      6. neg-sub0 19.6%

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

      7. +-commutative 19.6%

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

      8. sub-neg 19.6%

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

      9. div-sub 19.6%

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

   3. Simplified 19.6%

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

   4. Taylor expanded in x around 0 5.6%

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

      1. log1p-def 5.6%

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

   6. Simplified 5.6%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 69.5%

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

   9. Simplified 69.5%

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

   if -25000 < y

   1. Initial program 95.1%

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

      1. sub-neg 95.1%

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

      2. log1p-def 95.1%

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

      3. neg-sub0 95.1%

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

      4. div-sub 95.1%

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

      5. associate--r- 95.1%

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

      6. neg-sub0 95.1%

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

      7. +-commutative 95.1%

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

      8. sub-neg 95.1%

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

      9. div-sub 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in y around 0 88.0%

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

      1. log1p-def 88.0%

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

      2. mul-1-neg 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 11: 63.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. neg-sub0 74.1%

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

   4. div-sub 74.1%

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

   5. associate--r- 74.1%

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

   6. neg-sub0 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

   9. div-sub 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in y around 0 67.1%

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

   1. log1p-def 67.1%

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

   2. mul-1-neg 67.1%

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

6. Simplified 67.1%

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

7. Final simplification 67.1%

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

Alternative 12: 41.2% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. neg-sub0 74.1%

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

   4. div-sub 74.1%

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

   5. associate--r- 74.1%

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

   6. neg-sub0 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

   9. div-sub 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in x around 0 42.9%

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

   1. log1p-def 42.9%

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

6. Simplified 42.9%

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

7. Taylor expanded in y around 0 42.9%

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

8. Final simplification 42.9%

   \[\leadsto 1 - y \]

Alternative 13: 43.8% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. neg-sub0 74.1%

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

   4. div-sub 74.1%

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

   5. associate--r- 74.1%

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

   6. neg-sub0 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

   9. div-sub 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in y around 0 67.1%

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

5. Taylor expanded in x around 0 44.5%

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

   1. mul-1-neg 44.5%

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

   2. unsub-neg 44.5%

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

   3. unpow2 44.5%

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

7. Simplified 44.5%

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

8. Taylor expanded in x around 0 45.5%

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

   1. neg-mul-1 45.5%

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

10. Simplified 45.5%

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

11. Final simplification 45.5%

    \[\leadsto x + 1 \]

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 2023200 
(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))))))