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

Percentage Accurate: 71.7% → 99.7%

Time: 12.1s

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: 71.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{x + -1}\\ \mathbf{if}\;y \leq -1600:\\ \;\;\;\;1 + \left(\left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-6, t_0, \mathsf{fma}\left(2, t_0, t_0 \cdot 6\right)\right)}{{y}^{3}}, \frac{0.5}{{y}^{2}}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) (+ x -1.0))))
   (if (<= y -1600.0)
     (+
      1.0
      (-
       (-
        (- (/ (- 1.0 x) (* y (+ x -1.0))) (log (/ -1.0 y)))
        (fma
         -0.16666666666666666
         (/ (fma -6.0 t_0 (fma 2.0 t_0 (* t_0 6.0))) (pow y 3.0))
         (/ 0.5 (pow y 2.0))))
       (log1p (- x))))
     (if (<= y 5e+16)
       (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))
       (log (/ E (/ (+ 1.0 x) y)))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / (x + -1.0);
	double tmp;
	if (y <= -1600.0) {
		tmp = 1.0 + (((((1.0 - x) / (y * (x + -1.0))) - log((-1.0 / y))) - fma(-0.16666666666666666, (fma(-6.0, t_0, fma(2.0, t_0, (t_0 * 6.0))) / pow(y, 3.0)), (0.5 / pow(y, 2.0)))) - log1p(-x));
	} else if (y <= 5e+16) {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = log((((double) M_E) / ((1.0 + x) / y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / Float64(x + -1.0))
	tmp = 0.0
	if (y <= -1600.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(1.0 - x) / Float64(y * Float64(x + -1.0))) - log(Float64(-1.0 / y))) - fma(-0.16666666666666666, Float64(fma(-6.0, t_0, fma(2.0, t_0, Float64(t_0 * 6.0))) / (y ^ 3.0)), Float64(0.5 / (y ^ 2.0)))) - log1p(Float64(-x))));
	elseif (y <= 5e+16)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(1.0 - y)) * Float64(y - x))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(1.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1600.0], N[(1.0 + N[(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[(-0.16666666666666666 * N[(N[(-6.0 * t$95$0 + N[(2.0 * t$95$0 + N[(t$95$0 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+16], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1600

   1. Initial program 17.2%

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

      1. sub-neg 17.2%

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

      2. log1p-def 17.2%

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

      3. distribute-neg-frac 17.2%

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

      4. sub-neg 17.2%

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

      5. distribute-neg-in 17.2%

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

      6. remove-double-neg 17.2%

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

      7. +-commutative 17.2%

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

      8. sub-neg 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in y around -inf 80.8%

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

   6. Simplified 99.6%

      \[\leadsto 1 - \color{blue}{\left(\mathsf{log1p}\left(-x\right) + \left(\left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \left(-1 + x\right)}\right) + \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-6, \frac{1 - x}{-1 + x}, \mathsf{fma}\left(2, 1 \cdot \frac{1 - x}{-1 + x}, 6 \cdot \frac{1 - x}{-1 + x}\right)\right)}{{y}^{3}}, \frac{0.5}{{y}^{2}}\right)\right)\right)} \]

   if -1600 < y < 5e16

   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. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 5e16 < y

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 51.9%

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

      3. distribute-neg-frac 51.9%

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

      4. sub-neg 51.9%

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

      5. distribute-neg-in 51.9%

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

      6. remove-double-neg 51.9%

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

      7. +-commutative 51.9%

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

      8. sub-neg 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in y around -inf 0.0%

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

      1. associate--r+ 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-lft-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-def 0.0%

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

      8. mul-1-neg 0.0%

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

   6. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. neg-log 0.0%

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

      10. clear-num 0.0%

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

      11. add-exp-log 0.0%

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

      12. div-inv 0.0%

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

      13. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 1.2%

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

   10. Applied egg-rr 60.9%

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

       1. expm1-def 99.3%

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

       2. expm1-log1p 99.3%

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

       3. exp-1-e 99.3%

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

       4. associate-*r/ 99.3%

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

       5. *-commutative 99.3%

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

       6. associate-/l* 99.3%

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

       7. exp-1-e 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1600:\\ \;\;\;\;1 + \left(\left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-6, \frac{1 - x}{x + -1}, \mathsf{fma}\left(2, \frac{1 - x}{x + -1}, \frac{1 - x}{x + -1} \cdot 6\right)\right)}{{y}^{3}}, \frac{0.5}{{y}^{2}}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11000:\\ \;\;\;\;1 + \left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \left(\mathsf{log1p}\left(-x\right) + \frac{0.5}{{y}^{2}}\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -11000.0)
   (+
    1.0
    (-
     (- (/ (- 1.0 x) (* y (+ x -1.0))) (log (/ -1.0 y)))
     (+ (log1p (- x)) (/ 0.5 (pow y 2.0)))))
   (if (<= y 2e+15)
     (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))
     (log (/ E (/ (+ 1.0 x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -11000.0) {
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - log((-1.0 / y))) - (log1p(-x) + (0.5 / pow(y, 2.0))));
	} else if (y <= 2e+15) {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = log((((double) M_E) / ((1.0 + x) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -11000.0) {
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - Math.log((-1.0 / y))) - (Math.log1p(-x) + (0.5 / Math.pow(y, 2.0))));
	} else if (y <= 2e+15) {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = Math.log((Math.E / ((1.0 + x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -11000.0:
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - math.log((-1.0 / y))) - (math.log1p(-x) + (0.5 / math.pow(y, 2.0))))
	elif y <= 2e+15:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	else:
		tmp = math.log((math.e / ((1.0 + x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -11000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - x) / Float64(y * Float64(x + -1.0))) - log(Float64(-1.0 / y))) - Float64(log1p(Float64(-x)) + Float64(0.5 / (y ^ 2.0)))));
	elseif (y <= 2e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(1.0 - y)) * Float64(y - x))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(1.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -11000.0], 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[(N[Log[1 + (-x)], $MachinePrecision] + N[(0.5 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+15], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -11000

   1. Initial program 17.2%

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

      1. sub-neg 17.2%

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

      2. log1p-def 17.2%

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

      3. distribute-neg-frac 17.2%

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

      4. sub-neg 17.2%

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

      5. distribute-neg-in 17.2%

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

      6. remove-double-neg 17.2%

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

      7. +-commutative 17.2%

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

      8. sub-neg 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in y around -inf 86.5%

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

   6. Simplified 99.5%

      \[\leadsto 1 - \color{blue}{\left(\left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \left(-1 + x\right)}\right) + \left(\frac{0.5}{{y}^{2}} + \mathsf{log1p}\left(-x\right)\right)\right)} \]

   if -11000 < y < 2e15

   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. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 2e15 < y

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 51.9%

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

      3. distribute-neg-frac 51.9%

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

      4. sub-neg 51.9%

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

      5. distribute-neg-in 51.9%

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

      6. remove-double-neg 51.9%

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

      7. +-commutative 51.9%

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

      8. sub-neg 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in y around -inf 0.0%

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

      1. associate--r+ 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-lft-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-def 0.0%

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

      8. mul-1-neg 0.0%

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

   6. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. neg-log 0.0%

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

      10. clear-num 0.0%

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

      11. add-exp-log 0.0%

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

      12. div-inv 0.0%

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

      13. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 1.2%

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

   10. Applied egg-rr 60.9%

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

       1. expm1-def 99.3%

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

       2. expm1-log1p 99.3%

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

       3. exp-1-e 99.3%

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

       4. associate-*r/ 99.3%

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

       5. *-commutative 99.3%

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

       6. associate-/l* 99.3%

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

       7. exp-1-e 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11000:\\ \;\;\;\;1 + \left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \left(\mathsf{log1p}\left(-x\right) + \frac{0.5}{{y}^{2}}\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \]

Alternative 3: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- y x) (- 1.0 y))) 0.0)
   (log (* y (- E)))
   (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((y - x) / (1.0 - y))) <= 0.0) {
		tmp = log((y * -((double) M_E)));
	} else {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 + ((y - x) / (1.0 - y))) <= 0.0) {
		tmp = Math.log((y * -Math.E));
	} else {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 + ((y - x) / (1.0 - y))) <= 0.0:
		tmp = math.log((y * -math.e))
	else:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-def 3.1%

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

      3. distribute-neg-frac 3.1%

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

      4. sub-neg 3.1%

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

      5. distribute-neg-in 3.1%

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

      6. remove-double-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around -inf 80.3%

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

      1. associate--r+ 80.3%

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

      2. sub-neg 80.3%

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

      3. metadata-eval 80.3%

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

      4. distribute-lft-in 80.3%

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

      5. metadata-eval 80.3%

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

      6. +-commutative 80.3%

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

      7. log1p-def 80.3%

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

      8. mul-1-neg 80.3%

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

   6. Simplified 80.3%

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

      1. add-log-exp 80.3%

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

      2. sub-neg 80.3%

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

      3. exp-sum 80.3%

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

      4. add-sqr-sqrt 45.6%

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

      5. sqrt-unprod 77.6%

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

      6. sqr-neg 77.6%

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

      7. sqrt-unprod 34.7%

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

      8. add-sqr-sqrt 64.1%

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

      9. neg-log 64.1%

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

      10. clear-num 64.1%

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

      11. add-exp-log 64.1%

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

      12. div-inv 64.1%

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

      13. metadata-eval 64.1%

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

   8. Applied egg-rr 64.1%

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

   9. Taylor expanded in x around 0 67.0%

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

       1. mul-1-neg 67.0%

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

       2. distribute-rgt-neg-in 67.0%

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

       3. exp-1-e 67.0%

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

   11. Simplified 67.0%

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

   if 0.0 < (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.7%

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

      3. distribute-neg-frac 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. sub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \end{array} \]

Alternative 4: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+16}:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+16)
   (log (* y (- E)))
   (if (<= y 500000000000.0)
     (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))
     (log (/ E (/ (+ 1.0 x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+16) {
		tmp = log((y * -((double) M_E)));
	} else if (y <= 500000000000.0) {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = log((((double) M_E) / ((1.0 + x) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+16) {
		tmp = Math.log((y * -Math.E));
	} else if (y <= 500000000000.0) {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = Math.log((Math.E / ((1.0 + x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+16:
		tmp = math.log((y * -math.e))
	elif y <= 500000000000.0:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	else:
		tmp = math.log((math.e / ((1.0 + x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+16)
		tmp = log(Float64(y * Float64(-exp(1))));
	elseif (y <= 500000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(1.0 - y)) * Float64(y - x))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(1.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+16], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 500000000000.0], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e16

   1. Initial program 14.9%

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

      1. sub-neg 14.9%

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

      2. log1p-def 14.9%

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

      3. distribute-neg-frac 14.9%

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

      4. sub-neg 14.9%

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

      5. distribute-neg-in 14.9%

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

      6. remove-double-neg 14.9%

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

      7. +-commutative 14.9%

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

      8. sub-neg 14.9%

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

   3. Simplified 14.9%

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

   4. Taylor expanded in y around -inf 99.6%

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

      1. associate--r+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

      7. log1p-def 99.6%

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

      8. mul-1-neg 99.6%

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

   6. Simplified 99.6%

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

      1. add-log-exp 99.6%

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

      2. sub-neg 99.6%

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

      3. exp-sum 99.6%

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

      4. add-sqr-sqrt 61.8%

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

      5. sqrt-unprod 86.5%

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

      6. sqr-neg 86.5%

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

      7. sqrt-unprod 37.9%

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

      8. add-sqr-sqrt 69.9%

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

      9. neg-log 69.9%

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

      10. clear-num 69.9%

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

      11. add-exp-log 70.0%

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

      12. div-inv 70.0%

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

      13. metadata-eval 70.0%

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

   8. Applied egg-rr 70.0%

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

   9. Taylor expanded in x around 0 73.3%

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

       1. mul-1-neg 73.3%

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

       2. distribute-rgt-neg-in 73.3%

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

       3. exp-1-e 73.3%

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

   11. Simplified 73.3%

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

   if -6.5e16 < y < 5e11

   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 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 5e11 < y

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 51.9%

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

      3. distribute-neg-frac 51.9%

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

      4. sub-neg 51.9%

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

      5. distribute-neg-in 51.9%

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

      6. remove-double-neg 51.9%

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

      7. +-commutative 51.9%

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

      8. sub-neg 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in y around -inf 0.0%

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

      1. associate--r+ 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-lft-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-def 0.0%

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

      8. mul-1-neg 0.0%

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

   6. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. neg-log 0.0%

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

      10. clear-num 0.0%

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

      11. add-exp-log 0.0%

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

      12. div-inv 0.0%

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

      13. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 1.2%

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

   10. Applied egg-rr 60.9%

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

       1. expm1-def 99.3%

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

       2. expm1-log1p 99.3%

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

       3. exp-1-e 99.3%

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

       4. associate-*r/ 99.3%

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

       5. *-commutative 99.3%

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

       6. associate-/l* 99.3%

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

       7. exp-1-e 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+16}:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+16}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+16)
   (- (- 1.0 (log1p (- x))) (log (/ -1.0 y)))
   (if (<= y 500000000000.0)
     (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))
     (log (/ E (/ (+ 1.0 x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+16) {
		tmp = (1.0 - log1p(-x)) - log((-1.0 / y));
	} else if (y <= 500000000000.0) {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = log((((double) M_E) / ((1.0 + x) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+16) {
		tmp = (1.0 - Math.log1p(-x)) - Math.log((-1.0 / y));
	} else if (y <= 500000000000.0) {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = Math.log((Math.E / ((1.0 + x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+16:
		tmp = (1.0 - math.log1p(-x)) - math.log((-1.0 / y))
	elif y <= 500000000000.0:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	else:
		tmp = math.log((math.e / ((1.0 + x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+16)
		tmp = Float64(Float64(1.0 - log1p(Float64(-x))) - log(Float64(-1.0 / y)));
	elseif (y <= 500000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(1.0 - y)) * Float64(y - x))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(1.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+16], N[(N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 500000000000.0], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e16

   1. Initial program 14.9%

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

      1. sub-neg 14.9%

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

      2. log1p-def 14.9%

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

      3. distribute-neg-frac 14.9%

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

      4. sub-neg 14.9%

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

      5. distribute-neg-in 14.9%

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

      6. remove-double-neg 14.9%

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

      7. +-commutative 14.9%

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

      8. sub-neg 14.9%

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

   3. Simplified 14.9%

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

   4. Taylor expanded in y around -inf 99.6%

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

      1. associate--r+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

      7. log1p-def 99.6%

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

      8. mul-1-neg 99.6%

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

   6. Simplified 99.6%

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

   if -6.5e16 < y < 5e11

   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 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 5e11 < y

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 51.9%

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

      3. distribute-neg-frac 51.9%

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

      4. sub-neg 51.9%

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

      5. distribute-neg-in 51.9%

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

      6. remove-double-neg 51.9%

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

      7. +-commutative 51.9%

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

      8. sub-neg 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in y around -inf 0.0%

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

      1. associate--r+ 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-lft-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-def 0.0%

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

      8. mul-1-neg 0.0%

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

   6. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. neg-log 0.0%

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

      10. clear-num 0.0%

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

      11. add-exp-log 0.0%

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

      12. div-inv 0.0%

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

      13. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 1.2%

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

   10. Applied egg-rr 60.9%

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

       1. expm1-def 99.3%

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

       2. expm1-log1p 99.3%

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

       3. exp-1-e 99.3%

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

       4. associate-*r/ 99.3%

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

       5. *-commutative 99.3%

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

       6. associate-/l* 99.3%

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

       7. exp-1-e 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+16}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{1 + x}{y}}\right)\\ \end{array} \]

Alternative 6: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- y x) (- 1.0 y))) 0.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((y - x) / (1.0 - y))) <= 0.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 + ((y - x) / (1.0 - y))) <= 0.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 + ((y - x) / (1.0 - y))) <= 0.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-def 3.1%

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

      3. distribute-neg-frac 3.1%

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

      4. sub-neg 3.1%

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

      5. distribute-neg-in 3.1%

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

      6. remove-double-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around -inf 80.3%

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

      1. associate--r+ 80.3%

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

      2. sub-neg 80.3%

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

      3. metadata-eval 80.3%

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

      4. distribute-lft-in 80.3%

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

      5. metadata-eval 80.3%

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

      6. +-commutative 80.3%

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

      7. log1p-def 80.3%

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

      8. mul-1-neg 80.3%

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

   6. Simplified 80.3%

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

   7. Taylor expanded in x around 0 67.0%

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

   if 0.0 < (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.7%

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

      3. distribute-neg-frac 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. sub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \end{array} \]

Alternative 7: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{1 - y}\\ \mathbf{if}\;1 + t_0 \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- 1.0 y))))
   (if (<= (+ 1.0 t_0) 0.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p t_0)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (1.0 - y);
	double tmp;
	if ((1.0 + t_0) <= 0.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (1.0 - y);
	double tmp;
	if ((1.0 + t_0) <= 0.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (1.0 - y)
	tmp = 0
	if (1.0 + t_0) <= 0.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(t_0)
	return tmp

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 + t$95$0), $MachinePrecision], 0.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-def 3.1%

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

      3. distribute-neg-frac 3.1%

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

      4. sub-neg 3.1%

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

      5. distribute-neg-in 3.1%

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

      6. remove-double-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around -inf 80.3%

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

      1. associate--r+ 80.3%

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

      2. sub-neg 80.3%

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

      3. metadata-eval 80.3%

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

      4. distribute-lft-in 80.3%

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

      5. metadata-eval 80.3%

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

      6. +-commutative 80.3%

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

      7. log1p-def 80.3%

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

      8. mul-1-neg 80.3%

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

   6. Simplified 80.3%

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

   7. Taylor expanded in x around 0 67.0%

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

   if 0.0 < (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.7%

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

      3. distribute-neg-frac 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. sub-neg 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \end{array} \]

Alternative 8: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+16}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+16)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+16) {
		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 <= -6.5e+16) {
		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 <= -6.5e+16:
		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 <= -6.5e+16)
		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, -6.5e+16], 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 < -6.5e16

   1. Initial program 14.9%

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

      1. sub-neg 14.9%

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

      2. log1p-def 14.9%

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

      3. distribute-neg-frac 14.9%

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

      4. sub-neg 14.9%

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

      5. distribute-neg-in 14.9%

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

      6. remove-double-neg 14.9%

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

      7. +-commutative 14.9%

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

      8. sub-neg 14.9%

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

   3. Simplified 14.9%

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

   4. Taylor expanded in y around -inf 99.6%

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

      1. associate--r+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

      7. log1p-def 99.6%

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

      8. mul-1-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 73.3%

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

   if -6.5e16 < y

   1. Initial program 92.6%

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

      1. sub-neg 92.6%

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

      2. log1p-def 92.6%

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

      3. distribute-neg-frac 92.6%

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

      4. sub-neg 92.6%

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

      5. distribute-neg-in 92.6%

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

      6. remove-double-neg 92.6%

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

      7. +-commutative 92.6%

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

      8. sub-neg 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in x around inf 91.4%

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

      1. neg-mul-1 91.4%

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

      2. distribute-neg-frac 91.4%

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

   6. Simplified 91.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+16}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternative 9: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8) (- 1.0 (log (/ -1.0 y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998

   1. Initial program 18.4%

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

      1. sub-neg 18.4%

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

      2. log1p-def 18.4%

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

      3. distribute-neg-frac 18.4%

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

      4. sub-neg 18.4%

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

      5. distribute-neg-in 18.4%

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

      6. remove-double-neg 18.4%

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

      7. +-commutative 18.4%

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

      8. sub-neg 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in y around -inf 97.7%

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

      1. associate--r+ 97.7%

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

      2. sub-neg 97.7%

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

      3. metadata-eval 97.7%

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

      4. distribute-lft-in 97.7%

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

      5. metadata-eval 97.7%

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

      6. +-commutative 97.7%

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

      7. log1p-def 97.7%

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

      8. mul-1-neg 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in x around 0 71.1%

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

   if -3.7999999999999998 < y

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. log1p-def 92.5%

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

      3. distribute-neg-frac 92.5%

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

      4. sub-neg 92.5%

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

      5. distribute-neg-in 92.5%

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

      6. remove-double-neg 92.5%

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

      7. +-commutative 92.5%

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

      8. sub-neg 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around 0 83.8%

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

      1. +-commutative 83.8%

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

      2. div-sub 83.8%

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

      3. mul-1-neg 83.8%

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

      4. sub-neg 83.8%

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

      5. *-inverses 83.8%

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

      6. *-rgt-identity 83.8%

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

      7. log1p-def 83.8%

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

      8. mul-1-neg 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 10: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.0) {
		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 <= -4.0) {
		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 <= -4.0:
		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 <= -4.0)
		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, -4.0], 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 < -4

   1. Initial program 18.4%

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

      1. sub-neg 18.4%

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

      2. log1p-def 18.4%

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

      3. distribute-neg-frac 18.4%

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

      4. sub-neg 18.4%

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

      5. distribute-neg-in 18.4%

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

      6. remove-double-neg 18.4%

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

      7. +-commutative 18.4%

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

      8. sub-neg 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in y around -inf 97.7%

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

      1. associate--r+ 97.7%

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

      2. sub-neg 97.7%

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

      3. metadata-eval 97.7%

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

      4. distribute-lft-in 97.7%

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

      5. metadata-eval 97.7%

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

      6. +-commutative 97.7%

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

      7. log1p-def 97.7%

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

      8. mul-1-neg 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in x around 0 71.1%

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

   if -4 < y

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. log1p-def 92.5%

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

      3. distribute-neg-frac 92.5%

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

      4. sub-neg 92.5%

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

      5. distribute-neg-in 92.5%

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

      6. remove-double-neg 92.5%

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

      7. +-commutative 92.5%

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

      8. sub-neg 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around 0 83.6%

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

      1. log1p-def 83.6%

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

      2. mul-1-neg 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 11: 62.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. log1p-def 72.6%

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

   3. distribute-neg-frac 72.6%

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

   4. sub-neg 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. remove-double-neg 72.6%

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

   7. +-commutative 72.6%

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

   8. sub-neg 72.6%

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

3. Simplified 72.6%

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

4. Taylor expanded in y around 0 64.5%

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

   1. log1p-def 64.6%

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

   2. mul-1-neg 64.6%

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

6. Simplified 64.6%

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

7. Final simplification 64.6%

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

Alternative 12: 42.8% 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(x))
end

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. log1p-def 72.6%

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

   3. distribute-neg-frac 72.6%

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

   4. sub-neg 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. remove-double-neg 72.6%

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

   7. +-commutative 72.6%

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

   8. sub-neg 72.6%

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

3. Simplified 72.6%

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

4. Taylor expanded in y around 0 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. log-prod 64.5%

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

   3. metadata-eval 64.5%

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

   4. log1p-def 64.6%

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

   5. neg-mul-1 64.6%

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

   6. add-sqr-sqrt 43.6%

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

   7. sqrt-unprod 53.5%

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

   8. sqr-neg 53.5%

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

   9. sqrt-unprod 21.7%

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

   10. add-sqr-sqrt 44.8%

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

6. Applied egg-rr 44.8%

   \[\leadsto 1 - \color{blue}{\left(0 + \mathsf{log1p}\left(x\right)\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 44.8%

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

8. Simplified 44.8%

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

9. Final simplification 44.8%

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

Alternative 13: 4.0% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. log1p-def 72.6%

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

   3. distribute-neg-frac 72.6%

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

   4. sub-neg 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. remove-double-neg 72.6%

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

   7. +-commutative 72.6%

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

   8. sub-neg 72.6%

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

3. Simplified 72.6%

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

4. Taylor expanded in y around -inf 28.4%

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

   1. associate--r+ 28.4%

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

   2. sub-neg 28.4%

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

   3. metadata-eval 28.4%

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

   4. distribute-lft-in 28.4%

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

   5. metadata-eval 28.4%

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

   6. +-commutative 28.4%

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

   7. log1p-def 28.4%

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

   8. mul-1-neg 28.4%

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

6. Simplified 28.4%

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

7. Taylor expanded in x around 0 19.6%

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

8. Taylor expanded in x around inf 3.9%

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

9. Final simplification 3.9%

   \[\leadsto x \]

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 2023311 
(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))))))