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

Percentage Accurate: 73.1% → 99.6%

Time: 17.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13600:\\ \;\;\;\;1 + \left(\frac{-0.5}{{y}^{2}} + \left(\frac{\frac{x}{1 - x} + \frac{-1}{1 - x}}{y} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot e^{1 - \mathsf{log1p}\left(x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13600.0)
   (+
    1.0
    (+
     (/ -0.5 (pow y 2.0))
     (-
      (/ (+ (/ x (- 1.0 x)) (/ -1.0 (- 1.0 x))) y)
      (+ (log1p (- x)) (log (/ -1.0 y))))))
   (if (<= y 1.25e+34)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (log (* y (exp (- 1.0 (log1p x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13600.0) {
		tmp = 1.0 + ((-0.5 / pow(y, 2.0)) + ((((x / (1.0 - x)) + (-1.0 / (1.0 - x))) / y) - (log1p(-x) + log((-1.0 / y)))));
	} else if (y <= 1.25e+34) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((y * exp((1.0 - log1p(x)))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -13600.0) {
		tmp = 1.0 + ((-0.5 / Math.pow(y, 2.0)) + ((((x / (1.0 - x)) + (-1.0 / (1.0 - x))) / y) - (Math.log1p(-x) + Math.log((-1.0 / y)))));
	} else if (y <= 1.25e+34) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((y * Math.exp((1.0 - Math.log1p(x)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13600.0:
		tmp = 1.0 + ((-0.5 / math.pow(y, 2.0)) + ((((x / (1.0 - x)) + (-1.0 / (1.0 - x))) / y) - (math.log1p(-x) + math.log((-1.0 / y)))))
	elif y <= 1.25e+34:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((y * math.exp((1.0 - math.log1p(x)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13600.0)
		tmp = Float64(1.0 + Float64(Float64(-0.5 / (y ^ 2.0)) + Float64(Float64(Float64(Float64(x / Float64(1.0 - x)) + Float64(-1.0 / Float64(1.0 - x))) / y) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))))));
	elseif (y <= 1.25e+34)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(y * exp(Float64(1.0 - log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13600.0], N[(1.0 + N[(N[(-0.5 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+34], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[Exp[N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -13600

   1. Initial program 18.2%

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

      1. sub-neg 18.2%

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

      2. log1p-define 18.2%

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

      3. distribute-neg-frac2 18.2%

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

      4. neg-sub0 18.2%

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

      5. associate--r- 18.2%

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

      6. metadata-eval 18.2%

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

      7. +-commutative 18.2%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in y around -inf 87.5%

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

   7. Simplified 99.6%

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

   if -13600 < y < 1.25e34

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 1.25e34 < y

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. log1p-define 45.7%

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

      3. distribute-neg-frac2 45.7%

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

      4. neg-sub0 45.7%

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

      5. associate--r- 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around -inf 0.0%

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

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

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)} \]
   8. 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. div-inv 0.0%

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

      12. metadata-eval 0.0%

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

      13. add-exp-log 0.0%

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

      14. metadata-eval 0.0%

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

      15. div-inv 0.0%

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

      16. clear-num 0.0%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 41.9%

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

      19. frac-times 38.3%

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

   9. Applied egg-rr 98.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13600:\\ \;\;\;\;1 + \left(\frac{-0.5}{{y}^{2}} + \left(\frac{\frac{x}{1 - x} + \frac{-1}{1 - x}}{y} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot e^{1 - \mathsf{log1p}\left(x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14600:\\ \;\;\;\;1 + \left(\left(\left(\frac{-1}{y} - \frac{0.5}{{y}^{2}}\right) - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+32}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot e^{1 - \mathsf{log1p}\left(x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -14600.0)
   (+
    1.0
    (- (- (- (/ -1.0 y) (/ 0.5 (pow y 2.0))) (log1p (- x))) (log (/ -1.0 y))))
   (if (<= y 3.15e+32)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (log (* y (exp (- 1.0 (log1p x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -14600.0) {
		tmp = 1.0 + ((((-1.0 / y) - (0.5 / pow(y, 2.0))) - log1p(-x)) - log((-1.0 / y)));
	} else if (y <= 3.15e+32) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((y * exp((1.0 - log1p(x)))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -14600.0) {
		tmp = 1.0 + ((((-1.0 / y) - (0.5 / Math.pow(y, 2.0))) - Math.log1p(-x)) - Math.log((-1.0 / y)));
	} else if (y <= 3.15e+32) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((y * Math.exp((1.0 - Math.log1p(x)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -14600.0:
		tmp = 1.0 + ((((-1.0 / y) - (0.5 / math.pow(y, 2.0))) - math.log1p(-x)) - math.log((-1.0 / y)))
	elif y <= 3.15e+32:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((y * math.exp((1.0 - math.log1p(x)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -14600.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(-1.0 / y) - Float64(0.5 / (y ^ 2.0))) - log1p(Float64(-x))) - log(Float64(-1.0 / y))));
	elseif (y <= 3.15e+32)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(y * exp(Float64(1.0 - log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -14600.0], N[(1.0 + N[(N[(N[(N[(-1.0 / y), $MachinePrecision] - N[(0.5 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.15e+32], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[Exp[N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -14600

   1. Initial program 18.2%

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

      1. sub-neg 18.2%

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

      2. log1p-define 18.2%

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

      3. distribute-neg-frac2 18.2%

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

      4. neg-sub0 18.2%

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

      5. associate--r- 18.2%

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

      6. metadata-eval 18.2%

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

      7. +-commutative 18.2%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in y around -inf 87.5%

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

   7. Simplified 99.6%

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

   if -14600 < y < 3.1500000000000001e32

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 3.1500000000000001e32 < y

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. log1p-define 45.7%

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

      3. distribute-neg-frac2 45.7%

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

      4. neg-sub0 45.7%

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

      5. associate--r- 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around -inf 0.0%

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

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

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)} \]
   8. 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. div-inv 0.0%

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

      12. metadata-eval 0.0%

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

      13. add-exp-log 0.0%

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

      14. metadata-eval 0.0%

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

      15. div-inv 0.0%

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

      16. clear-num 0.0%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 41.9%

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

      19. frac-times 38.3%

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

   9. Applied egg-rr 98.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14600:\\ \;\;\;\;1 + \left(\left(\left(\frac{-1}{y} - \frac{0.5}{{y}^{2}}\right) - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+32}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot e^{1 - \mathsf{log1p}\left(x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -520000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+33}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot e^{1 - \mathsf{log1p}\left(x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -520000.0)
   (+ 1.0 (- (- (/ -1.0 y) (log1p (- x))) (log (/ -1.0 y))))
   (if (<= y 1.62e+33)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (log (* y (exp (- 1.0 (log1p x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -520000.0) {
		tmp = 1.0 + (((-1.0 / y) - log1p(-x)) - log((-1.0 / y)));
	} else if (y <= 1.62e+33) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((y * exp((1.0 - log1p(x)))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -520000.0) {
		tmp = 1.0 + (((-1.0 / y) - Math.log1p(-x)) - Math.log((-1.0 / y)));
	} else if (y <= 1.62e+33) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((y * Math.exp((1.0 - Math.log1p(x)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -520000.0:
		tmp = 1.0 + (((-1.0 / y) - math.log1p(-x)) - math.log((-1.0 / y)))
	elif y <= 1.62e+33:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((y * math.exp((1.0 - math.log1p(x)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -520000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 / y) - log1p(Float64(-x))) - log(Float64(-1.0 / y))));
	elseif (y <= 1.62e+33)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(y * exp(Float64(1.0 - log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -520000.0], N[(1.0 + N[(N[(N[(-1.0 / y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62e+33], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[Exp[N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e5

   1. Initial program 16.5%

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

      1. sub-neg 16.5%

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

      2. log1p-define 16.5%

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

      3. distribute-neg-frac2 16.5%

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

      4. neg-sub0 16.5%

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

      5. associate--r- 16.5%

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

      6. metadata-eval 16.5%

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

      7. +-commutative 16.5%

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

   3. Simplified 16.5%

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

   5. Taylor expanded in y around -inf 99.6%

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

   7. Simplified 99.6%

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

   if -5.2e5 < y < 1.62e33

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 1.62e33 < y

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. log1p-define 45.7%

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

      3. distribute-neg-frac2 45.7%

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

      4. neg-sub0 45.7%

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

      5. associate--r- 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around -inf 0.0%

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

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

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)} \]
   8. 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. div-inv 0.0%

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

      12. metadata-eval 0.0%

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

      13. add-exp-log 0.0%

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

      14. metadata-eval 0.0%

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

      15. div-inv 0.0%

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

      16. clear-num 0.0%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 41.9%

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

      19. frac-times 38.3%

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

   9. Applied egg-rr 98.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+33}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot e^{1 - \mathsf{log1p}\left(x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -640000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(x\right)\right) + \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -640000.0)
   (+ 1.0 (- (- (/ -1.0 y) (log1p (- x))) (log (/ -1.0 y))))
   (if (<= y 9e+34)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ (- 1.0 (log1p x)) (log y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -640000.0) {
		tmp = 1.0 + (((-1.0 / y) - log1p(-x)) - log((-1.0 / y)));
	} else if (y <= 9e+34) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 - log1p(x)) + log(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -640000.0) {
		tmp = 1.0 + (((-1.0 / y) - Math.log1p(-x)) - Math.log((-1.0 / y)));
	} else if (y <= 9e+34) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 - Math.log1p(x)) + Math.log(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -640000.0:
		tmp = 1.0 + (((-1.0 / y) - math.log1p(-x)) - math.log((-1.0 / y)))
	elif y <= 9e+34:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = (1.0 - math.log1p(x)) + math.log(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -640000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 / y) - log1p(Float64(-x))) - log(Float64(-1.0 / y))));
	elseif (y <= 9e+34)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 - log1p(x)) + log(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -640000.0], N[(1.0 + N[(N[(N[(-1.0 / y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+34], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4e5

   1. Initial program 16.5%

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

      1. sub-neg 16.5%

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

      2. log1p-define 16.5%

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

      3. distribute-neg-frac2 16.5%

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

      4. neg-sub0 16.5%

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

      5. associate--r- 16.5%

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

      6. metadata-eval 16.5%

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

      7. +-commutative 16.5%

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

   3. Simplified 16.5%

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

   5. Taylor expanded in y around -inf 99.6%

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

   7. Simplified 99.6%

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

   if -6.4e5 < y < 9.0000000000000001e34

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 9.0000000000000001e34 < y

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. log1p-define 45.7%

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

      3. distribute-neg-frac2 45.7%

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

      4. neg-sub0 45.7%

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

      5. associate--r- 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around -inf 0.0%

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

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

   7. Simplified 0.0%

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

      1. associate--l- 0.0%

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

      2. sub-neg 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 41.8%

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

      10. frac-times 38.2%

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

      11. metadata-eval 38.2%

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

      12. metadata-eval 38.2%

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

      13. frac-times 41.8%

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

      14. sqrt-unprod 98.6%

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

      15. add-sqr-sqrt 98.6%

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

      16. log-rec 98.6%

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

   9. Applied egg-rr 98.6%

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

       1. sub-neg 98.6%

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

       2. unsub-neg 98.6%

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

       3. *-rgt-identity 98.6%

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

       4. associate-+l- 98.6%

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

       5. *-rgt-identity 98.6%

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

   11. Simplified 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -640000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(x\right)\right) + \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+14}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(x\right)\right) + \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.9e+14)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 8.2e+34)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ (- 1.0 (log1p x)) (log y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.9e+14) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 8.2e+34) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 - log1p(x)) + log(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.9e+14) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 8.2e+34) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 - Math.log1p(x)) + Math.log(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.9e+14:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 8.2e+34:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = (1.0 - math.log1p(x)) + math.log(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.9e+14)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 8.2e+34)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 - log1p(x)) + log(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.9e+14], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+34], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9e14

   1. Initial program 14.4%

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

      1. sub-neg 14.4%

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

      2. log1p-define 14.4%

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

      3. distribute-neg-frac2 14.4%

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

      4. neg-sub0 14.4%

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

      5. associate--r- 14.4%

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

      6. metadata-eval 14.4%

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

      7. +-commutative 14.4%

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

   3. Simplified 14.4%

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

   5. Taylor expanded in y around -inf 99.6%

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

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 72.6%

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

   if -4.9e14 < y < 8.1999999999999997e34

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. log1p-define 99.2%

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

      3. distribute-neg-frac2 99.2%

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

      4. neg-sub0 99.2%

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

      5. associate--r- 99.2%

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

      6. metadata-eval 99.2%

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

      7. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   if 8.1999999999999997e34 < y

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. log1p-define 45.7%

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

      3. distribute-neg-frac2 45.7%

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

      4. neg-sub0 45.7%

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

      5. associate--r- 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around -inf 0.0%

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

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

   7. Simplified 0.0%

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

      1. associate--l- 0.0%

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

      2. sub-neg 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 41.8%

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

      10. frac-times 38.2%

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

      11. metadata-eval 38.2%

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

      12. metadata-eval 38.2%

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

      13. frac-times 41.8%

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

      14. sqrt-unprod 98.6%

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

      15. add-sqr-sqrt 98.6%

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

      16. log-rec 98.6%

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

   9. Applied egg-rr 98.6%

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

       1. sub-neg 98.6%

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

       2. unsub-neg 98.6%

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

       3. *-rgt-identity 98.6%

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

       4. associate-+l- 98.6%

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

       5. *-rgt-identity 98.6%

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

   11. Simplified 98.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+14}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(x\right)\right) + \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -345000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(x\right)\right) + \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -345000000.0)
   (- (- 1.0 (log1p (- x))) (log (/ -1.0 y)))
   (if (<= y 2.6e+34)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ (- 1.0 (log1p x)) (log y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -345000000.0) {
		tmp = (1.0 - log1p(-x)) - log((-1.0 / y));
	} else if (y <= 2.6e+34) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 - log1p(x)) + log(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -345000000.0) {
		tmp = (1.0 - Math.log1p(-x)) - Math.log((-1.0 / y));
	} else if (y <= 2.6e+34) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 - Math.log1p(x)) + Math.log(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -345000000.0:
		tmp = (1.0 - math.log1p(-x)) - math.log((-1.0 / y))
	elif y <= 2.6e+34:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = (1.0 - math.log1p(x)) + math.log(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -345000000.0)
		tmp = Float64(Float64(1.0 - log1p(Float64(-x))) - log(Float64(-1.0 / y)));
	elseif (y <= 2.6e+34)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 - log1p(x)) + log(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -345000000.0], N[(N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+34], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.45e8

   1. Initial program 16.5%

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

      1. sub-neg 16.5%

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

      2. log1p-define 16.5%

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

      3. distribute-neg-frac2 16.5%

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

      4. neg-sub0 16.5%

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

      5. associate--r- 16.5%

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

      6. metadata-eval 16.5%

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

      7. +-commutative 16.5%

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

   3. Simplified 16.5%

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

   5. Taylor expanded in y around -inf 99.1%

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

      1. associate--r+ 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. distribute-lft-in 99.1%

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

      5. metadata-eval 99.1%

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

      6. +-commutative 99.1%

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

      7. log1p-define 99.1%

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

      8. mul-1-neg 99.1%

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

   7. Simplified 99.1%

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

   if -3.45e8 < y < 2.59999999999999997e34

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 2.59999999999999997e34 < y

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. log1p-define 45.7%

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

      3. distribute-neg-frac2 45.7%

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

      4. neg-sub0 45.7%

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

      5. associate--r- 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in y around -inf 0.0%

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

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

   7. Simplified 0.0%

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

      1. associate--l- 0.0%

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

      2. sub-neg 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 41.8%

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

      10. frac-times 38.2%

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

      11. metadata-eval 38.2%

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

      12. metadata-eval 38.2%

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

      13. frac-times 41.8%

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

      14. sqrt-unprod 98.6%

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

      15. add-sqr-sqrt 98.6%

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

      16. log-rec 98.6%

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

   9. Applied egg-rr 98.6%

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

       1. sub-neg 98.6%

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

       2. unsub-neg 98.6%

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

       3. *-rgt-identity 98.6%

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

       4. associate-+l- 98.6%

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

       5. *-rgt-identity 98.6%

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

   11. Simplified 98.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -345000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+34}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(x\right)\right) + \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{1}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99999)
   (- 1.0 (log1p (* (- x y) (/ 1.0 (+ y -1.0)))))
   (- 1.0 (log (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - log1p(((x - y) * (1.0 / (y + -1.0))));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - Math.log1p(((x - y) * (1.0 / (y + -1.0))));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99999:
		tmp = 1.0 - math.log1p(((x - y) * (1.0 / (y + -1.0))))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99999)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) * Float64(1.0 / Float64(y + -1.0)))));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99999], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] * N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999990000000000046

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 0.999990000000000046 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.4%

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

      1. sub-neg 4.4%

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

      2. log1p-define 4.4%

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

      3. distribute-neg-frac2 4.4%

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

      4. neg-sub0 4.4%

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

      5. associate--r- 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

   3. Simplified 4.4%

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

   5. Taylor expanded in y around -inf 81.7%

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

      1. associate--r+ 81.7%

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

      2. sub-neg 81.7%

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

      3. metadata-eval 81.7%

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

      4. distribute-lft-in 81.7%

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

      5. metadata-eval 81.7%

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

      6. +-commutative 81.7%

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

      7. log1p-define 81.7%

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

      8. mul-1-neg 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in x around 0 67.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{1}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99999)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99999:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99999)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99999], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999990000000000046

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 0.999990000000000046 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.4%

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

      1. sub-neg 4.4%

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

      2. log1p-define 4.4%

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

      3. distribute-neg-frac2 4.4%

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

      4. neg-sub0 4.4%

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

      5. associate--r- 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

   3. Simplified 4.4%

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

   5. Taylor expanded in y around -inf 81.7%

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

      1. associate--r+ 81.7%

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

      2. sub-neg 81.7%

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

      3. metadata-eval 81.7%

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

      4. distribute-lft-in 81.7%

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

      5. metadata-eval 81.7%

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

      6. +-commutative 81.7%

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

      7. log1p-define 81.7%

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

      8. mul-1-neg 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in x around 0 67.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -25.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -25.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -25.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -25.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -25.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -25.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -25

   1. Initial program 20.1%

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

      1. sub-neg 20.1%

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

      2. log1p-define 20.1%

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

      3. distribute-neg-frac2 20.1%

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

      4. neg-sub0 20.1%

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

      5. associate--r- 20.1%

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

      6. metadata-eval 20.1%

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

      7. +-commutative 20.1%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in y around -inf 96.2%

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

      1. associate--r+ 96.2%

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

      2. sub-neg 96.2%

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

      3. metadata-eval 96.2%

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

      4. distribute-lft-in 96.2%

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

      5. metadata-eval 96.2%

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

      6. +-commutative 96.2%

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

      7. log1p-define 96.2%

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

      8. mul-1-neg 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in x around 0 70.3%

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

   if -25 < y

   1. Initial program 91.5%

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

      1. sub-neg 91.5%

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

      2. log1p-define 91.5%

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

      3. distribute-neg-frac2 91.5%

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

      4. neg-sub0 91.5%

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

      5. associate--r- 91.5%

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

      6. metadata-eval 91.5%

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

      7. +-commutative 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around inf 90.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18.0) {
		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 <= -18.0) {
		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 <= -18.0:
		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 <= -18.0)
		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, -18.0], 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 < -18

   1. Initial program 20.1%

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

      1. sub-neg 20.1%

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

      2. log1p-define 20.1%

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

      3. distribute-neg-frac2 20.1%

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

      4. neg-sub0 20.1%

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

      5. associate--r- 20.1%

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

      6. metadata-eval 20.1%

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

      7. +-commutative 20.1%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in y around -inf 96.2%

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

      1. associate--r+ 96.2%

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

      2. sub-neg 96.2%

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

      3. metadata-eval 96.2%

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

      4. distribute-lft-in 96.2%

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

      5. metadata-eval 96.2%

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

      6. +-commutative 96.2%

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

      7. log1p-define 96.2%

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

      8. mul-1-neg 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in x around 0 70.3%

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

   if -18 < y

   1. Initial program 91.5%

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

      1. sub-neg 91.5%

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

      2. log1p-define 91.5%

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

      3. distribute-neg-frac2 91.5%

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

      4. neg-sub0 91.5%

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

      5. associate--r- 91.5%

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

      6. metadata-eval 91.5%

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

      7. +-commutative 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. div-sub 80.2%

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

      3. *-commutative 80.2%

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

      4. mul-1-neg 80.2%

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

      5. sub-neg 80.2%

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

      6. *-inverses 80.2%

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

      7. metadata-eval 80.2%

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

      8. distribute-lft-neg-in 80.2%

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

      9. neg-mul-1 80.2%

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

      10. remove-double-neg 80.2%

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

      11. log1p-define 80.2%

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

      12. mul-1-neg 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -14.8) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -14.8) {
		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 <= -14.8) {
		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 <= -14.8:
		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 <= -14.8)
		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, -14.8], 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 < -14.800000000000001

   1. Initial program 20.1%

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

      1. sub-neg 20.1%

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

      2. log1p-define 20.1%

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

      3. distribute-neg-frac2 20.1%

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

      4. neg-sub0 20.1%

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

      5. associate--r- 20.1%

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

      6. metadata-eval 20.1%

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

      7. +-commutative 20.1%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in y around -inf 96.2%

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

      1. associate--r+ 96.2%

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

      2. sub-neg 96.2%

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

      3. metadata-eval 96.2%

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

      4. distribute-lft-in 96.2%

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

      5. metadata-eval 96.2%

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

      6. +-commutative 96.2%

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

      7. log1p-define 96.2%

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

      8. mul-1-neg 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in x around 0 70.3%

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

   if -14.800000000000001 < y

   1. Initial program 91.5%

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

      1. sub-neg 91.5%

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

      2. log1p-define 91.5%

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

      3. distribute-neg-frac2 91.5%

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

      4. neg-sub0 91.5%

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

      5. associate--r- 91.5%

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

      6. metadata-eval 91.5%

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

      7. +-commutative 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 79.3%

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

      1. log1p-define 79.3%

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

      2. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. log1p-define 68.1%

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

   3. distribute-neg-frac2 68.1%

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

   4. neg-sub0 68.1%

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

   5. associate--r- 68.1%

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

   6. metadata-eval 68.1%

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

   7. +-commutative 68.1%

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

3. Simplified 68.1%

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

5. Taylor expanded in y around 0 57.4%

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

   1. log1p-define 57.4%

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

   2. mul-1-neg 57.4%

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

7. Simplified 57.4%

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

8. Final simplification 57.4%

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

Alternative 13: 41.8% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. log1p-define 68.1%

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

   3. distribute-neg-frac2 68.1%

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

   4. neg-sub0 68.1%

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

   5. associate--r- 68.1%

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

   6. metadata-eval 68.1%

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

   7. +-commutative 68.1%

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

3. Simplified 68.1%

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

5. Taylor expanded in y around 0 55.5%

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

   1. +-commutative 55.5%

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

   2. div-sub 55.5%

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

   3. *-commutative 55.5%

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

   4. mul-1-neg 55.5%

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

   5. sub-neg 55.5%

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

   6. *-inverses 55.5%

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

   7. metadata-eval 55.5%

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

   8. distribute-lft-neg-in 55.5%

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

   9. neg-mul-1 55.5%

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

   10. remove-double-neg 55.5%

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

   11. log1p-define 55.5%

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

   12. mul-1-neg 55.5%

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

7. Simplified 55.5%

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

8. Taylor expanded in x around 0 38.6%

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

9. Final simplification 38.6%

   \[\leadsto \left(1 + x\right) - y \]
10. Add Preprocessing

Alternative 14: 40.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. log1p-define 68.1%

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

   3. distribute-neg-frac2 68.1%

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

   4. neg-sub0 68.1%

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

   5. associate--r- 68.1%

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

   6. metadata-eval 68.1%

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

   7. +-commutative 68.1%

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

3. Simplified 68.1%

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

5. Taylor expanded in y around 0 55.5%

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

   1. +-commutative 55.5%

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

   2. div-sub 55.5%

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

   3. *-commutative 55.5%

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

   4. mul-1-neg 55.5%

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

   5. sub-neg 55.5%

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

   6. *-inverses 55.5%

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

   7. metadata-eval 55.5%

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

   8. distribute-lft-neg-in 55.5%

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

   9. neg-mul-1 55.5%

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

   10. remove-double-neg 55.5%

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

   11. log1p-define 55.5%

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

   12. mul-1-neg 55.5%

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

7. Simplified 55.5%

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

8. Taylor expanded in x around 0 37.9%

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

9. Final simplification 37.9%

   \[\leadsto 1 - y \]
10. Add Preprocessing

Alternative 15: 4.0% accurate, 55.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-y)

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. log1p-define 68.1%

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

   3. distribute-neg-frac2 68.1%

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

   4. neg-sub0 68.1%

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

   5. associate--r- 68.1%

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

   6. metadata-eval 68.1%

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

   7. +-commutative 68.1%

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

3. Simplified 68.1%

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

5. Taylor expanded in y around 0 55.5%

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

   1. +-commutative 55.5%

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

   2. div-sub 55.5%

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

   3. *-commutative 55.5%

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

   4. mul-1-neg 55.5%

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

   5. sub-neg 55.5%

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

   6. *-inverses 55.5%

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

   7. metadata-eval 55.5%

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

   8. distribute-lft-neg-in 55.5%

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

   9. neg-mul-1 55.5%

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

   10. remove-double-neg 55.5%

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

   11. log1p-define 55.5%

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

   12. mul-1-neg 55.5%

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

7. Simplified 55.5%

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

8. Taylor expanded in y around inf 4.2%

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

   1. neg-mul-1 4.2%

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

10. Simplified 4.2%

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

11. Final simplification 4.2%

    \[\leadsto -y \]
12. Add Preprocessing

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024067 
(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))))))