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

Average Error: 18.2 → 0.1

Time: 10.4s

Precision: binary64

Cost: 14084

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} + \frac{-0.5}{y \cdot y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -100000.0)
   (+
    1.0
    (- (+ (/ -1.0 y) (/ -0.5 (* y y))) (+ (log1p (- x)) (log (/ -1.0 y)))))
   (if (<= y 1e+20)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ 1.0 (log (/ y (+ x -1.0)))))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (y <= -100000.0) {
		tmp = 1.0 + (((-1.0 / y) + (-0.5 / (y * y))) - (log1p(-x) + log((-1.0 / y))));
	} else if (y <= 1e+20) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + log((y / (x + -1.0)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y) {
	double tmp;
	if (y <= -100000.0) {
		tmp = 1.0 + (((-1.0 / y) + (-0.5 / (y * y))) - (Math.log1p(-x) + Math.log((-1.0 / y))));
	} else if (y <= 1e+20) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + Math.log((y / (x + -1.0)));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if y <= -100000.0:
		tmp = 1.0 + (((-1.0 / y) + (-0.5 / (y * y))) - (math.log1p(-x) + math.log((-1.0 / y))))
	elif y <= 1e+20:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + math.log((y / (x + -1.0)))
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (y <= -100000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 / y) + Float64(-0.5 / Float64(y * y))) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y)))));
	elseif (y <= 1e+20)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + log(Float64(y / Float64(x + -1.0))));
	end
	return tmp
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]

↓

code[x_, y_] := If[LessEqual[y, -100000.0], N[(1.0 + N[(N[(N[(-1.0 / y), $MachinePrecision] + N[(-0.5 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+20], 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[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.2
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -1e5

   1. Initial program 52.1

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

   2. Simplified 52.1

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      Proof
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 y -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> +-commutative_binary64 (+.f64 -1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 y) -1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (-.f64 x y) (-.f64 1 y)) -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))) -1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 -1 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 2 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 2 points decrease in error
      (-.f64 1 (log1p.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (-.f64 x y) (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 50.8

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

   4. Taylor expanded in y around -inf 10.8

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

   5. Simplified 0.3

      \[\leadsto 1 - \color{blue}{\left(\left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right) + \left(\frac{1}{y} + \frac{0.5}{y \cdot y}\right)\right)} \]
      Proof
      (+.f64 (+.f64 (log1p.f64 (neg.f64 x)) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log1p.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (*.f64 -1 x)))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 -1 -1)) (*.f64 -1 x))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 -1 (+.f64 -1 x)))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 x -1)))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 1 y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (Rewrite<= metadata-eval (*.f64 -1 1))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 -1 (Rewrite<= *-inverses_binary64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 1 x) 2))))) y) (/.f64 1/2 (*.f64 y y)))): 23 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (pow.f64 (-.f64 1 x) 2)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (pow.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x))) 2))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (pow.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) 2))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (Rewrite=> unpow2_binary64 (*.f64 (+.f64 1 (*.f64 -1 x)) (+.f64 1 (*.f64 -1 x)))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)) (+.f64 1 (*.f64 -1 x))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1) (+.f64 1 (*.f64 -1 x))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1))) (+.f64 1 (*.f64 -1 x))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1))) (+.f64 1 (*.f64 -1 x))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (+.f64 1 (*.f64 -1 x))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (+.f64 1 (*.f64 -1 x))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1)))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1)))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (Rewrite=> sqr-neg_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 (-.f64 x 1)) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 -1)) (-.f64 x 1)) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 -1 (-.f64 x 1)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (*.f64 -1 (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 1))))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (*.f64 -1 (+.f64 x (Rewrite=> metadata-eval -1)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (*.f64 -1 (Rewrite=> +-commutative_binary64 (+.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 -1 -1) (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 (Rewrite=> metadata-eval 1) (*.f64 -1 x))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 (-.f64 1 x) (-.f64 1 x))) (+.f64 1 (*.f64 -1 x))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (/.f64 (-.f64 1 x) (-.f64 1 x)) (+.f64 1 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (/.f64 (-.f64 1 x) (-.f64 1 x)) (Rewrite<= sub-neg_binary64 (-.f64 1 x))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 1 x) (*.f64 (-.f64 1 x) (-.f64 1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 1 x) 2))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 1 x) (pow.f64 (-.f64 1 x) 2)) (*.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1))))) y) (/.f64 1/2 (*.f64 y y)))): 5 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 1 x))) (*.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 1 x) 2)) (/.f64 (-.f64 1 x) (-.f64 x 1))))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 5 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (*.f64 (Rewrite=> *-inverses_binary64 1) (/.f64 (-.f64 1 x) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 23 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1 (-.f64 1 x)) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (neg.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 (-.f64 1 x)) (-.f64 x 1))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (-.f64 1 x)) (-.f64 x 1))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (neg.f64 x))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 -1 x)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (Rewrite=> remove-double-neg_binary64 (-.f64 x 1)) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 x 1)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 1))))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (neg.f64 (+.f64 x (Rewrite=> metadata-eval -1)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 -1 x)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 -1) (neg.f64 x)))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (+.f64 (Rewrite=> metadata-eval 1) (neg.f64 x))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 x))) (-.f64 x 1)) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (/.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (-.f64 1 x) (-.f64 x 1)))) y) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (/.f64 (-.f64 1 x) (-.f64 x 1)) y))) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (neg.f64 (/.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1)))) y)) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y))) (/.f64 1/2 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (Rewrite=> associate-/l*_binary64 (/.f64 1/2 (/.f64 (pow.f64 y 2) 1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (Rewrite<= metadata-eval (+.f64 2 -1)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (Rewrite<= metadata-eval (*.f64 -1 1))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (*.f64 -1 (Rewrite<= *-inverses_binary64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 1 x) 2))))))))): 23 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (pow.f64 (-.f64 1 x) 2)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 -1 (Rewrite=> unpow2_binary64 (*.f64 (-.f64 1 x) (-.f64 1 x)))) (pow.f64 (-.f64 1 x) 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x))) (pow.f64 (-.f64 1 x) 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (pow.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x))) 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (pow.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (Rewrite=> unpow2_binary64 (*.f64 (+.f64 1 (*.f64 -1 x)) (+.f64 1 (*.f64 -1 x)))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)) (+.f64 1 (*.f64 -1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1) (+.f64 1 (*.f64 -1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1))) (+.f64 1 (*.f64 -1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1))) (+.f64 1 (*.f64 -1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (+.f64 1 (*.f64 -1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (+.f64 1 (*.f64 -1 x))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1)))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1)))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (/.f64 (*.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 1 x)) (Rewrite=> sqr-neg_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (*.f64 -1 (-.f64 1 x)) (-.f64 x 1)) (/.f64 (-.f64 1 x) (-.f64 x 1))))))))): 0 points increase in error, 23 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 1 x) (-.f64 x 1)))) (/.f64 (-.f64 1 x) (-.f64 x 1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (*.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (-.f64 1 x) (-.f64 x 1)))) (/.f64 (-.f64 1 x) (-.f64 x 1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 2 (*.f64 (/.f64 (-.f64 1 x) (-.f64 x 1)) (/.f64 (-.f64 1 x) (-.f64 x 1))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (-.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 1 x) (-.f64 1 x)) (*.f64 (-.f64 x 1) (-.f64 x 1))))))))): 23 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (-.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 1 x) 2)) (*.f64 (-.f64 x 1) (-.f64 x 1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (-.f64 2 (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 x 1) 2)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (Rewrite<= unsub-neg_binary64 (+.f64 2 (neg.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (/.f64 1/2 (/.f64 (pow.f64 y 2) (+.f64 2 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 (+.f64 2 (*.f64 -1 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2))))) (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (+.f64 2 (*.f64 -1 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2)))) (pow.f64 y 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (log.f64 (/.f64 -1 y))) (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y))) (*.f64 1/2 (/.f64 (+.f64 2 (*.f64 -1 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2)))) (pow.f64 y 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (+.f64 (log.f64 (/.f64 -1 y)) (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y))))) (*.f64 1/2 (/.f64 (+.f64 2 (*.f64 -1 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2)))) (pow.f64 y 2)))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (+.f64 2 (*.f64 -1 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2)))) (pow.f64 y 2))) (+.f64 (log.f64 (*.f64 -1 (-.f64 x 1))) (+.f64 (log.f64 (/.f64 -1 y)) (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)))))): 0 points increase in error, 0 points decrease in error

   if -1e5 < y < 1e20

   1. Initial program 0.1

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      Proof
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 y -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> +-commutative_binary64 (+.f64 -1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 y) -1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (-.f64 x y) (-.f64 1 y)) -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))) -1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 -1 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 2 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 2 points decrease in error
      (-.f64 1 (log1p.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (-.f64 x y) (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error

   if 1e20 < y

   1. Initial program 31.6

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

   2. Simplified 31.6

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      Proof
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 y -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> +-commutative_binary64 (+.f64 -1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 y) -1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (-.f64 x y) (-.f64 1 y)) -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))) -1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 -1 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 2 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 2 points decrease in error
      (-.f64 1 (log1p.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (-.f64 x y) (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 0.9

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

   4. Simplified 0.9

      \[\leadsto 1 - \color{blue}{\left(\log \left(-1 + x\right) - \log y\right)} \]
      Proof
      (-.f64 (log.f64 (+.f64 -1 x)) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1))) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 (-.f64 x 1)) (neg.f64 (log.f64 y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (-.f64 x 1)) (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (log.f64 (/.f64 1 y)) (log.f64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.0

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

   6. Applied egg-rr 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} + \frac{-0.5}{y \cdot y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	13700

\[\begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13700

\[\begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+18}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 10^{+20}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]

Alternative 7
Error	6.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 8
Error	13.3
Cost	6984

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	6.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 10
Error	23.6
Cost	6656

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

Alternative 11
Error	35.2
Cost	448

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

Alternative 12
Error	36.1
Cost	192

\[1 + x \]

Alternative 13
Error	36.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(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))))))