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

Average Error: 18.5 → 0.1

Time: 12.4s

Precision: binary64

Cost: 13700

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1000000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 10000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1000000.0)
   (+ 1.0 (- (- (/ -1.0 y) (log1p (- x))) (log (/ -1.0 y))))
   (if (<= y 10000000000.0)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (log (/ (+ -1.0 x) y))))))

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 <= -1000000.0) {
		tmp = 1.0 + (((-1.0 / y) - log1p(-x)) - log((-1.0 / y)));
	} else if (y <= 10000000000.0) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((-1.0 + x) / y));
	}
	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 <= -1000000.0) {
		tmp = 1.0 + (((-1.0 / y) - Math.log1p(-x)) - Math.log((-1.0 / y)));
	} else if (y <= 10000000000.0) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((-1.0 + x) / y));
	}
	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 <= -1000000.0:
		tmp = 1.0 + (((-1.0 / y) - math.log1p(-x)) - math.log((-1.0 / y)))
	elif y <= 10000000000.0:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((-1.0 + x) / y))
	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 <= -1000000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 / y) - log1p(Float64(-x))) - log(Float64(-1.0 / y))));
	elseif (y <= 10000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	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, -1000000.0], 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, 10000000000.0], 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[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.5
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 < -1e6

   1. Initial program 52.6

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

   2. Simplified 52.6

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      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))))))): 3 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 3 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))))))): 8 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 0.3

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

   4. Simplified 0.3

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{-1}{y}\right) + \left(\frac{1}{y} + \mathsf{log1p}\left(-x\right)\right)\right)} \]
      Proof
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 1 y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 1 x) 2))) y) (log1p.f64 (neg.f64 x)))): 24 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x))) 2)) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) 2)) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite=> unpow2_binary64 (*.f64 (+.f64 1 (*.f64 -1 x)) (+.f64 1 (*.f64 -1 x))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)) (+.f64 1 (*.f64 -1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1) (+.f64 1 (*.f64 -1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1))) (+.f64 1 (*.f64 -1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1))) (+.f64 1 (*.f64 -1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (+.f64 1 (*.f64 -1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (+.f64 1 (*.f64 -1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite=> sqr-neg_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 x 1) 2))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (Rewrite=> unpow2_binary64 (*.f64 (-.f64 1 x) (-.f64 1 x))) (pow.f64 (-.f64 x 1) 2)) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 24 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (neg.f64 x)) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 -1 x))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 (-.f64 x 1) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 x 1)))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 1))))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (neg.f64 (neg.f64 (+.f64 x (Rewrite=> metadata-eval -1)))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (neg.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 -1) (neg.f64 x)))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (neg.f64 (+.f64 (Rewrite=> metadata-eval 1) (neg.f64 x))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 x))) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 x 1) 2) (-.f64 1 x)))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 1 x) (-.f64 1 x)) (pow.f64 (-.f64 x 1) 2))))) y) (log1p.f64 (neg.f64 x)))): 24 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (neg.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 1 x) 2)) (pow.f64 (-.f64 x 1) 2)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (pow.f64 (-.f64 1 x) 2)) (pow.f64 (-.f64 x 1) 2)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2))) (pow.f64 (-.f64 x 1) 2))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 -1 (pow.f64 (-.f64 1 x) 2)) (Rewrite=> unpow2_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 1 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 -1 x) (*.f64 -1 -1)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 (*.f64 -1 x) (Rewrite=> metadata-eval 1))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 5 points decrease in error
      (+.f64 (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) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (pow.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x))) 2)) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (pow.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) 2)) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (Rewrite=> unpow2_binary64 (*.f64 (+.f64 1 (*.f64 -1 x)) (+.f64 1 (*.f64 -1 x))))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)) (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1) (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1))) (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1))) (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (+.f64 1 (*.f64 -1 x)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) 1))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (-.f64 x 1)) (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 -1))))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (-.f64 x 1)) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x -1))))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (Rewrite=> sqr-neg_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1)))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (*.f64 (/.f64 (-.f64 1 x) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 x 1) 2))) (/.f64 (pow.f64 (-.f64 1 x) 2) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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 x 1) 2) (-.f64 x 1))))) y) (log1p.f64 (neg.f64 x)))): 15 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (pow.f64 (-.f64 1 x) 2)) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1))) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (pow.f64 (-.f64 1 x) 2)) (Rewrite<= unpow3_binary64 (pow.f64 (-.f64 x 1) 3)))) y) (log1p.f64 (neg.f64 x)))): 4 points increase in error, 3 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (pow.f64 (-.f64 1 x) 2)) (Rewrite=> cube-mult_binary64 (*.f64 (-.f64 x 1) (*.f64 (-.f64 x 1) (-.f64 x 1)))))) y) (log1p.f64 (neg.f64 x)))): 3 points increase in error, 4 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (pow.f64 (-.f64 1 x) 2)) (*.f64 (-.f64 x 1) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 x 1) 2))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (-.f64 1 x) (-.f64 x 1)) (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2))))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 10 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (-.f64 x 1) 2))) (-.f64 x 1)))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite=> unpow2_binary64 (*.f64 (-.f64 x 1) (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite<= sqr-neg_binary64 (*.f64 (neg.f64 (-.f64 x 1)) (neg.f64 (-.f64 x 1)))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 1)))) (neg.f64 (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (neg.f64 (+.f64 x (Rewrite=> metadata-eval -1))) (neg.f64 (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 x) (neg.f64 -1))) (neg.f64 (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) (neg.f64 -1)) (neg.f64 (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 (*.f64 -1 x) (Rewrite=> metadata-eval 1)) (neg.f64 (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1 x))) (neg.f64 (-.f64 x 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 1 (*.f64 -1 x)) (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 1))))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 1 (*.f64 -1 x)) (neg.f64 (+.f64 x (Rewrite=> metadata-eval -1)))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 1 (*.f64 -1 x)) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 x) (neg.f64 -1)))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 1 (*.f64 -1 x)) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) (neg.f64 -1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 1 (*.f64 -1 x)) (+.f64 (*.f64 -1 x) (Rewrite=> metadata-eval 1))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (*.f64 (+.f64 1 (*.f64 -1 x)) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1 x)))))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 1 (*.f64 -1 x)) 2)))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (+.f64 1 (Rewrite=> mul-1-neg_binary64 (neg.f64 x))) 2))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (/.f64 (pow.f64 (-.f64 1 x) 2) (pow.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 x)) 2))) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) (Rewrite=> *-inverses_binary64 1)) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 24 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (/.f64 (neg.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (-.f64 1 x)) (-.f64 x 1))) y) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (/.f64 (-.f64 1 x) (-.f64 x 1)) y))) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (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)) (log1p.f64 (neg.f64 x)))): 2 points increase in error, 0 points decrease in error
      (+.f64 (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))) (log1p.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (log1p.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (*.f64 -1 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (log.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 -1 -1)) (*.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (log.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 -1 (+.f64 -1 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (log.f64 (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 x -1)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (log.f64 (*.f64 -1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y)) (log.f64 (*.f64 -1 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (log.f64 (/.f64 -1 y)) (*.f64 -1 (/.f64 (-.f64 (/.f64 1 (-.f64 x 1)) (/.f64 x (-.f64 x 1))) y))) (log.f64 (*.f64 -1 (-.f64 x 1))))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_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))))): 0 points increase in error, 0 points decrease in error

   if -1e6 < y < 1e10

   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))))))): 3 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 3 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))))))): 8 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 1e10 < y

   1. Initial program 30.7

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

   2. Simplified 30.7

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      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))))))): 3 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 3 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))))))): 8 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 1.0

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

   4. Simplified 1.0

      \[\leadsto 1 - \color{blue}{\left(\log \left(-1 + x\right) - \log y\right)} \]
      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 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 10000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7312

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.9 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.054:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.9
Cost	7248

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.9 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -17000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0092:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7240

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{if}\;y \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 46:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{if}\;y \leq -1000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -17000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 7
Error	24.1
Cost	6656

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

Alternative 8
Error	35.3
Cost	448

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

Alternative 9
Error	36.3
Cost	192

\[1 + x \]

Alternative 10
Error	36.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(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))))))