2isqrt (example 3.6)

Average Error: 20.1 → 0.1

Time: 11.4s

Precision: binary64

Cost: 20676

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 279.12834226240955:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{1 + x}} \cdot {\left(x \cdot \left(1 + x\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{x}^{-0.5}}{x}, \frac{1}{{x}^{2.5}} \cdot \left(-0.375 + \left(\frac{-0.2734375}{x \cdot x} + \frac{0.3125}{x}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x 279.12834226240955)
   (*
    (/ (+ x (- 1.0 x)) (+ (sqrt x) (sqrt (+ 1.0 x))))
    (pow (* x (+ 1.0 x)) -0.5))
   (fma
    0.5
    (/ (pow x -0.5) x)
    (*
     (/ 1.0 (pow x 2.5))
     (+ -0.375 (+ (/ -0.2734375 (* x x)) (/ 0.3125 x)))))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

↓

double code(double x) {
	double tmp;
	if (x <= 279.12834226240955) {
		tmp = ((x + (1.0 - x)) / (sqrt(x) + sqrt((1.0 + x)))) * pow((x * (1.0 + x)), -0.5);
	} else {
		tmp = fma(0.5, (pow(x, -0.5) / x), ((1.0 / pow(x, 2.5)) * (-0.375 + ((-0.2734375 / (x * x)) + (0.3125 / x)))));
	}
	return tmp;
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

↓

function code(x)
	tmp = 0.0
	if (x <= 279.12834226240955)
		tmp = Float64(Float64(Float64(x + Float64(1.0 - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) * (Float64(x * Float64(1.0 + x)) ^ -0.5));
	else
		tmp = fma(0.5, Float64((x ^ -0.5) / x), Float64(Float64(1.0 / (x ^ 2.5)) * Float64(-0.375 + Float64(Float64(-0.2734375 / Float64(x * x)) + Float64(0.3125 / x)))));
	end
	return tmp
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, 279.12834226240955], N[(N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 2.5], $MachinePrecision]), $MachinePrecision] * N[(-0.375 + N[(N[(-0.2734375 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.3125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	20.1
Target	0.6
Herbie	0.1

\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

1. Split input into 2 regimes

2. if x < 279.12834226240955

   1. Initial program 0.3

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 0.1

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

   if 279.12834226240955 < x

   1. Initial program 39.9

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

   2. Applied egg-rr 53.0

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{{\left(\sqrt[3]{1 + x}\right)}^{-1.5}} \]

   3. Applied egg-rr 39.9

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

   4. Taylor expanded in x around -inf 64.0

      \[\leadsto \color{blue}{-0.2734375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{4}} + \left(0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}} + \left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right)\right)} \]

   5. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{-0.5}}{x}, \frac{1}{{x}^{2.5}} \cdot \left(-0.375 + \left(\frac{-0.2734375}{x \cdot x} + \frac{0.3125}{x}\right)\right)\right)} \]
      Proof
      (fma.f64 1/2 (/.f64 (pow.f64 x -1/2) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 x) -1/2))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 70 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (log.f64 x)))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 -1/2) (log.f64 x))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 25 points increase in error, 30 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= +-lft-identity_binary64 (+.f64 0 (log.f64 x)))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (log.f64 -1) (log.f64 -1))) (log.f64 x))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 146 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= associate--r-_binary64 (-.f64 (log.f64 -1) (-.f64 (log.f64 -1) (log.f64 x))))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (-.f64 (log.f64 -1) (Rewrite<= log-div_binary64 (log.f64 (/.f64 -1 x))))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 -1) (neg.f64 (log.f64 (/.f64 -1 x)))))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (+.f64 (log.f64 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 -1 x)))))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) x) (*.f64 (/.f64 1 (pow.f64 x 5/2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (pow.f64 x (Rewrite<= metadata-eval (/.f64 5 2)))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (pow.f64 x (Rewrite=> metadata-eval 5/2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (pow.f64 x (Rewrite<= metadata-eval (+.f64 3/2 1)))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (pow.f64 x (+.f64 (Rewrite<= metadata-eval (/.f64 3 2)) 1))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 x (/.f64 3 2)) x))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (*.f64 (pow.f64 x (Rewrite=> metadata-eval 3/2)) x)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (*.f64 (pow.f64 x (Rewrite<= metadata-eval (+.f64 1/2 1))) x)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (*.f64 (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 x 1/2) x)) x)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (*.f64 (*.f64 (pow.f64 x (Rewrite<= metadata-eval (/.f64 1 2))) x) x)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (*.f64 x x)))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 1 (*.f64 (pow.f64 x (/.f64 1 2)) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 1 (pow.f64 x 2)) (pow.f64 x (/.f64 1 2)))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 1 (Rewrite=> unpow2_binary64 (*.f64 x x))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 x) x)) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (/.f64 1 (Rewrite<= unpow1_binary64 (pow.f64 x 1))) x) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (/.f64 1 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) x) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 (pow.f64 x (/.f64 1 2))) (pow.f64 x (/.f64 1 2)))) x) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 1 (pow.f64 x (/.f64 1 2))) (*.f64 x (pow.f64 x (/.f64 1 2))))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (/.f64 1 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 (pow.f64 x (/.f64 1 2)))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (log.f64 (pow.f64 x (/.f64 1 2)))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (neg.f64 (log.f64 (pow.f64 x (Rewrite=> metadata-eval 1/2))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (neg.f64 (Rewrite=> log-pow_binary64 (*.f64 1/2 (log.f64 x))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 1/2) (log.f64 x)))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (*.f64 (Rewrite=> metadata-eval -1/2) (log.f64 x))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 -1/2) (log.f64 x))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= +-lft-identity_binary64 (+.f64 0 (log.f64 x)))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (log.f64 -1) (log.f64 -1))) (log.f64 x))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= associate--r-_binary64 (-.f64 (log.f64 -1) (-.f64 (log.f64 -1) (log.f64 x))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (-.f64 (log.f64 -1) (Rewrite<= log-div_binary64 (log.f64 (/.f64 -1 x))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 -1) (neg.f64 (log.f64 (/.f64 -1 x)))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (+.f64 (log.f64 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 -1 x)))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 x (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) x))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (pow.f64 x (/.f64 1 2)))) (pow.f64 x (/.f64 1 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (Rewrite=> unpow1_binary64 x)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (*.f64 x x))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (*.f64 x x)) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)) (+.f64 -3/8 (+.f64 (/.f64 -35/128 (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (/.f64 5/16 x))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)) (+.f64 -3/8 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 5/16 x) (/.f64 -35/128 (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)) -3/8) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)) (+.f64 (/.f64 5/16 x) (/.f64 -35/128 (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)) (+.f64 (/.f64 5/16 x) (/.f64 -35/128 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 5/16 x) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (*.f64 (/.f64 -35/128 (pow.f64 x 2)) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 5/16 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 x (pow.f64 x 2)))) (*.f64 (/.f64 -35/128 (pow.f64 x 2)) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (/.f64 (*.f64 5/16 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 x (Rewrite=> unpow2_binary64 (*.f64 x x)))) (*.f64 (/.f64 -35/128 (pow.f64 x 2)) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (/.f64 (*.f64 5/16 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (Rewrite<= cube-mult_binary64 (pow.f64 x 3))) (*.f64 (/.f64 -35/128 (pow.f64 x 2)) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (*.f64 (/.f64 -35/128 (pow.f64 x 2)) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -35/128 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 (pow.f64 x 2) (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (/.f64 (*.f64 -35/128 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (Rewrite=> pow-sqr_binary64 (pow.f64 x (*.f64 2 2))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (/.f64 (*.f64 -35/128 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (pow.f64 x (Rewrite=> metadata-eval 4)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (Rewrite<= associate-*r/_binary64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))) (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (+.f64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))) (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -35/128 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 4))) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 279.12834226240955:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{1 + x}} \cdot {\left(x \cdot \left(1 + x\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{x}^{-0.5}}{x}, \frac{1}{{x}^{2.5}} \cdot \left(-0.375 + \left(\frac{-0.2734375}{x \cdot x} + \frac{0.3125}{x}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	38912

\[{\left(\left(\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-1} \]

Alternative 2
Error	0.2
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	20484

\[\begin{array}{l} \mathbf{if}\;x \leq 123040.3876905906:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{1 + x}} \cdot {\left(x \cdot \left(1 + x\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7556

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 - x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.8
Cost	7556

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.375 - x \cdot -0.3125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \end{array} \]

Alternative 6
Error	1.2
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 - x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\frac{x}{0.5}}\\ \end{array} \]

Alternative 7
Error	1.0
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 - x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \end{array} \]

Alternative 8
Error	1.2
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\frac{x}{0.5}}\\ \end{array} \]

Alternative 9
Error	1.3
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\frac{x}{0.5}}\\ \end{array} \]

Alternative 10
Error	30.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7555410670793095 \cdot 10^{-6}:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 11
Error	31.8
Cost	6528

\[{x}^{-0.5} \]

Alternative 12
Error	59.3
Cost	192

\[\frac{1}{x} \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))