Hyperbolic arc-cosine

Average Accuracy: 50.2% → 100.0%

Time: 4.1s

Precision: binary64

Cost: 26048

Math

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

↓

\[\log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (log (fma (sqrt (+ x 1.0)) (sqrt (+ x -1.0)) x)))

C

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

↓

double code(double x) {
	return log(fma(sqrt((x + 1.0)), sqrt((x + -1.0)), x));
}

Julia

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

↓

function code(x)
	return log(fma(sqrt(Float64(x + 1.0)), sqrt(Float64(x + -1.0)), x))
end

Wolfram

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

↓

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

Derivation

1. Initial program 55.1%

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

2. Applied egg-rr 100.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right)} \]
   Step-by-step derivation

   [Start] 55.1	\[ \log \left(x + \sqrt{x \cdot x - 1}\right) \]
   +-commutative [=>] 55.1	\[ \log \color{blue}{\left(\sqrt{x \cdot x - 1} + x\right)} \]
   difference-of-sqr-1 [=>] 55.1	\[ \log \left(\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} + x\right) \]
   sqrt-prod [=>] 100.0	\[ \log \left(\color{blue}{\sqrt{x + 1} \cdot \sqrt{x - 1}} + x\right) \]
   fma-def [=>] 100.0	\[ \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x - 1}, x\right)\right)} \]
   sub-neg [=>] 100.0	\[ \log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{\color{blue}{x + \left(-1\right)}}, x\right)\right) \]
   metadata-eval [=>] 100.0	\[ \log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + \color{blue}{-1}}, x\right)\right) \]

3. Final simplification 100.0%

   \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	6848

\[\log \left(x \cdot 2 - \frac{0.5}{x}\right) \]

Alternative 2
Accuracy	99.1%
Cost	6592

\[\log \left(x + x\right) \]

Alternative 3
Accuracy	31.4%
Cost	6464

\[\log x \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))