Rust f64::acosh

Average Error: 49.8% → 99.4%

Time: 9.3s

Precision: binary64

Cost: 14144.00

\[x \geq 1\]

Math

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

↓

\[\begin{array}{l} t_0 := x + \frac{0.5}{x}\\ \log \left(\mathsf{fma}\left(\frac{x}{t_0}, x, x\right) + \frac{\frac{\frac{-0.25}{x}}{x}}{t_0}\right) \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (/ 0.5 x))))
   (log (+ (fma (/ x t_0) x x) (/ (/ (/ -0.25 x) x) t_0)))))

C

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

↓

double code(double x) {
	double t_0 = x + (0.5 / x);
	return log((fma((x / t_0), x, x) + (((-0.25 / x) / x) / t_0)));
}

Julia

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

↓

function code(x)
	t_0 = Float64(x + Float64(0.5 / x))
	return log(Float64(fma(Float64(x / t_0), x, x) + Float64(Float64(Float64(-0.25 / x) / x) / t_0)))
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, N[Log[N[(N[(N[(x / t$95$0), $MachinePrecision] * x + x), $MachinePrecision] + N[(N[(N[(-0.25 / x), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	49.8%
Target	99.9%
Herbie	99.4%

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

Derivation

1. Initial program 49.8

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

2. Taylor expanded in x around inf 99.4

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

3. Simplified 99.4

   \[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right) \]
   Proof

   [Start] 99.4	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   associate-*r/ [=>] 99.4	\[ \log \left(x + \left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
   metadata-eval [=>] 99.4	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Applied egg-rr 99.4

   \[\leadsto \log \color{blue}{\left(\frac{x}{\frac{x + \frac{0.5}{x}}{x}} - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right)} \]

5. Simplified 99.4

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{x}{x + \frac{0.5}{x}}, x, x\right) - \frac{\frac{\frac{0.25}{x}}{x}}{x + \frac{0.5}{x}}\right)} \]
   Proof

   [Start] 99.4	\[ \log \left(\frac{x}{\frac{x + \frac{0.5}{x}}{x}} - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right) \]
   associate--r- [=>] 99.4	\[ \log \color{blue}{\left(\left(\frac{x}{\frac{x + \frac{0.5}{x}}{x}} - \frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)}\right) + x\right)} \]
   +-commutative [<=] 99.4	\[ \log \color{blue}{\left(x + \left(\frac{x}{\frac{x + \frac{0.5}{x}}{x}} - \frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)}\right)\right)} \]
   associate--l+ [<=] 99.4	\[ \log \color{blue}{\left(\left(x + \frac{x}{\frac{x + \frac{0.5}{x}}{x}}\right) - \frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)}\right)} \]
   +-commutative [=>] 99.4	\[ \log \left(\color{blue}{\left(\frac{x}{\frac{x + \frac{0.5}{x}}{x}} + x\right)} - \frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)}\right) \]
   associate-/r/ [=>] 99.4	\[ \log \left(\left(\color{blue}{\frac{x}{x + \frac{0.5}{x}} \cdot x} + x\right) - \frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)}\right) \]
   fma-def [=>] 99.4	\[ \log \left(\color{blue}{\mathsf{fma}\left(\frac{x}{x + \frac{0.5}{x}}, x, x\right)} - \frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)}\right) \]
   *-commutative [=>] 99.4	\[ \log \left(\mathsf{fma}\left(\frac{x}{x + \frac{0.5}{x}}, x, x\right) - \frac{0.25}{\color{blue}{\left(x \cdot x\right) \cdot \left(x + \frac{0.5}{x}\right)}}\right) \]
   associate-/r* [=>] 99.4	\[ \log \left(\mathsf{fma}\left(\frac{x}{x + \frac{0.5}{x}}, x, x\right) - \color{blue}{\frac{\frac{0.25}{x \cdot x}}{x + \frac{0.5}{x}}}\right) \]
   associate-/r* [=>] 99.4	\[ \log \left(\mathsf{fma}\left(\frac{x}{x + \frac{0.5}{x}}, x, x\right) - \frac{\color{blue}{\frac{\frac{0.25}{x}}{x}}}{x + \frac{0.5}{x}}\right) \]

6. Final simplification 99.4

   \[\leadsto \log \left(\mathsf{fma}\left(\frac{x}{x + \frac{0.5}{x}}, x, x\right) + \frac{\frac{\frac{-0.25}{x}}{x}}{x + \frac{0.5}{x}}\right) \]

Alternatives

Alternative 1
Error	99.4%
Cost	6848.00

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

Alternative 2
Error	98.9%
Cost	6592.00

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

Alternative 3
Error	31.4%
Cost	6464.00

\[\log x \]

Alternative 4
Error	1.6%
Cost	64.00

\[-0.03125 \]

Alternative 5
Error	3.1%
Cost	64.00

\[0 \]

Reproduce

herbie shell --seed 2023093 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

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

  (log (+ x (sqrt (- (* x x) 1.0)))))