Average Error: 0.6 → 0.6
Time: 11.2s
Precision: binary64
Cost: 33472
\[\left(10 \leq v \land v \leq 1000000000\right) \land \left(10 \leq x \land x \leq 1000\right)\]
\[e^{v \cdot \log \left(\frac{x}{v + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}}\right) + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{v \cdot v - x \cdot x}\\ e^{3 \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(v, \log \left(\frac{x}{v + t_0}\right), t_0\right)\right)} \end{array} \]
(FPCore (v x)
 :precision binary64
 (exp
  (+
   (* v (log (/ x (+ v (sqrt (* (+ v x) (- v x)))))))
   (sqrt (* (+ v x) (- v x))))))
↓
(FPCore (v x)
 :precision binary64
 (let* ((t_0 (sqrt (- (* v v) (* x x)))))
   (exp (* 3.0 (* 0.3333333333333333 (fma v (log (/ x (+ v t_0))) t_0))))))
double code(double v, double x) {
	return exp(((v * log((x / (v + sqrt(((v + x) * (v - x))))))) + sqrt(((v + x) * (v - x)))));
}
↓
double code(double v, double x) {
	double t_0 = sqrt(((v * v) - (x * x)));
	return exp((3.0 * (0.3333333333333333 * fma(v, log((x / (v + t_0))), t_0))));
}

function code(v, x)
	return exp(Float64(Float64(v * log(Float64(x / Float64(v + sqrt(Float64(Float64(v + x) * Float64(v - x))))))) + sqrt(Float64(Float64(v + x) * Float64(v - x)))))
end
↓
function code(v, x)
	t_0 = sqrt(Float64(Float64(v * v) - Float64(x * x)))
	return exp(Float64(3.0 * Float64(0.3333333333333333 * fma(v, log(Float64(x / Float64(v + t_0))), t_0))))
end

code[v_, x_] := N[Exp[N[(N[(v * N[Log[N[(x / N[(v + N[Sqrt[N[(N[(v + x), $MachinePrecision] * N[(v - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(v + x), $MachinePrecision] * N[(v - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[v_, x_] := Block[{t$95$0 = N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Exp[N[(3.0 * N[(0.3333333333333333 * N[(v * N[Log[N[(x / N[(v + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

e^{v \cdot \log \left(\frac{x}{v + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}}\right) + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}}
↓
\begin{array}{l}
t_0 := \sqrt{v \cdot v - x \cdot x}\\
e^{3 \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(v, \log \left(\frac{x}{v + t_0}\right), t_0\right)\right)}
\end{array}

Error

Derivation

  1. Initial program 0.6

    \[e^{v \cdot \log \left(\frac{x}{v + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}}\right) + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}} \]

  2. Applied egg-rr0.6

    \[\leadsto e^{\color{blue}{3 \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(v, \log \left(\frac{x}{v + \sqrt{v \cdot v - x \cdot x}}\right), \sqrt{v \cdot v - x \cdot x}\right)\right)}} \]

  3. Final simplification0.6

    \[\leadsto e^{3 \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(v, \log \left(\frac{x}{v + \sqrt{v \cdot v - x \cdot x}}\right), \sqrt{v \cdot v - x \cdot x}\right)\right)} \]

Alternatives

Alternative 1
Error0.6
Cost33216
\[\begin{array}{l} t_0 := \sqrt{v \cdot v - x \cdot x}\\ e^{\mathsf{fma}\left(v, \log \left(\frac{x}{v + t_0}\right), t_0\right)} \end{array} \]
Alternative 2
Error0.7
Cost27072
\[e^{v \cdot \log \left(x \cdot \frac{1}{v + \sqrt{v \cdot v - x \cdot x}}\right) + \sqrt{\left(v + x\right) \cdot \left(v - x\right)}} \]
Alternative 3
Error1.8
Cost27012
\[\begin{array}{l} t_0 := \sqrt{v \cdot v - x \cdot x}\\ \mathbf{if}\;v \leq 500:\\ \;\;\;\;{\left(\frac{x}{v + t_0}\right)}^{v} \cdot e^{t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{v + \mathsf{fma}\left(v, \log \left(\frac{0.5}{v}\right) + \log x, -0.25 \cdot \frac{x}{\frac{v}{x}}\right)}\\ \end{array} \]
Alternative 4
Error0.6
Cost26944
\[\begin{array}{l} t_0 := \sqrt{\left(v + x\right) \cdot \left(v - x\right)}\\ e^{t_0 + v \cdot \log \left(\frac{x}{v + t_0}\right)} \end{array} \]
Alternative 5
Error5.2
Cost26560
\[e^{v + \mathsf{fma}\left(v, \log \left(\frac{0.5}{v}\right) + \log x, -0.25 \cdot \frac{x}{\frac{v}{x}}\right)} \]
Alternative 6
Error5.8
Cost20160
\[e^{\mathsf{fma}\left(v, \log \left(0.5 \cdot \frac{x}{v}\right), v\right) + -0.5 \cdot \left(x \cdot \frac{x}{v}\right)} \]
Alternative 7
Error5.8
Cost20160
\[e^{v \cdot \log \left(0.5 \cdot \frac{x}{v}\right) + \mathsf{fma}\left(-0.5, x \cdot \frac{x}{v}, v\right)} \]
Alternative 8
Error5.8
Cost13376
\[e^{v + v \cdot \log \left(0.5 \cdot \frac{x}{v}\right)} \]
Alternative 9
Error61.9
Cost13312
\[e^{v} \cdot {\left(0.5 \cdot \frac{x}{v}\right)}^{v} \]
Alternative 10
Error64.0
Cost12992
\[e^{x \cdot \sqrt{-1}} \]

Error

Reproduce

herbie shell --seed 1 
(FPCore (v x)
  :name "exp(v * log(x / (v + sqrt((v + x) * (v - x)))) + sqrt((v + x) * (v - x)))"
  :precision binary64
  :pre (and (and (<= 10.0 v) (<= v 1000000000.0)) (and (<= 10.0 x) (<= x 1000.0)))
  (exp (+ (* v (log (/ x (+ v (sqrt (* (+ v x) (- v x))))))) (sqrt (* (+ v x) (- v x))))))