Average Error: 4.3 → 0.1
Time: 4.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[e^{\log \left(\sqrt{1 \cdot \mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
e^{\log \left(\sqrt{1 \cdot \mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}\right)}
double f(double x) {
        double r13324 = 2.0;
        double r13325 = x;
        double r13326 = r13324 * r13325;
        double r13327 = exp(r13326);
        double r13328 = 1.0;
        double r13329 = r13327 - r13328;
        double r13330 = exp(r13325);
        double r13331 = r13330 - r13328;
        double r13332 = r13329 / r13331;
        double r13333 = sqrt(r13332);
        return r13333;
}
↓
double f(double x) {
        double r13334 = 1.0;
        double r13335 = x;
        double r13336 = exp(r13335);
        double r13337 = cbrt(r13336);
        double r13338 = r13337 * r13337;
        double r13339 = fma(r13338, r13337, r13334);
        double r13340 = r13334 * r13339;
        double r13341 = sqrt(r13340);
        double r13342 = log(r13341);
        double r13343 = exp(r13342);
        return r13343;
}

Error

Bits error versus x

Derivation

  1. Initial program 4.3

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
  2. Using strategy rm

  3. Applied flip--3.9

    \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

  4. Applied associate-/r/3.9

    \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]

  5. Simplified2.8

    \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]

  6. Taylor expanded around 0 0.1

    \[\leadsto \sqrt{\color{blue}{1} \cdot \left(e^{x} + 1\right)}\]
  7. Using strategy rm

  8. Applied add-exp-log0.1

    \[\leadsto \color{blue}{e^{\log \left(\sqrt{1 \cdot \left(e^{x} + 1\right)}\right)}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.1

    \[\leadsto e^{\log \left(\sqrt{1 \cdot \left(\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}} + 1\right)}\right)}\]

  11. Applied fma-def0.1

    \[\leadsto e^{\log \left(\sqrt{1 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}}\right)}\]

  12. Final simplification0.1

    \[\leadsto e^{\log \left(\sqrt{1 \cdot \mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}\right)}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))