Average Error: 4.7 → 0.1
Time: 5.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[e^{\log \left(\sqrt{1 \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
e^{\log \left(\sqrt{1 \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\right)}
double f(double x) {
        double r12712 = 2.0;
        double r12713 = x;
        double r12714 = r12712 * r12713;
        double r12715 = exp(r12714);
        double r12716 = 1.0;
        double r12717 = r12715 - r12716;
        double r12718 = exp(r12713);
        double r12719 = r12718 - r12716;
        double r12720 = r12717 / r12719;
        double r12721 = sqrt(r12720);
        return r12721;
}
↓
double f(double x) {
        double r12722 = 1.0;
        double r12723 = x;
        double r12724 = exp(r12723);
        double r12725 = sqrt(r12724);
        double r12726 = fma(r12725, r12725, r12722);
        double r12727 = r12722 * r12726;
        double r12728 = sqrt(r12727);
        double r12729 = log(r12728);
        double r12730 = exp(r12729);
        return r12730;
}

Error

Bits error versus x

Derivation

  1. Initial program 4.7

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

  3. Applied flip--4.2

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

  4. Applied associate-/r/4.2

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

  5. Simplified3.0

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

  6. Taylor expanded around 0 0.0

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied fma-def0.0

    \[\leadsto \sqrt{1 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}}\]
  10. Using strategy rm

  11. Applied add-exp-log0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))