Average Error: 4.4 → 0.1
Time: 18.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}
double f(double x) {
        double r19762 = 2.0;
        double r19763 = x;
        double r19764 = r19762 * r19763;
        double r19765 = exp(r19764);
        double r19766 = 1.0;
        double r19767 = r19765 - r19766;
        double r19768 = exp(r19763);
        double r19769 = r19768 - r19766;
        double r19770 = r19767 / r19769;
        double r19771 = sqrt(r19770);
        return r19771;
}
↓
double f(double x) {
        double r19772 = 1.0;
        double r19773 = sqrt(r19772);
        double r19774 = x;
        double r19775 = exp(r19774);
        double r19776 = sqrt(r19775);
        double r19777 = fma(r19776, r19776, r19772);
        double r19778 = sqrt(r19777);
        double r19779 = r19773 * r19778;
        return r19779;
}

Error

Bits error versus x

Derivation

  1. Initial program 4.4

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
  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. Applied sqrt-prod3.9

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

  6. Simplified2.8

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

  7. Taylor expanded around 0 0.1

    \[\leadsto \sqrt{\color{blue}{1}} \cdot \sqrt{e^{x} + 1}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.1

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

  10. Applied fma-def0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))