Average Error: 4.5 → 0.1
Time: 5.0s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{1 \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1 \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}
double f(double x) {
        double r11168 = 2.0;
        double r11169 = x;
        double r11170 = r11168 * r11169;
        double r11171 = exp(r11170);
        double r11172 = 1.0;
        double r11173 = r11171 - r11172;
        double r11174 = exp(r11169);
        double r11175 = r11174 - r11172;
        double r11176 = r11173 / r11175;
        double r11177 = sqrt(r11176);
        return r11177;
}
↓
double f(double x) {
        double r11178 = 1.0;
        double r11179 = x;
        double r11180 = exp(r11179);
        double r11181 = sqrt(r11180);
        double r11182 = fma(r11181, r11181, r11178);
        double r11183 = r11178 * r11182;
        double r11184 = sqrt(r11183);
        return r11184;
}

Error

Bits error versus x

Derivation

  1. Initial program 4.5

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

  3. Applied flip--4.1

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

  4. Applied associate-/r/4.1

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

  5. Simplified2.9

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

  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-sqr-sqrt0.1

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

  9. Applied fma-def0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))