Average Error: 4.4 → 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 r11163 = 2.0;
        double r11164 = x;
        double r11165 = r11163 * r11164;
        double r11166 = exp(r11165);
        double r11167 = 1.0;
        double r11168 = r11166 - r11167;
        double r11169 = exp(r11164);
        double r11170 = r11169 - r11167;
        double r11171 = r11168 / r11170;
        double r11172 = sqrt(r11171);
        return r11172;
}
↓
double f(double x) {
        double r11173 = 1.0;
        double r11174 = x;
        double r11175 = exp(r11174);
        double r11176 = sqrt(r11175);
        double r11177 = fma(r11176, r11176, r11173);
        double r11178 = r11173 * r11177;
        double r11179 = sqrt(r11178);
        return r11179;
}

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. Simplified2.7

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

  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 2020049 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))