Average Error: 4.4 → 0.7
Time: 23.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.0314812521352058 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.0314812521352058 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\

\end{array}
double f(double x) {
        double r28129 = 2.0;
        double r28130 = x;
        double r28131 = r28129 * r28130;
        double r28132 = exp(r28131);
        double r28133 = 1.0;
        double r28134 = r28132 - r28133;
        double r28135 = exp(r28130);
        double r28136 = r28135 - r28133;
        double r28137 = r28134 / r28136;
        double r28138 = sqrt(r28137);
        return r28138;
}
↓
double f(double x) {
        double r28139 = x;
        double r28140 = -3.031481252135206e-11;
        bool r28141 = r28139 <= r28140;
        double r28142 = exp(r28139);
        double r28143 = 1.0;
        double r28144 = r28142 + r28143;
        double r28145 = 2.0;
        double r28146 = r28145 * r28139;
        double r28147 = exp(r28146);
        double r28148 = r28147 - r28143;
        double r28149 = r28144 * r28148;
        double r28150 = 2.0;
        double r28151 = r28150 * r28139;
        double r28152 = exp(r28151);
        double r28153 = r28152 - r28143;
        double r28154 = r28149 / r28153;
        double r28155 = sqrt(r28154);
        double r28156 = 0.5;
        double r28157 = pow(r28139, r28150);
        double r28158 = fma(r28143, r28139, r28145);
        double r28159 = fma(r28156, r28157, r28158);
        double r28160 = sqrt(r28159);
        double r28161 = r28141 ? r28155 : r28160;
        return r28161;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3.031481252135206e-11

    1. Initial program 0.5

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

    3. Applied flip--0.3

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

    4. Simplified0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied *-un-lft-identity0.0

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

    8. Applied times-frac0.0

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

    9. Applied *-un-lft-identity0.0

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

    10. Applied times-frac0.0

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

    11. Simplified0.0

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

    12. Taylor expanded around inf 0.0

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

    if -3.031481252135206e-11 < x

    1. Initial program 37.1

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

    2. Taylor expanded around 0 6.6

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified6.6

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.0314812521352058 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))