Average Error: 4.5 → 0.1
Time: 18.4s
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 r21509 = 2.0;
        double r21510 = x;
        double r21511 = r21509 * r21510;
        double r21512 = exp(r21511);
        double r21513 = 1.0;
        double r21514 = r21512 - r21513;
        double r21515 = exp(r21510);
        double r21516 = r21515 - r21513;
        double r21517 = r21514 / r21516;
        double r21518 = sqrt(r21517);
        return r21518;
}
↓
double f(double x) {
        double r21519 = 1.0;
        double r21520 = x;
        double r21521 = exp(r21520);
        double r21522 = sqrt(r21521);
        double r21523 = fma(r21522, r21522, r21519);
        double r21524 = r21519 * r21523;
        double r21525 = sqrt(r21524);
        return r21525;
}

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--4.0

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

  4. Applied associate-/r/4.0

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

  5. Simplified2.9

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

  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 2019208 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))