Average Error: 61.8 → 0.3
Time: 14.1s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot t\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)\]
\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot t\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)
double f(double t) {
        double r45546 = 1.0;
        double r45547 = t;
        double r45548 = 2e-16;
        double r45549 = r45547 * r45548;
        double r45550 = r45546 + r45549;
        double r45551 = r45550 * r45550;
        double r45552 = -1.0;
        double r45553 = 2.0;
        double r45554 = r45553 * r45549;
        double r45555 = r45552 - r45554;
        double r45556 = r45551 + r45555;
        return r45556;
}
↓
double f(double t) {
        double r45557 = 3.9999999999999997e-32;
        double r45558 = sqrt(r45557);
        double r45559 = t;
        double r45560 = r45558 * r45559;
        double r45561 = 2.0;
        double r45562 = r45561 / r45561;
        double r45563 = pow(r45559, r45562);
        double r45564 = r45560 * r45563;
        double r45565 = r45558 * r45564;
        return r45565;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.3
\[\mathsf{fma}\left(1 + t \cdot 2 \cdot 10^{-16}, 1 + t \cdot 2 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Simplified50.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, 2 \cdot 10^{-16}, 1\right), \mathsf{fma}\left(t, 2 \cdot 10^{-16}, 1\right), -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)} \cdot {t}^{2}\]

  6. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)}\]
  7. Using strategy rm

  8. Applied sqr-pow0.4

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{\left({t}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\right)\]

  9. Applied associate-*r*0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{\left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{\left(\frac{2}{2}\right)}\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\]

  10. Simplified0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\color{blue}{\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot t\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)\]

  11. Final simplification0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot t\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16)) (- -1 (* 2 (* t 2e-16))))

  (+ (* (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16))) (- -1 (* 2 (* t 2e-16)))))