Average Error: 61.8 → 0.4
Time: 17.2s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(t \cdot t\right)\right)\]
\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(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(t \cdot t\right)\right)
double f(double t) {
        double r2061630 = 1.0;
        double r2061631 = t;
        double r2061632 = 2e-16;
        double r2061633 = r2061631 * r2061632;
        double r2061634 = r2061630 + r2061633;
        double r2061635 = r2061634 * r2061634;
        double r2061636 = -1.0;
        double r2061637 = 2.0;
        double r2061638 = r2061637 * r2061633;
        double r2061639 = r2061636 - r2061638;
        double r2061640 = r2061635 + r2061639;
        return r2061640;
}
↓
double f(double t) {
        double r2061641 = 3.9999999999999997e-32;
        double r2061642 = sqrt(r2061641);
        double r2061643 = t;
        double r2061644 = r2061643 * r2061643;
        double r2061645 = r2061642 * r2061644;
        double r2061646 = r2061642 * r2061645;
        return r2061646;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.4
\[\mathsf{fma}\left(1 + t \cdot 2 \cdot 10^{-16}, 1 + t \cdot 2 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

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. Simplified0.4

    \[\leadsto \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)}\]
  3. Using strategy rm

  4. Applied associate-*l*0.3

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

  5. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]

  6. Simplified0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied associate-*r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :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)))))