Average Error: 61.8 → 0.3
Time: 10.6s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]
\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)
↓
\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t
double f(double t) {
        double r1341421 = 1.0;
        double r1341422 = t;
        double r1341423 = 2e-16;
        double r1341424 = r1341422 * r1341423;
        double r1341425 = r1341421 + r1341424;
        double r1341426 = r1341425 * r1341425;
        double r1341427 = -1.0;
        double r1341428 = 2.0;
        double r1341429 = r1341428 * r1341424;
        double r1341430 = r1341427 - r1341429;
        double r1341431 = r1341426 + r1341430;
        return r1341431;
}
↓
double f(double t) {
        double r1341432 = 3.9999999999999997e-32;
        double r1341433 = t;
        double r1341434 = r1341432 * r1341433;
        double r1341435 = r1341434 * r1341433;
        return r1341435;
}

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. Simplified61.8

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

  4. Applied associate-*l*61.8

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

  5. Simplified0.3

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

  6. Taylor expanded around inf 0.4

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 
(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)))))