Average Error: 61.8 → 0.3
Time: 16.9s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right) \cdot t\]
\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(t \cdot 3.9999999999999997 \cdot 10^{-32}\right) \cdot t
double f(double t) {
        double r57035 = 1.0;
        double r57036 = t;
        double r57037 = 2e-16;
        double r57038 = r57036 * r57037;
        double r57039 = r57035 + r57038;
        double r57040 = r57039 * r57039;
        double r57041 = -1.0;
        double r57042 = 2.0;
        double r57043 = r57042 * r57038;
        double r57044 = r57041 - r57043;
        double r57045 = r57040 + r57044;
        return r57045;
}
↓
double f(double t) {
        double r57046 = t;
        double r57047 = 3.9999999999999997e-32;
        double r57048 = r57046 * r57047;
        double r57049 = r57048 * r57046;
        return r57049;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.5
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. Simplified50.5

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

  3. Taylor expanded around 0 0.3

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

  5. Applied unpow20.3

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

  6. Applied associate-*r*0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020043 +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)))))