Average Error: 61.8 → 0.3
Time: 10.6s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\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)
↓
t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)
double f(double t) {
        double r4615012 = 1.0;
        double r4615013 = t;
        double r4615014 = 2e-16;
        double r4615015 = r4615013 * r4615014;
        double r4615016 = r4615012 + r4615015;
        double r4615017 = r4615016 * r4615016;
        double r4615018 = -1.0;
        double r4615019 = 2.0;
        double r4615020 = r4615019 * r4615015;
        double r4615021 = r4615018 - r4615020;
        double r4615022 = r4615017 + r4615021;
        return r4615022;
}
↓
double f(double t) {
        double r4615023 = t;
        double r4615024 = 3.9999999999999997e-32;
        double r4615025 = r4615023 * r4615024;
        double r4615026 = r4615023 * r4615025;
        return r4615026;
}

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

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. Simplified50.3

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

  3. Taylor expanded around -inf 0.3

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

  4. Simplified0.3

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

  6. Applied associate-*l*0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 +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)))))