Average Error: 61.8 → 0.3
Time: 11.3s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right) \cdot t\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right) \cdot t
double f(double t) {
        double r4306514 = 1.0;
        double r4306515 = t;
        double r4306516 = 2e-16;
        double r4306517 = r4306515 * r4306516;
        double r4306518 = r4306514 + r4306517;
        double r4306519 = r4306518 * r4306518;
        double r4306520 = -1.0;
        double r4306521 = 2.0;
        double r4306522 = r4306521 * r4306517;
        double r4306523 = r4306520 - r4306522;
        double r4306524 = r4306519 + r4306523;
        return r4306524;
}
↓
double f(double t) {
        double r4306525 = 3.9999999999999997e-32;
        double r4306526 = t;
        double r4306527 = r4306525 * r4306526;
        double r4306528 = r4306527 * r4306526;
        return r4306528;
}

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
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(t \cdot t\right) \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]

  4. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  5. Simplified0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right) \cdot t}\]

  6. Final simplification0.3

    \[\leadsto \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right) \cdot t\]

Reproduce

herbie shell --seed 2019192 
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))