Average Error: 61.8 → 0.3
Time: 35.3s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\]
\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)
↓
t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)
double f(double t) {
        double r3689473 = 1.0;
        double r3689474 = t;
        double r3689475 = 2e-16;
        double r3689476 = r3689474 * r3689475;
        double r3689477 = r3689473 + r3689476;
        double r3689478 = r3689477 * r3689477;
        double r3689479 = -1.0;
        double r3689480 = 2.0;
        double r3689481 = r3689480 * r3689476;
        double r3689482 = r3689479 - r3689481;
        double r3689483 = r3689478 + r3689482;
        return r3689483;
}
↓
double f(double t) {
        double r3689484 = t;
        double r3689485 = 3.9999999999999997e-32;
        double r3689486 = r3689485 * r3689484;
        double r3689487 = r3689484 * r3689486;
        return r3689487;
}

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

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. Simplified50.6

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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)))))