Average Error: 61.8 → 0.3
Time: 10.3s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)\]
\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)
↓
3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)
double f(double t) {
        double r1228668 = 1.0;
        double r1228669 = t;
        double r1228670 = 2e-16;
        double r1228671 = r1228669 * r1228670;
        double r1228672 = r1228668 + r1228671;
        double r1228673 = r1228672 * r1228672;
        double r1228674 = -1.0;
        double r1228675 = 2.0;
        double r1228676 = r1228675 * r1228671;
        double r1228677 = r1228674 - r1228676;
        double r1228678 = r1228673 + r1228677;
        return r1228678;
}
↓
double f(double t) {
        double r1228679 = 3.9999999999999997e-32;
        double r1228680 = t;
        double r1228681 = r1228680 * r1228680;
        double r1228682 = r1228679 * r1228681;
        return r1228682;
}

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. Taylor expanded around 0 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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 
(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)))))