Average Error: 61.8 → 0.3
Time: 13.7s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\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 r64324 = 1.0;
        double r64325 = t;
        double r64326 = 2e-16;
        double r64327 = r64325 * r64326;
        double r64328 = r64324 + r64327;
        double r64329 = r64328 * r64328;
        double r64330 = -1.0;
        double r64331 = 2.0;
        double r64332 = r64331 * r64327;
        double r64333 = r64330 - r64332;
        double r64334 = r64329 + r64333;
        return r64334;
}
↓
double f(double t) {
        double r64335 = t;
        double r64336 = 3.9999999999999997e-32;
        double r64337 = r64335 * r64336;
        double r64338 = r64335 * r64337;
        return r64338;
}

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. Taylor expanded around 0 0.4

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

  4. Applied sqr-pow0.4

    \[\leadsto 3.9999999999999997 \cdot 10^{-32} \cdot \color{blue}{\left({t}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\]

  5. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{\left(\frac{2}{2}\right)}\right) \cdot {t}^{\left(\frac{2}{2}\right)}}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\]

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019198 
(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)))))