Average Error: 61.8 → 0.3
Time: 7.2s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]
\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)
↓
\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t
double f(double t) {
        double r1219097 = 1.0;
        double r1219098 = t;
        double r1219099 = 2e-16;
        double r1219100 = r1219098 * r1219099;
        double r1219101 = r1219097 + r1219100;
        double r1219102 = r1219101 * r1219101;
        double r1219103 = -1.0;
        double r1219104 = 2.0;
        double r1219105 = r1219104 * r1219100;
        double r1219106 = r1219103 - r1219105;
        double r1219107 = r1219102 + r1219106;
        return r1219107;
}
↓
double f(double t) {
        double r1219108 = 3.9999999999999997e-32;
        double r1219109 = t;
        double r1219110 = r1219108 * r1219109;
        double r1219111 = r1219110 * r1219109;
        return r1219111;
}

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. Using strategy rm

  4. Applied associate-*l*61.8

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

  5. Simplified0.3

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

  6. Taylor expanded around 0 0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 
(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)))))