Average Error: 61.8 → 0.3
Time: 15.6s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(t \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \cdot 2 \cdot 10^{-16}\]
\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(t \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \cdot 2 \cdot 10^{-16}
double f(double t) {
        double r3368863 = 1.0;
        double r3368864 = t;
        double r3368865 = 2e-16;
        double r3368866 = r3368864 * r3368865;
        double r3368867 = r3368863 + r3368866;
        double r3368868 = r3368867 * r3368867;
        double r3368869 = -1.0;
        double r3368870 = 2.0;
        double r3368871 = r3368870 * r3368866;
        double r3368872 = r3368869 - r3368871;
        double r3368873 = r3368868 + r3368872;
        return r3368873;
}
↓
double f(double t) {
        double r3368874 = t;
        double r3368875 = 2e-16;
        double r3368876 = r3368874 * r3368875;
        double r3368877 = r3368874 * r3368876;
        double r3368878 = r3368877 * r3368875;
        return r3368878;
}

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. Simplified0.4

    \[\leadsto \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 
(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)))))