Average Error: 61.8 → 0.3
Time: 15.5s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(2 \cdot 10^{-16} \cdot \left(2 \cdot 10^{-16} \cdot t\right)\right) \cdot t\]
\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(2 \cdot 10^{-16} \cdot \left(2 \cdot 10^{-16} \cdot t\right)\right) \cdot t
double f(double t) {
        double r1731731 = 1.0;
        double r1731732 = t;
        double r1731733 = 2e-16;
        double r1731734 = r1731732 * r1731733;
        double r1731735 = r1731731 + r1731734;
        double r1731736 = r1731735 * r1731735;
        double r1731737 = -1.0;
        double r1731738 = 2.0;
        double r1731739 = r1731738 * r1731734;
        double r1731740 = r1731737 - r1731739;
        double r1731741 = r1731736 + r1731740;
        return r1731741;
}
↓
double f(double t) {
        double r1731742 = 2e-16;
        double r1731743 = t;
        double r1731744 = r1731742 * r1731743;
        double r1731745 = r1731742 * r1731744;
        double r1731746 = r1731745 * r1731743;
        return r1731746;
}

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

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

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

  4. Applied associate-*r*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(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)))))