Average Error: 61.8 → 0.3
Time: 18.8s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\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 r2687627 = 1.0;
        double r2687628 = t;
        double r2687629 = 2e-16;
        double r2687630 = r2687628 * r2687629;
        double r2687631 = r2687627 + r2687630;
        double r2687632 = r2687631 * r2687631;
        double r2687633 = -1.0;
        double r2687634 = 2.0;
        double r2687635 = r2687634 * r2687630;
        double r2687636 = r2687633 - r2687635;
        double r2687637 = r2687632 + r2687636;
        return r2687637;
}
↓
double f(double t) {
        double r2687638 = t;
        double r2687639 = 3.9999999999999997e-32;
        double r2687640 = r2687638 * r2687639;
        double r2687641 = r2687638 * r2687640;
        return r2687641;
}

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-*l*0.3

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

  5. Taylor expanded around 0 0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +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)))))