Average Error: 61.8 → 0.3
Time: 10.1s
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 r4137272 = 1.0;
        double r4137273 = t;
        double r4137274 = 2e-16;
        double r4137275 = r4137273 * r4137274;
        double r4137276 = r4137272 + r4137275;
        double r4137277 = r4137276 * r4137276;
        double r4137278 = -1.0;
        double r4137279 = 2.0;
        double r4137280 = r4137279 * r4137275;
        double r4137281 = r4137278 - r4137280;
        double r4137282 = r4137277 + r4137281;
        return r4137282;
}
↓
double f(double t) {
        double r4137283 = t;
        double r4137284 = 3.9999999999999997e-32;
        double r4137285 = r4137283 * r4137284;
        double r4137286 = r4137283 * r4137285;
        return r4137286;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.5
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. Simplified50.3

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

    \[\leadsto \color{blue}{\left(t \cdot t\right) \cdot 3.9999999999999997 \cdot 10^{-32}}\]
  5. Using strategy rm

  6. Applied associate-*l*0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 +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)))))