Average Error: 61.8 → 0.3
Time: 11.2s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)\]
\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)
↓
3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)
double f(double t) {
        double r3101344 = 1.0;
        double r3101345 = t;
        double r3101346 = 2e-16;
        double r3101347 = r3101345 * r3101346;
        double r3101348 = r3101344 + r3101347;
        double r3101349 = r3101348 * r3101348;
        double r3101350 = -1.0;
        double r3101351 = 2.0;
        double r3101352 = r3101351 * r3101347;
        double r3101353 = r3101350 - r3101352;
        double r3101354 = r3101349 + r3101353;
        return r3101354;
}
↓
double f(double t) {
        double r3101355 = 3.9999999999999997e-32;
        double r3101356 = t;
        double r3101357 = r3101356 * r3101356;
        double r3101358 = r3101355 * r3101357;
        return r3101358;
}

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
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

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. Taylor expanded around -inf 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019104 
(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)))))