Average Error: 61.8 → 0.3
Time: 12.6s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)
double f(double t) {
        double r62422 = 1.0;
        double r62423 = t;
        double r62424 = 2e-16;
        double r62425 = r62423 * r62424;
        double r62426 = r62422 + r62425;
        double r62427 = r62426 * r62426;
        double r62428 = -1.0;
        double r62429 = 2.0;
        double r62430 = r62429 * r62425;
        double r62431 = r62428 - r62430;
        double r62432 = r62427 + r62431;
        return r62432;
}
↓
double f(double t) {
        double r62433 = t;
        double r62434 = 3.9999999999999997e-32;
        double r62435 = r62434 * r62433;
        double r62436 = r62433 * r62435;
        return r62436;
}

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
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Simplified50.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), \mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]
  4. Using strategy rm

  5. Applied sqr-pow0.3

    \[\leadsto 3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \color{blue}{\left({t}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\]

  6. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{\left(\frac{2}{2}\right)}\right) \cdot {t}^{\left(\frac{2}{2}\right)}}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\]

  8. Final simplification0.3

    \[\leadsto t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.900000000000000022 t 1.1000000000000001)

  :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)))))