Average Error: 61.8 → 0.3
Time: 16.9s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right) \cdot t\]
\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)
↓
\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right) \cdot t
double f(double t) {
        double r4216413 = 1.0;
        double r4216414 = t;
        double r4216415 = 2e-16;
        double r4216416 = r4216414 * r4216415;
        double r4216417 = r4216413 + r4216416;
        double r4216418 = r4216417 * r4216417;
        double r4216419 = -1.0;
        double r4216420 = 2.0;
        double r4216421 = r4216420 * r4216416;
        double r4216422 = r4216419 - r4216421;
        double r4216423 = r4216418 + r4216422;
        return r4216423;
}
↓
double f(double t) {
        double r4216424 = 3.9999999999999997e-32;
        double r4216425 = t;
        double r4216426 = r4216424 * r4216425;
        double r4216427 = r4216426 * r4216425;
        return r4216427;
}

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

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

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.3

    \[\leadsto \color{blue}{\left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right) \cdot t}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.6

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

  7. Taylor expanded around 0 0.4

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

  8. Taylor expanded around 0 0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))