Average Error: 61.8 → 0.3
Time: 11.3s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)\]
\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(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)
double f(double t) {
        double r63310 = 1.0;
        double r63311 = t;
        double r63312 = 2e-16;
        double r63313 = r63311 * r63312;
        double r63314 = r63310 + r63313;
        double r63315 = r63314 * r63314;
        double r63316 = -1.0;
        double r63317 = 2.0;
        double r63318 = r63317 * r63313;
        double r63319 = r63316 - r63318;
        double r63320 = r63315 + r63319;
        return r63320;
}
↓
double f(double t) {
        double r63321 = t;
        double r63322 = 3.9999999999999997e-32;
        double r63323 = r63321 * r63322;
        double r63324 = r63321 * r63323;
        return r63324;
}

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 0.4

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

  4. Simplified0.4

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

  6. Applied pow10.4

    \[\leadsto 3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(t \cdot \color{blue}{{t}^{1}}\right)\]

  7. Applied pow10.4

    \[\leadsto 3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(\color{blue}{{t}^{1}} \cdot {t}^{1}\right)\]

  8. Applied pow-prod-down0.4

    \[\leadsto 3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \color{blue}{{\left(t \cdot t\right)}^{1}}\]

  9. Applied pow10.4

    \[\leadsto \color{blue}{{\left( 3.999999999999999676487027278085939408227 \cdot 10^{-32} \right)}^{1}} \cdot {\left(t \cdot t\right)}^{1}\]

  10. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(t \cdot t\right)\right)}^{1}}\]

  11. Simplified0.3

    \[\leadsto {\color{blue}{\left(t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\right)}}^{1}\]

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +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)))))