Average Error: 61.8 → 0.3
Time: 5.0s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(-1 + 1 \cdot 1\right) + \frac{3653754093327257}{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot {t}^{2}\]
\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(-1 + 1 \cdot 1\right) + \frac{3653754093327257}{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot {t}^{2}
double f(double t) {
        double r62399 = 1.0;
        double r62400 = t;
        double r62401 = 2e-16;
        double r62402 = r62400 * r62401;
        double r62403 = r62399 + r62402;
        double r62404 = r62403 * r62403;
        double r62405 = -1.0;
        double r62406 = 2.0;
        double r62407 = r62406 * r62402;
        double r62408 = r62405 - r62407;
        double r62409 = r62404 + r62408;
        return r62409;
}
↓
double f(double t) {
        double r62410 = -1.0;
        double r62411 = 1.0;
        double r62412 = r62411 * r62411;
        double r62413 = r62410 + r62412;
        double r62414 = 3653754093327257.0;
        double r62415 = 9.134385233318143e+46;
        double r62416 = r62414 / r62415;
        double r62417 = t;
        double r62418 = 2.0;
        double r62419 = pow(r62417, r62418);
        double r62420 = r62416 * r62419;
        double r62421 = r62413 + r62420;
        return r62421;
}

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. Simplified61.8

    \[\leadsto \color{blue}{\left(-1 + 1 \cdot 1\right) + \left(t \cdot \frac{2028240960365167}{10141204801825835211973625643008}\right) \cdot \left(\left(\left(1 + t \cdot \frac{2028240960365167}{10141204801825835211973625643008}\right) + 1\right) - 2\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \color{blue}{\frac{3653754093327257}{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot {t}^{2}}\]

  5. Final simplification0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \frac{3653754093327257}{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot {t}^{2}\]

Reproduce

herbie shell --seed 2019304 
(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)))))