Average Error: 61.8 → 0.3
Time: 15.5s
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 r6080581 = 1.0;
        double r6080582 = t;
        double r6080583 = 2e-16;
        double r6080584 = r6080582 * r6080583;
        double r6080585 = r6080581 + r6080584;
        double r6080586 = r6080585 * r6080585;
        double r6080587 = -1.0;
        double r6080588 = 2.0;
        double r6080589 = r6080588 * r6080584;
        double r6080590 = r6080587 - r6080589;
        double r6080591 = r6080586 + r6080590;
        return r6080591;
}
↓
double f(double t) {
        double r6080592 = t;
        double r6080593 = 3.9999999999999997e-32;
        double r6080594 = r6080592 * r6080593;
        double r6080595 = r6080592 * r6080594;
        return r6080595;
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 
(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)))))