Average Error: 61.8 → 0.3
Time: 9.1s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(t \cdot t\right)\]
\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)
↓
3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(t \cdot t\right)
double f(double t) {
        double r48995 = 1.0;
        double r48996 = t;
        double r48997 = 2e-16;
        double r48998 = r48996 * r48997;
        double r48999 = r48995 + r48998;
        double r49000 = r48999 * r48999;
        double r49001 = -1.0;
        double r49002 = 2.0;
        double r49003 = r49002 * r48998;
        double r49004 = r49001 - r49003;
        double r49005 = r49000 + r49004;
        return r49005;
}
↓
double f(double t) {
        double r49006 = 3.9999999999999997e-32;
        double r49007 = t;
        double r49008 = r49007 * r49007;
        double r49009 = r49006 * r49008;
        return r49009;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(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)))))