Average Error: 61.8 → 0.3
Time: 21.0s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}\]
\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)
↓
3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}
double f(double t) {
        double r46821 = 1.0;
        double r46822 = t;
        double r46823 = 2e-16;
        double r46824 = r46822 * r46823;
        double r46825 = r46821 + r46824;
        double r46826 = r46825 * r46825;
        double r46827 = -1.0;
        double r46828 = 2.0;
        double r46829 = r46828 * r46824;
        double r46830 = r46827 - r46829;
        double r46831 = r46826 + r46830;
        return r46831;
}
↓
double f(double t) {
        double r46832 = 3.9999999999999997e-32;
        double r46833 = t;
        double r46834 = 2.0;
        double r46835 = pow(r46833, r46834);
        double r46836 = r46832 * r46835;
        return r46836;
}

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. Final simplification0.3

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

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)))))