Average Error: 61.8 → 0.3
Time: 21.3s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]
\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(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}
double f(double t) {
        double r101749 = 1.0;
        double r101750 = t;
        double r101751 = 2e-16;
        double r101752 = r101750 * r101751;
        double r101753 = r101749 + r101752;
        double r101754 = r101753 * r101753;
        double r101755 = -1.0;
        double r101756 = 2.0;
        double r101757 = r101756 * r101752;
        double r101758 = r101755 - r101757;
        double r101759 = r101754 + r101758;
        return r101759;
}
↓
double f(double t) {
        double r101760 = 3.9999999999999997e-32;
        double r101761 = sqrt(r101760);
        double r101762 = t;
        double r101763 = 2.0;
        double r101764 = pow(r101762, r101763);
        double r101765 = r101761 * r101764;
        double r101766 = r101765 * r101761;
        return r101766;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt0.3

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

  5. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right)}\]
  6. Using strategy rm

  7. Applied *-commutative0.3

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

  8. Final simplification0.3

    \[\leadsto \left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]

Reproduce

herbie shell --seed 2019303 
(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)))))