Average Error: 61.8 → 0.3
Time: 34.3s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(t \cdot \left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right)\right)\]
\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)
↓
\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(t \cdot \left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right)\right)
double f(double t) {
        double r5727871 = 1.0;
        double r5727872 = t;
        double r5727873 = 2e-16;
        double r5727874 = r5727872 * r5727873;
        double r5727875 = r5727871 + r5727874;
        double r5727876 = r5727875 * r5727875;
        double r5727877 = -1.0;
        double r5727878 = 2.0;
        double r5727879 = r5727878 * r5727874;
        double r5727880 = r5727877 - r5727879;
        double r5727881 = r5727876 + r5727880;
        return r5727881;
}
↓
double f(double t) {
        double r5727882 = 3.9999999999999997e-32;
        double r5727883 = sqrt(r5727882);
        double r5727884 = t;
        double r5727885 = r5727884 * r5727883;
        double r5727886 = r5727884 * r5727885;
        double r5727887 = r5727883 * r5727886;
        return r5727887;
}

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

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.4

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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)))))