Average Error: 61.8 → 0.3
Time: 2.2s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)\]
\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(\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)
double f(double t) {
        double r74789 = 1.0;
        double r74790 = t;
        double r74791 = 2e-16;
        double r74792 = r74790 * r74791;
        double r74793 = r74789 + r74792;
        double r74794 = r74793 * r74793;
        double r74795 = -1.0;
        double r74796 = 2.0;
        double r74797 = r74796 * r74792;
        double r74798 = r74795 - r74797;
        double r74799 = r74794 + r74798;
        return r74799;
}
↓
double f(double t) {
        double r74800 = 3.9999999999999997e-32;
        double r74801 = sqrt(r74800);
        double r74802 = t;
        double r74803 = fabs(r74802);
        double r74804 = r74801 * r74803;
        double r74805 = 2.0;
        double r74806 = pow(r74802, r74805);
        double r74807 = sqrt(r74806);
        double r74808 = r74804 * r74807;
        double r74809 = r74801 * r74808;
        return r74809;
}

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. Simplified57.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, \mathsf{fma}\left(\mathsf{fma}\left(1.999999999999999958195573448069207123682 \cdot 10^{-16}, t, 1\right), \mathsf{fma}\left(1.999999999999999958195573448069207123682 \cdot 10^{-16}, t, 1\right), -1\right)\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.3

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

  6. Applied associate-*l*0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied associate-*r*0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.9 t 1.1)

  :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)))))