Average Error: 61.8 → 0.4
Time: 19.1s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \left(\left({\left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\right)}^{3} \cdot t\right) \cdot t\right)\]
\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)
↓
\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \left(\left({\left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\right)}^{3} \cdot t\right) \cdot t\right)
double f(double t) {
        double r42676 = 1.0;
        double r42677 = t;
        double r42678 = 2e-16;
        double r42679 = r42677 * r42678;
        double r42680 = r42676 + r42679;
        double r42681 = r42680 * r42680;
        double r42682 = -1.0;
        double r42683 = 2.0;
        double r42684 = r42683 * r42679;
        double r42685 = r42682 - r42684;
        double r42686 = r42681 + r42685;
        return r42686;
}
↓
double f(double t) {
        double r42687 = 3.9999999999999997e-32;
        double r42688 = sqrt(r42687);
        double r42689 = sqrt(r42688);
        double r42690 = 3.0;
        double r42691 = pow(r42689, r42690);
        double r42692 = t;
        double r42693 = r42691 * r42692;
        double r42694 = r42693 * r42692;
        double r42695 = r42689 * r42694;
        return r42695;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.4
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

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. Simplified50.6

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

  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*0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied sqrt-prod0.4

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

  10. Applied associate-*l*0.4

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

  11. Simplified0.4

    \[\leadsto \sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \color{blue}{\left({\left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\right)}^{3} \cdot \left(t \cdot t\right)\right)}\]
  12. Using strategy rm

  13. Applied associate-*r*0.4

    \[\leadsto \sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \color{blue}{\left(\left({\left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\right)}^{3} \cdot t\right) \cdot t\right)}\]

  14. Final simplification0.4

    \[\leadsto \sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \left(\left({\left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\right)}^{3} \cdot t\right) \cdot t\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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)))))