Average Error: 61.8 → 0.3
Time: 13.0s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)\]
\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(\left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)
double f(double t) {
        double r224661 = 1.0;
        double r224662 = t;
        double r224663 = 2e-16;
        double r224664 = r224662 * r224663;
        double r224665 = r224661 + r224664;
        double r224666 = r224665 * r224665;
        double r224667 = -1.0;
        double r224668 = 2.0;
        double r224669 = r224668 * r224664;
        double r224670 = r224667 - r224669;
        double r224671 = r224666 + r224670;
        return r224671;
}
↓
double f(double t) {
        double r224672 = 3.9999999999999997e-32;
        double r224673 = sqrt(r224672);
        double r224674 = t;
        double r224675 = r224674 * r224673;
        double r224676 = 2.0;
        double r224677 = r224676 / r224676;
        double r224678 = pow(r224674, r224677);
        double r224679 = r224675 * r224678;
        double r224680 = r224673 * r224679;
        return r224680;
}

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*0.4

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

  7. Applied sqr-pow0.4

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

  8. Applied associate-*r*0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019350 
(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)))))