Average Error: 61.8 → 0.3
Time: 1.8s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{1} \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)\]
\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
\sqrt{1} \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)
double f(double t) {
        double r60339 = 1.0;
        double r60340 = t;
        double r60341 = 2e-16;
        double r60342 = r60340 * r60341;
        double r60343 = r60339 + r60342;
        double r60344 = r60343 * r60343;
        double r60345 = -1.0;
        double r60346 = 2.0;
        double r60347 = r60346 * r60342;
        double r60348 = r60345 - r60347;
        double r60349 = r60344 + r60348;
        return r60349;
}
↓
double f(double t) {
        double r60350 = 1.0;
        double r60351 = sqrt(r60350);
        double r60352 = 3.9999999999999997e-32;
        double r60353 = t;
        double r60354 = 2.0;
        double r60355 = pow(r60353, r60354);
        double r60356 = r60352 * r60355;
        double r60357 = r60351 * r60356;
        return r60357;
}

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

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)} \cdot {t}^{2}\]

  5. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.4

    \[\leadsto \sqrt{\color{blue}{1 \cdot 3.9999999999999997 \cdot 10^{-32}}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)\]

  8. Applied sqrt-prod0.4

    \[\leadsto \color{blue}{\left(\sqrt{1} \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)\]

  9. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\sqrt{1} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)\right)}\]

  10. Simplified0.3

    \[\leadsto \sqrt{1} \cdot \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)}\]

  11. Final simplification0.3

    \[\leadsto \sqrt{1} \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)\]

Reproduce

herbie shell --seed 2020059 
(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)))))