Average Error: 61.8 → 0.3
Time: 18.1s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right) \cdot t\right)\]
\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{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right) \cdot t\right)
double f(double t) {
        double r56280 = 1.0;
        double r56281 = t;
        double r56282 = 2e-16;
        double r56283 = r56281 * r56282;
        double r56284 = r56280 + r56283;
        double r56285 = r56284 * r56284;
        double r56286 = -1.0;
        double r56287 = 2.0;
        double r56288 = r56287 * r56283;
        double r56289 = r56286 - r56288;
        double r56290 = r56285 + r56289;
        return r56290;
}
↓
double f(double t) {
        double r56291 = 3.9999999999999997e-32;
        double r56292 = sqrt(r56291);
        double r56293 = t;
        double r56294 = r56293 * r56292;
        double r56295 = r56294 * r56293;
        double r56296 = r56292 * r56295;
        return r56296;
}

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. Simplified50.6

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

  3. Taylor expanded around 0 0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied associate-*l*0.4

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

  8. Applied unpow20.4

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

  9. Applied associate-*r*0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(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)))))