Average Error: 61.8 → 0.3
Time: 2.1s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right) \cdot t\right) \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\]
\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)
↓
\left(\left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right) \cdot t\right) \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}
double f(double t) {
        double r70466 = 1.0;
        double r70467 = t;
        double r70468 = 2e-16;
        double r70469 = r70467 * r70468;
        double r70470 = r70466 + r70469;
        double r70471 = r70470 * r70470;
        double r70472 = -1.0;
        double r70473 = 2.0;
        double r70474 = r70473 * r70469;
        double r70475 = r70472 - r70474;
        double r70476 = r70471 + r70475;
        return r70476;
}
↓
double f(double t) {
        double r70477 = t;
        double r70478 = 3.9999999999999997e-32;
        double r70479 = sqrt(r70478);
        double r70480 = r70477 * r70479;
        double r70481 = r70480 * r70477;
        double r70482 = r70481 * r70479;
        return r70482;
}

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. Simplified57.6

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

  3. Taylor expanded around 0 0.3

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

  5. Applied add-sqr-sqrt1.0

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

  6. Applied unpow-prod-down1.0

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

  7. Applied add-sqr-sqrt1.0

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

  8. Applied unswap-sqr1.0

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

  9. Simplified0.7

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

  10. Simplified0.4

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

  12. Applied associate-*r*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020027 +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)))))