Error

Target

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. Simplified50.3

   \[\leadsto \color{blue}{(\left(2 \cdot 10^{-16} \cdot t\right) \cdot -2 + \left((\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right) + \left(2 \cdot 10^{-16} \cdot t\right))_*\right))_* + 2 \cdot 10^{-16} \cdot t}\]

3. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]

4. Simplified0.3

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

6. Applied add-sqr-sqrt0.3

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

7. Applied associate-*r*0.4

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

9. Applied associate-*l*0.3

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

10. 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 2019094 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :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)))))