Average Error: 61.8 → 0.4
Time: 39.8s
Precision: 64
Internal Precision: 576
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left|t\right| \cdot \sqrt{\left(t \cdot t\right) \cdot 3.9999999999999997 \cdot 10^{-32}}\right)\]

Error

Bits error versus t

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

Target

Original61.8
Target50.6
Herbie0.4
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

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-sqrt0.6

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

  6. Applied sqrt-prod0.4

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

  7. Applied associate-*l*0.4

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

  8. Applied simplify0.4

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

Runtime

Time bar (total: 39.8s)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(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)))))