Average Error: 61.8 → 0.3

Time: 5.0s

Precision: 64

Internal Precision: 576

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `t`

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In
Out

Enter valid numbers for all inputs

Target

Original	61.8
Target	50.6
Herbie	0.3

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

Derivation

Initial program 61.8
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
Initial simplification57.6
\[\leadsto (-2 \cdot \left(2 \cdot 10^{-16} \cdot t\right) + \left((\left((t \cdot \left( 2 \cdot 10^{-16} \right) + 1)_*\right) \cdot \left((t \cdot \left( 2 \cdot 10^{-16} \right) + 1)_*\right) + -1)_*\right))_*\]
Taylor expanded around inf 0.3
\[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
Using strategy rm
Applied unpow20.3
\[\leadsto 3.9999999999999997 \cdot 10^{-32} \cdot \color{blue}{\left(t \cdot t\right)}\]
Applied associate-*r*0.3
\[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t}\]
Final simplification0.3
\[\leadsto \left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.3	0.3	0.0	0.3	0%

herbie shell --seed 2018295 +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)))))