Average Error: 61.8 → 0.3
Time: 2.2s
Precision: binary64
\[0.9 \leq t \land t \leq 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\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)
↓
t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)
(FPCore (t)
 :precision binary64
 (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))
↓
(FPCore (t) :precision binary64 (* t (* t 3.9999999999999997e-32)))
double code(double t) {
	return ((double) (((double) (((double) (1.0 + ((double) (t * 2e-16)))) * ((double) (1.0 + ((double) (t * 2e-16)))))) + ((double) (-1.0 - ((double) (2.0 * ((double) (t * 2e-16))))))));
}
↓
double code(double t) {
	return ((double) (t * ((double) (t * 3.9999999999999997e-32))));
}

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 Error: 61.8 bits

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

  2. Taylor expanded around 0 Error: 0.3 bits

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

  3. SimplifiedError: 0.3 bits

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

  4. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :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)))))