Average Error: 61.8 → 0.2
Time: 5.2s
Precision: binary64
\[0.9 \leq t \land t \leq 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
↓
\[{\left({t}^{4} \cdot 1.6 \cdot 10^{-63}\right)}^{0.5} \]
\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({t}^{4} \cdot 1.6 \cdot 10^{-63}\right)}^{0.5}

(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 (pow (* (pow t 4.0) 1.6e-63) 0.5))
double code(double t) {
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
}
↓
double code(double t) {
	return pow((pow(t, 4.0) * 1.6e-63), 0.5);
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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. Simplified0.4

    \[\leadsto \color{blue}{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)} \]

  3. Applied egg-rr0.3

    \[\leadsto t \cdot \color{blue}{\sqrt{\left(t \cdot t\right) \cdot 1.6 \cdot 10^{-63}}} \]

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{{\left({t}^{4} \cdot 1.6 \cdot 10^{-63}\right)}^{0.5}} \]

  5. Final simplification0.2

    \[\leadsto {\left({t}^{4} \cdot 1.6 \cdot 10^{-63}\right)}^{0.5} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (and (<= 0.9 t) (<= 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)))))