Average Error: 61.8 → 0.3
Time: 39.5s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(2 \cdot 10^{-16} \cdot \left(2 \cdot 10^{-16} \cdot t\right)\right) \cdot t\]
\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(2 \cdot 10^{-16} \cdot \left(2 \cdot 10^{-16} \cdot t\right)\right) \cdot t
double f(double t) {
        double r1572238 = 1.0;
        double r1572239 = t;
        double r1572240 = 2e-16;
        double r1572241 = r1572239 * r1572240;
        double r1572242 = r1572238 + r1572241;
        double r1572243 = r1572242 * r1572242;
        double r1572244 = -1.0;
        double r1572245 = 2.0;
        double r1572246 = r1572245 * r1572241;
        double r1572247 = r1572244 - r1572246;
        double r1572248 = r1572243 + r1572247;
        return r1572248;
}
↓
double f(double t) {
        double r1572249 = 2e-16;
        double r1572250 = t;
        double r1572251 = r1572249 * r1572250;
        double r1572252 = r1572249 * r1572251;
        double r1572253 = r1572252 * r1572250;
        return r1572253;
}

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 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

    \[\leadsto \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 
(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)))))