Average Error: 61.8 → 0.3
Time: 21.8s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)
double f(double t) {
        double r44334 = 1.0;
        double r44335 = t;
        double r44336 = 2e-16;
        double r44337 = r44335 * r44336;
        double r44338 = r44334 + r44337;
        double r44339 = r44338 * r44338;
        double r44340 = -1.0;
        double r44341 = 2.0;
        double r44342 = r44341 * r44337;
        double r44343 = r44340 - r44342;
        double r44344 = r44339 + r44343;
        return r44344;
}
↓
double f(double t) {
        double r44345 = t;
        double r44346 = 3.9999999999999997e-32;
        double r44347 = r44346 * r44345;
        double r44348 = r44345 * r44347;
        return r44348;
}

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
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]
  3. Using strategy rm

  4. Applied sqr-pow0.3

    \[\leadsto 3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \color{blue}{\left({t}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\]

  5. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{\left(\frac{2}{2}\right)}\right) \cdot {t}^{\left(\frac{2}{2}\right)}}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\]

  7. Final simplification0.3

    \[\leadsto t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\]

Reproduce

herbie shell --seed 2019326 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :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)))))