Average Error: 61.8 → 0.3
Time: 25.3s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\left|t\right| \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right) \cdot \sqrt{{t}^{2}}\]
\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)
↓
\left(\left|t\right| \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right) \cdot \sqrt{{t}^{2}}
double f(double t) {
        double r43160 = 1.0;
        double r43161 = t;
        double r43162 = 2e-16;
        double r43163 = r43161 * r43162;
        double r43164 = r43160 + r43163;
        double r43165 = r43164 * r43164;
        double r43166 = -1.0;
        double r43167 = 2.0;
        double r43168 = r43167 * r43163;
        double r43169 = r43166 - r43168;
        double r43170 = r43165 + r43169;
        return r43170;
}
↓
double f(double t) {
        double r43171 = t;
        double r43172 = fabs(r43171);
        double r43173 = 3.9999999999999997e-32;
        double r43174 = r43172 * r43173;
        double r43175 = 2.0;
        double r43176 = pow(r43171, r43175);
        double r43177 = sqrt(r43176);
        double r43178 = r43174 * r43177;
        return r43178;
}

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. Simplified50.6

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

  3. Taylor expanded around 0 0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied associate-*r*0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

    \[\leadsto \left(\left|t\right| \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right) \cdot \sqrt{{t}^{2}}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)))))