Average Error: 61.8 → 0.3
Time: 11.4s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]
\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 \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}
double f(double t) {
        double r2925346 = 1.0;
        double r2925347 = t;
        double r2925348 = 2e-16;
        double r2925349 = r2925347 * r2925348;
        double r2925350 = r2925346 + r2925349;
        double r2925351 = r2925350 * r2925350;
        double r2925352 = -1.0;
        double r2925353 = 2.0;
        double r2925354 = r2925353 * r2925349;
        double r2925355 = r2925352 - r2925354;
        double r2925356 = r2925351 + r2925355;
        return r2925356;
}
↓
double f(double t) {
        double r2925357 = t;
        double r2925358 = 3.9999999999999997e-32;
        double r2925359 = sqrt(r2925358);
        double r2925360 = r2925357 * r2925359;
        double r2925361 = r2925360 * r2925357;
        double r2925362 = r2925361 * r2925359;
        return r2925362;
}

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. Simplified0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(\left(t \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.4

    \[\leadsto \left(\left(t \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot \sqrt{\color{blue}{1 \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\]

  9. Applied sqrt-prod0.4

    \[\leadsto \left(\left(t \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot \color{blue}{\left(\sqrt{1} \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right)}\]

  10. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(\left(\left(t \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot \sqrt{1}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}}\]

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(FPCore (t)
  :name "fma_test1"
  :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)))))