Average Error: 61.8 → 0.3
Time: 12.1s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot t\right) \cdot t\right)\]
\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)
↓
\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot t\right) \cdot t\right)
double f(double t) {
        double r42360 = 1.0;
        double r42361 = t;
        double r42362 = 2e-16;
        double r42363 = r42361 * r42362;
        double r42364 = r42360 + r42363;
        double r42365 = r42364 * r42364;
        double r42366 = -1.0;
        double r42367 = 2.0;
        double r42368 = r42367 * r42363;
        double r42369 = r42366 - r42368;
        double r42370 = r42365 + r42369;
        return r42370;
}
↓
double f(double t) {
        double r42371 = 3.9999999999999997e-32;
        double r42372 = sqrt(r42371);
        double r42373 = t;
        double r42374 = r42372 * r42373;
        double r42375 = r42374 * r42373;
        double r42376 = r42372 * r42375;
        return r42376;
}

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 add-sqr-sqrt0.3

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

  5. Applied associate-*l*0.4

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

  6. Taylor expanded around 0 0.4

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.900000000000000022 t 1.1000000000000001)

  :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)))))