Average Error: 61.8 → 0.3
Time: 12.1s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[{t}^{2} \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\]
\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)
↓
{t}^{2} \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}
double f(double t) {
        double r37409 = 1.0;
        double r37410 = t;
        double r37411 = 2e-16;
        double r37412 = r37410 * r37411;
        double r37413 = r37409 + r37412;
        double r37414 = r37413 * r37413;
        double r37415 = -1.0;
        double r37416 = 2.0;
        double r37417 = r37416 * r37412;
        double r37418 = r37415 - r37417;
        double r37419 = r37414 + r37418;
        return r37419;
}
↓
double f(double t) {
        double r37420 = t;
        double r37421 = 2.0;
        double r37422 = pow(r37420, r37421);
        double r37423 = 3.9999999999999997e-32;
        double r37424 = r37422 * r37423;
        return r37424;
}

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

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

  5. Applied associate-*r*0.3

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

  6. Simplified0.3

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

  8. Applied *-commutative0.3

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

  9. Final simplification0.3

    \[\leadsto {t}^{2} \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\]

Reproduce

herbie shell --seed 2019298 
(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)))))