Average Error: 61.8 → 0.4
Time: 21.9s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \left(t \cdot \left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot t\right)\right)\right)\]
\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(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot \left(t \cdot \left(\sqrt{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}} \cdot t\right)\right)\right)
double f(double t) {
        double r62043 = 1.0;
        double r62044 = t;
        double r62045 = 2e-16;
        double r62046 = r62044 * r62045;
        double r62047 = r62043 + r62046;
        double r62048 = r62047 * r62047;
        double r62049 = -1.0;
        double r62050 = 2.0;
        double r62051 = r62050 * r62046;
        double r62052 = r62049 - r62051;
        double r62053 = r62048 + r62052;
        return r62053;
}
↓
double f(double t) {
        double r62054 = 3.9999999999999997e-32;
        double r62055 = sqrt(r62054);
        double r62056 = sqrt(r62055);
        double r62057 = t;
        double r62058 = r62056 * r62057;
        double r62059 = r62057 * r62058;
        double r62060 = r62056 * r62059;
        double r62061 = r62055 * r62060;
        return r62061;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.4
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

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. Using strategy rm

  7. Applied add-sqr-sqrt0.4

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

  8. Applied sqrt-prod0.4

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(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)))))