Average Error: 61.8 → 0.3
Time: 2.0s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(t \cdot \left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)\right) \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\]
\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
\left(t \cdot \left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)\right) \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}
double f(double t) {
        double r57307 = 1.0;
        double r57308 = t;
        double r57309 = 2e-16;
        double r57310 = r57308 * r57309;
        double r57311 = r57307 + r57310;
        double r57312 = r57311 * r57311;
        double r57313 = -1.0;
        double r57314 = 2.0;
        double r57315 = r57314 * r57310;
        double r57316 = r57313 - r57315;
        double r57317 = r57312 + r57316;
        return r57317;
}
↓
double f(double t) {
        double r57318 = t;
        double r57319 = 3.9999999999999997e-32;
        double r57320 = sqrt(r57319);
        double r57321 = r57318 * r57320;
        double r57322 = r57318 * r57321;
        double r57323 = r57322 * r57320;
        return r57323;
}

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

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)} \cdot {t}^{2}\]

  5. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)}\]
  6. Using strategy rm

  7. Applied sqr-pow0.4

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{\left({t}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\right)\]

  8. Applied associate-*r*0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{\left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{\left(\frac{2}{2}\right)}\right) \cdot {t}^{\left(\frac{2}{2}\right)}\right)}\]

  9. Simplified0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\color{blue}{\left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)} \cdot {t}^{\left(\frac{2}{2}\right)}\right)\]

  10. Final simplification0.3

    \[\leadsto \left(t \cdot \left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)\right) \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\]

Reproduce

herbie shell --seed 2020020 
(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)))))