Average Error: 61.8 → 0.3
Time: 2.1s
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 r61290 = 1.0;
        double r61291 = t;
        double r61292 = 2e-16;
        double r61293 = r61291 * r61292;
        double r61294 = r61290 + r61293;
        double r61295 = r61294 * r61294;
        double r61296 = -1.0;
        double r61297 = 2.0;
        double r61298 = r61297 * r61293;
        double r61299 = r61296 - r61298;
        double r61300 = r61295 + r61299;
        return r61300;
}
↓
double f(double t) {
        double r61301 = t;
        double r61302 = 3.9999999999999997e-32;
        double r61303 = sqrt(r61302);
        double r61304 = r61301 * r61303;
        double r61305 = r61301 * r61304;
        double r61306 = r61305 * r61303;
        return r61306;
}

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 2020039 
(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)))))