fma_test1

Average Error: 61.8 → 0.3

Time: 2.7s

Precision: 64

\[0.900000000000000022 \le t \le 1.1000000000000001\]

Math

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

↓

\[{\left(3.9999999999999997 \cdot 10^{-32} \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{4}\right)\right)}^{\frac{1}{2}}\]

C

double f(double t) {
        double r58033 = 1.0;
        double r58034 = t;
        double r58035 = 2e-16;
        double r58036 = r58034 * r58035;
        double r58037 = r58033 + r58036;
        double r58038 = r58037 * r58037;
        double r58039 = -1.0;
        double r58040 = 2.0;
        double r58041 = r58040 * r58036;
        double r58042 = r58039 - r58041;
        double r58043 = r58038 + r58042;
        return r58043;
}

↓

double f(double t) {
        double r58044 = 3.9999999999999997e-32;
        double r58045 = t;
        double r58046 = 4.0;
        double r58047 = pow(r58045, r58046);
        double r58048 = r58044 * r58047;
        double r58049 = r58044 * r58048;
        double r58050 = 0.5;
        double r58051 = pow(r58049, r58050);
        return r58051;
}

Error

Bits error versus t

Target

Original	61.8
Target	50.6
Herbie	0.3

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

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 0.4

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

4. Applied add-sqr-sqrt 0.4

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

5. Applied add-sqr-sqrt 0.4

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

6. Applied unswap-sqr 0.4

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

8. Applied pow1/2 0.4

   \[\leadsto \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{{t}^{2}}\right) \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{{\left({t}^{2}\right)}^{\frac{1}{2}}}\right)\]

9. Applied pow1/2 0.4

   \[\leadsto \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{{t}^{2}}\right) \cdot \left(\color{blue}{{\left( 3.9999999999999997 \cdot 10^{-32} \right)}^{\frac{1}{2}}} \cdot {\left({t}^{2}\right)}^{\frac{1}{2}}\right)\]

10. Applied pow-prod-down 0.4

    \[\leadsto \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{{t}^{2}}\right) \cdot \color{blue}{{\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)}^{\frac{1}{2}}}\]

11. Applied pow1/2 0.4

    \[\leadsto \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{{\left({t}^{2}\right)}^{\frac{1}{2}}}\right) \cdot {\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)}^{\frac{1}{2}}\]

12. Applied pow1/2 0.4

    \[\leadsto \left(\color{blue}{{\left( 3.9999999999999997 \cdot 10^{-32} \right)}^{\frac{1}{2}}} \cdot {\left({t}^{2}\right)}^{\frac{1}{2}}\right) \cdot {\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)}^{\frac{1}{2}}\]

13. Applied pow-prod-down 0.6

    \[\leadsto \color{blue}{{\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)}^{\frac{1}{2}}} \cdot {\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)}^{\frac{1}{2}}\]

14. Applied pow-prod-down 0.4

    \[\leadsto \color{blue}{{\left(\left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right) \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\right)\right)}^{\frac{1}{2}}}\]

15. Simplified 0.3

    \[\leadsto {\color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{4}\right)\right)}}^{\frac{1}{2}}\]

16. Final simplification 0.3

    \[\leadsto {\left(3.9999999999999997 \cdot 10^{-32} \cdot \left(3.9999999999999997 \cdot 10^{-32} \cdot {t}^{4}\right)\right)}^{\frac{1}{2}}\]

Reproduce

herbie shell --seed 2020034 
(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)))))