Average Error: 61.8 → 0.3
Time: 11.7s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{1} \cdot \left(t \cdot \left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)\right)\right)\]
\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)
↓
\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{1} \cdot \left(t \cdot \left(t \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)\right)\right)
double f(double t) {
        double r105050 = 1.0;
        double r105051 = t;
        double r105052 = 2e-16;
        double r105053 = r105051 * r105052;
        double r105054 = r105050 + r105053;
        double r105055 = r105054 * r105054;
        double r105056 = -1.0;
        double r105057 = 2.0;
        double r105058 = r105057 * r105053;
        double r105059 = r105056 - r105058;
        double r105060 = r105055 + r105059;
        return r105060;
}
↓
double f(double t) {
        double r105061 = 3.9999999999999997e-32;
        double r105062 = sqrt(r105061);
        double r105063 = 1.0;
        double r105064 = sqrt(r105063);
        double r105065 = t;
        double r105066 = r105065 * r105062;
        double r105067 = r105065 * r105066;
        double r105068 = r105064 * r105067;
        double r105069 = r105062 * r105068;
        return r105069;
}

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 *-un-lft-identity0.4

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

  8. Applied sqrt-prod0.4

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 
(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)))))