Average Error: 61.8 → 0.3
Time: 2.5s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)\]
\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(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)
double f(double t) {
        double r91744 = 1.0;
        double r91745 = t;
        double r91746 = 2e-16;
        double r91747 = r91745 * r91746;
        double r91748 = r91744 + r91747;
        double r91749 = r91748 * r91748;
        double r91750 = -1.0;
        double r91751 = 2.0;
        double r91752 = r91751 * r91747;
        double r91753 = r91750 - r91752;
        double r91754 = r91749 + r91753;
        return r91754;
}
↓
double f(double t) {
        double r91755 = 3.9999999999999997e-32;
        double r91756 = sqrt(r91755);
        double r91757 = t;
        double r91758 = fabs(r91757);
        double r91759 = r91756 * r91758;
        double r91760 = 2.0;
        double r91761 = pow(r91757, r91760);
        double r91762 = sqrt(r91761);
        double r91763 = r91759 * r91762;
        double r91764 = r91756 * r91763;
        return r91764;
}

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. Simplified57.6

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

  3. Taylor expanded around 0 0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied associate-*l*0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied associate-*r*0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(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)))))