Average Error: 61.8 → 0.3
Time: 9.2s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]
\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|t\right|\right) \cdot \sqrt{{t}^{2}}
double f(double t) {
        double r59874 = 1.0;
        double r59875 = t;
        double r59876 = 2e-16;
        double r59877 = r59875 * r59876;
        double r59878 = r59874 + r59877;
        double r59879 = r59878 * r59878;
        double r59880 = -1.0;
        double r59881 = 2.0;
        double r59882 = r59881 * r59877;
        double r59883 = r59880 - r59882;
        double r59884 = r59879 + r59883;
        return r59884;
}
↓
double f(double t) {
        double r59885 = 3.9999999999999997e-32;
        double r59886 = t;
        double r59887 = fabs(r59886);
        double r59888 = r59885 * r59887;
        double r59889 = 2.0;
        double r59890 = pow(r59886, r59889);
        double r59891 = sqrt(r59890);
        double r59892 = r59888 * r59891;
        return r59892;
}

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

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

  5. Applied associate-*r*0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(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)))))