Average Error: 61.8 → 0.3
Time: 22.2s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}
double f(double t) {
        double r44123 = 1.0;
        double r44124 = t;
        double r44125 = 2e-16;
        double r44126 = r44124 * r44125;
        double r44127 = r44123 + r44126;
        double r44128 = r44127 * r44127;
        double r44129 = -1.0;
        double r44130 = 2.0;
        double r44131 = r44130 * r44126;
        double r44132 = r44129 - r44131;
        double r44133 = r44128 + r44132;
        return r44133;
}
↓
double f(double t) {
        double r44134 = 3.9999999999999997e-32;
        double r44135 = t;
        double r44136 = fabs(r44135);
        double r44137 = r44134 * r44136;
        double r44138 = 2.0;
        double r44139 = pow(r44135, r44138);
        double r44140 = sqrt(r44139);
        double r44141 = r44137 * r44140;
        return r44141;
}

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
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.4

    \[\leadsto 3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{{t}^{2}}\right)}\]

  5. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \sqrt{{t}^{2}}\right) \cdot \sqrt{{t}^{2}}}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left|t\right|\right)} \cdot \sqrt{{t}^{2}}\]

  7. Final simplification0.3

    \[\leadsto \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]

Reproduce

herbie shell --seed 2019322 
(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)))))