Average Error: 0.2 → 0.2
Time: 30.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r6961575 = a;
        double r6961576 = r6961575 * r6961575;
        double r6961577 = b;
        double r6961578 = r6961577 * r6961577;
        double r6961579 = r6961576 + r6961578;
        double r6961580 = 2.0;
        double r6961581 = pow(r6961579, r6961580);
        double r6961582 = 4.0;
        double r6961583 = 1.0;
        double r6961584 = r6961583 + r6961575;
        double r6961585 = r6961576 * r6961584;
        double r6961586 = 3.0;
        double r6961587 = r6961586 * r6961575;
        double r6961588 = r6961583 - r6961587;
        double r6961589 = r6961578 * r6961588;
        double r6961590 = r6961585 + r6961589;
        double r6961591 = r6961582 * r6961590;
        double r6961592 = r6961581 + r6961591;
        double r6961593 = r6961592 - r6961583;
        return r6961593;
}
↓
double f(double a, double b) {
        double r6961594 = 4.0;
        double r6961595 = a;
        double r6961596 = r6961595 * r6961595;
        double r6961597 = 1.0;
        double r6961598 = r6961597 + r6961595;
        double r6961599 = b;
        double r6961600 = r6961599 * r6961599;
        double r6961601 = 3.0;
        double r6961602 = r6961601 * r6961595;
        double r6961603 = r6961597 - r6961602;
        double r6961604 = r6961600 * r6961603;
        double r6961605 = fma(r6961596, r6961598, r6961604);
        double r6961606 = fma(r6961595, r6961595, r6961600);
        double r6961607 = 2.0;
        double r6961608 = pow(r6961606, r6961607);
        double r6961609 = fma(r6961594, r6961605, r6961608);
        double r6961610 = r6961609 - r6961597;
        return r6961610;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))