Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.5

Time: 4.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r323376 = a;
        double r323377 = r323376 * r323376;
        double r323378 = b;
        double r323379 = r323378 * r323378;
        double r323380 = r323377 + r323379;
        double r323381 = 2.0;
        double r323382 = pow(r323380, r323381);
        double r323383 = 4.0;
        double r323384 = r323383 * r323379;
        double r323385 = r323382 + r323384;
        double r323386 = 1.0;
        double r323387 = r323385 - r323386;
        return r323387;
}

↓

double f(double a, double b) {
        double r323388 = a;
        double r323389 = r323388 * r323388;
        double r323390 = b;
        double r323391 = r323390 * r323390;
        double r323392 = r323389 + r323391;
        double r323393 = 2.0;
        double r323394 = pow(r323392, r323393);
        double r323395 = sqrt(r323394);
        double r323396 = cbrt(r323395);
        double r323397 = r323396 * r323396;
        double r323398 = cbrt(r323394);
        double r323399 = r323397 * r323398;
        double r323400 = 4.0;
        double r323401 = r323400 * r323391;
        double r323402 = fma(r323399, r323398, r323401);
        double r323403 = 1.0;
        double r323404 = r323402 - r323403;
        return r323404;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.5

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

4. Applied fma-def 0.5

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied cbrt-prod 0.5

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

8. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))