Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B

Average Error: 0.1 → 0.3

Time: 4.9s

Precision: 64

Math

\[x \cdot \sin y + z \cdot \cos y\]

↓

\[\mathsf{fma}\left(x, \sin y, \left(\left(z \cdot \sqrt[3]{{\left({\left(\cos y\right)}^{2}\right)}^{\frac{2}{3}}}\right) \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)\]

C

double f(double x, double y, double z) {
        double r151662 = x;
        double r151663 = y;
        double r151664 = sin(r151663);
        double r151665 = r151662 * r151664;
        double r151666 = z;
        double r151667 = cos(r151663);
        double r151668 = r151666 * r151667;
        double r151669 = r151665 + r151668;
        return r151669;
}

↓

double f(double x, double y, double z) {
        double r151670 = x;
        double r151671 = y;
        double r151672 = sin(r151671);
        double r151673 = z;
        double r151674 = cos(r151671);
        double r151675 = 2.0;
        double r151676 = pow(r151674, r151675);
        double r151677 = 0.6666666666666666;
        double r151678 = pow(r151676, r151677);
        double r151679 = cbrt(r151678);
        double r151680 = r151673 * r151679;
        double r151681 = cbrt(r151676);
        double r151682 = 0.3333333333333333;
        double r151683 = pow(r151681, r151682);
        double r151684 = r151680 * r151683;
        double r151685 = cbrt(r151674);
        double r151686 = r151684 * r151685;
        double r151687 = fma(r151670, r151672, r151686);
        return r151687;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Initial program 0.1

   \[x \cdot \sin y + z \cdot \cos y\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\]
3. Using strategy rm

4. Applied add-cube-cbrt 0.4

   \[\leadsto \mathsf{fma}\left(x, \sin y, z \cdot \color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) \cdot \sqrt[3]{\cos y}\right)}\right)\]

5. Applied associate-*r* 0.4

   \[\leadsto \mathsf{fma}\left(x, \sin y, \color{blue}{\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}}\right)\]
6. Using strategy rm

7. Applied pow1/3 15.6

   \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\cos y}\right)\]

8. Applied pow1/3 15.5

   \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}} \cdot {\left(\cos y\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\cos y}\right)\]

9. Applied pow-prod-down 0.2

   \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \color{blue}{{\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\cos y}\right)\]

10. Simplified 0.2

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot {\color{blue}{\left({\left(\cos y\right)}^{2}\right)}}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot {\color{blue}{\left(\left(\sqrt[3]{{\left(\cos y\right)}^{2}} \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right)}}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)\]

13. Applied unpow-prod-down 0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(\cos y\right)}^{2}} \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}} \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}}\right)}\right) \cdot \sqrt[3]{\cos y}\right)\]

14. Applied associate-*r* 0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \color{blue}{\left(\left(z \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}} \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}}\right) \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}}\right)} \cdot \sqrt[3]{\cos y}\right)\]

15. Simplified 0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(\color{blue}{\left(z \cdot \sqrt[3]{{\left({\left(\cos y\right)}^{2}\right)}^{\frac{2}{3}}}\right)} \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)\]

16. Final simplification 0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(\left(z \cdot \sqrt[3]{{\left({\left(\cos y\right)}^{2}\right)}^{\frac{2}{3}}}\right) \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))