Average Error: 0.1 → 0.2
Time: 19.9s
Precision: 64
\[x \cdot \sin y + z \cdot \cos y\]
↓
\[\mathsf{fma}\left(x, \sin y, \sqrt[3]{\cos y} \cdot \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right)\right)\]
x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(x, \sin y, \sqrt[3]{\cos y} \cdot \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right)\right)
double f(double x, double y, double z) {
        double r7719234 = x;
        double r7719235 = y;
        double r7719236 = sin(r7719235);
        double r7719237 = r7719234 * r7719236;
        double r7719238 = z;
        double r7719239 = cos(r7719235);
        double r7719240 = r7719238 * r7719239;
        double r7719241 = r7719237 + r7719240;
        return r7719241;
}
↓
double f(double x, double y, double z) {
        double r7719242 = x;
        double r7719243 = y;
        double r7719244 = sin(r7719243);
        double r7719245 = cos(r7719243);
        double r7719246 = cbrt(r7719245);
        double r7719247 = z;
        double r7719248 = r7719245 * r7719245;
        double r7719249 = 0.3333333333333333;
        double r7719250 = pow(r7719248, r7719249);
        double r7719251 = r7719247 * r7719250;
        double r7719252 = r7719246 * r7719251;
        double r7719253 = fma(r7719242, r7719244, r7719252);
        return r7719253;
}

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. Simplified0.1

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

  3. Taylor expanded around inf 0.1

    \[\leadsto \color{blue}{z \cdot \cos y + x \cdot \sin y}\]

  4. Simplified0.1

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

  6. Applied add-cube-cbrt0.4

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

  7. Applied associate-*r*0.4

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

  9. Applied cbrt-unprod0.3

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

  11. Applied pow1/30.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  (+ (* x (sin y)) (* z (cos y))))