Average Error: 0.4 → 0.4
Time: 45.8s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \left(y - x\right) \cdot \left(\left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right) \cdot 6.0\right) + \mathsf{fma}\left(z, -1, z\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)\right)\]
x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)
↓
\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \left(y - x\right) \cdot \left(\left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right) \cdot 6.0\right) + \mathsf{fma}\left(z, -1, z\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)\right)
double f(double x, double y, double z) {
        double r12079082 = x;
        double r12079083 = y;
        double r12079084 = r12079083 - r12079082;
        double r12079085 = 6.0;
        double r12079086 = r12079084 * r12079085;
        double r12079087 = 2.0;
        double r12079088 = 3.0;
        double r12079089 = r12079087 / r12079088;
        double r12079090 = z;
        double r12079091 = r12079089 - r12079090;
        double r12079092 = r12079086 * r12079091;
        double r12079093 = r12079082 + r12079092;
        return r12079093;
}
↓
double f(double x, double y, double z) {
        double r12079094 = x;
        double r12079095 = cbrt(r12079094);
        double r12079096 = r12079095 * r12079095;
        double r12079097 = cbrt(r12079096);
        double r12079098 = cbrt(r12079095);
        double r12079099 = r12079097 * r12079098;
        double r12079100 = y;
        double r12079101 = r12079100 - r12079094;
        double r12079102 = 2.0;
        double r12079103 = 3.0;
        double r12079104 = sqrt(r12079103);
        double r12079105 = r12079102 / r12079104;
        double r12079106 = r12079105 / r12079104;
        double r12079107 = z;
        double r12079108 = r12079106 - r12079107;
        double r12079109 = 6.0;
        double r12079110 = r12079108 * r12079109;
        double r12079111 = r12079101 * r12079110;
        double r12079112 = -1.0;
        double r12079113 = fma(r12079107, r12079112, r12079107);
        double r12079114 = r12079101 * r12079109;
        double r12079115 = r12079113 * r12079114;
        double r12079116 = r12079111 + r12079115;
        double r12079117 = fma(r12079096, r12079099, r12079116);
        return r12079117;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.4

    \[x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.8

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\right)\]

  4. Applied add-sqr-sqrt1.1

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{\color{blue}{\sqrt{3.0} \cdot \sqrt{3.0}}} - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)\]

  5. Applied *-un-lft-identity1.1

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{\color{blue}{1 \cdot 2.0}}{\sqrt{3.0} \cdot \sqrt{3.0}} - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)\]

  6. Applied times-frac1.5

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\color{blue}{\frac{1}{\sqrt{3.0}} \cdot \frac{2.0}{\sqrt{3.0}}} - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)\]

  7. Applied prod-diff1.5

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{1}{\sqrt{3.0}}, \frac{2.0}{\sqrt{3.0}}, -\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)}\]

  8. Applied distribute-lft-in1.5

    \[\leadsto x + \color{blue}{\left(\left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{3.0}}, \frac{2.0}{\sqrt{3.0}}, -\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)}\]

  9. Simplified0.3

    \[\leadsto x + \left(\color{blue}{\left(y - x\right) \cdot \left(6.0 \cdot \left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right)\right)} + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)\]

  10. Simplified0.3

    \[\leadsto x + \left(\left(y - x\right) \cdot \left(6.0 \cdot \left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right)\right) + \color{blue}{\left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(z, -1, z\right)}\right)\]
  11. Using strategy rm

  12. Applied add-cube-cbrt0.4

    \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} + \left(\left(y - x\right) \cdot \left(6.0 \cdot \left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right)\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(z, -1, z\right)\right)\]

  13. Applied fma-def0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(y - x\right) \cdot \left(6.0 \cdot \left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right)\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(z, -1, z\right)\right)}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt0.4

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}, \left(y - x\right) \cdot \left(6.0 \cdot \left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right)\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(z, -1, z\right)\right)\]

  16. Applied cbrt-prod0.4

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}, \left(y - x\right) \cdot \left(6.0 \cdot \left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right)\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(z, -1, z\right)\right)\]

  17. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \left(y - x\right) \cdot \left(\left(\frac{\frac{2.0}{\sqrt{3.0}}}{\sqrt{3.0}} - z\right) \cdot 6.0\right) + \mathsf{fma}\left(z, -1, z\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))