Average Error: 6.7 → 2.1
Time: 24.5s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[x + \left(\frac{z - x}{\frac{t}{y}} + \frac{\mathsf{fma}\left(x, -1, x\right)}{\frac{t}{y}}\right)\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
x + \left(\frac{z - x}{\frac{t}{y}} + \frac{\mathsf{fma}\left(x, -1, x\right)}{\frac{t}{y}}\right)
double f(double x, double y, double z, double t) {
        double r14114745 = x;
        double r14114746 = y;
        double r14114747 = z;
        double r14114748 = r14114747 - r14114745;
        double r14114749 = r14114746 * r14114748;
        double r14114750 = t;
        double r14114751 = r14114749 / r14114750;
        double r14114752 = r14114745 + r14114751;
        return r14114752;
}
↓
double f(double x, double y, double z, double t) {
        double r14114753 = x;
        double r14114754 = z;
        double r14114755 = r14114754 - r14114753;
        double r14114756 = t;
        double r14114757 = y;
        double r14114758 = r14114756 / r14114757;
        double r14114759 = r14114755 / r14114758;
        double r14114760 = -1.0;
        double r14114761 = fma(r14114753, r14114760, r14114753);
        double r14114762 = r14114761 / r14114758;
        double r14114763 = r14114759 + r14114762;
        double r14114764 = r14114753 + r14114763;
        return r14114764;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.1
Herbie2.1
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Initial program 6.7

    \[x + \frac{y \cdot \left(z - x\right)}{t}\]

  2. Simplified6.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)}\]
  3. Using strategy rm

  4. Applied clear-num6.8

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t}{z - x}}}, y, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef6.8

    \[\leadsto \color{blue}{\frac{1}{\frac{t}{z - x}} \cdot y + x}\]

  7. Simplified2.1

    \[\leadsto \color{blue}{\frac{y}{t} \cdot \left(z - x\right)} + x\]
  8. Using strategy rm

  9. Applied add-cube-cbrt2.3

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

  10. Applied add-sqr-sqrt32.8

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

  11. Applied prod-diff32.8

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

  12. Applied distribute-lft-in32.8

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

  13. Simplified2.2

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

  14. Simplified2.1

    \[\leadsto \left(\frac{z - x}{\frac{t}{y}} + \color{blue}{\frac{\mathsf{fma}\left(x, -1, x\right)}{\frac{t}{y}}}\right) + x\]

  15. Final simplification2.1

    \[\leadsto x + \left(\frac{z - x}{\frac{t}{y}} + \frac{\mathsf{fma}\left(x, -1, x\right)}{\frac{t}{y}}\right)\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))