Average Error: 10.4 → 3.4
Time: 27.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\left(\left(\sqrt[3]{-y} \cdot \sqrt[3]{-y}\right) \cdot \frac{\sqrt[3]{-y}}{\frac{t}{z} - a} + \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right) + \frac{z}{t - a \cdot z} \cdot \left(\left(-y\right) + y\right)\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\left(\left(\sqrt[3]{-y} \cdot \sqrt[3]{-y}\right) \cdot \frac{\sqrt[3]{-y}}{\frac{t}{z} - a} + \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right) + \frac{z}{t - a \cdot z} \cdot \left(\left(-y\right) + y\right)
double f(double x, double y, double z, double t, double a) {
        double r473578 = x;
        double r473579 = y;
        double r473580 = z;
        double r473581 = r473579 * r473580;
        double r473582 = r473578 - r473581;
        double r473583 = t;
        double r473584 = a;
        double r473585 = r473584 * r473580;
        double r473586 = r473583 - r473585;
        double r473587 = r473582 / r473586;
        return r473587;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r473588 = y;
        double r473589 = -r473588;
        double r473590 = cbrt(r473589);
        double r473591 = r473590 * r473590;
        double r473592 = t;
        double r473593 = z;
        double r473594 = r473592 / r473593;
        double r473595 = a;
        double r473596 = r473594 - r473595;
        double r473597 = r473590 / r473596;
        double r473598 = r473591 * r473597;
        double r473599 = x;
        double r473600 = -r473593;
        double r473601 = fma(r473600, r473595, r473592);
        double r473602 = r473599 / r473601;
        double r473603 = r473598 + r473602;
        double r473604 = r473595 * r473593;
        double r473605 = r473592 - r473604;
        double r473606 = r473593 / r473605;
        double r473607 = r473589 + r473588;
        double r473608 = r473606 * r473607;
        double r473609 = r473603 + r473608;
        return r473609;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.4
Target1.8
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.4

    \[\frac{x - y \cdot z}{t - a \cdot z}\]
  2. Using strategy rm

  3. Applied div-sub10.4

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

  4. Simplified7.8

    \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{y \cdot \frac{z}{t - a \cdot z}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt8.3

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

  7. Applied prod-diff8.3

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

  8. Simplified7.8

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

  9. Simplified7.8

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

  11. Applied fma-udef7.8

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

  12. Simplified2.8

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

  14. Applied *-un-lft-identity2.8

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

  15. Applied add-cube-cbrt3.4

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

  16. Applied times-frac3.4

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

  17. Simplified3.4

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

  18. Final simplification3.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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