Average Error: 10.8 → 10.8
Time: 14.0s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x - z \cdot y}{t - a \cdot z}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x - z \cdot y}{t - a \cdot z}
double f(double x, double y, double z, double t, double a) {
        double r735208 = x;
        double r735209 = y;
        double r735210 = z;
        double r735211 = r735209 * r735210;
        double r735212 = r735208 - r735211;
        double r735213 = t;
        double r735214 = a;
        double r735215 = r735214 * r735210;
        double r735216 = r735213 - r735215;
        double r735217 = r735212 / r735216;
        return r735217;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r735218 = x;
        double r735219 = z;
        double r735220 = y;
        double r735221 = r735219 * r735220;
        double r735222 = r735218 - r735221;
        double r735223 = t;
        double r735224 = a;
        double r735225 = r735224 * r735219;
        double r735226 = r735223 - r735225;
        double r735227 = r735222 / r735226;
        return r735227;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.8
Target1.7
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.8

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

  3. Applied add-cube-cbrt11.3

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

  4. Applied fma-neg11.3

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

  5. Simplified11.3

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

  7. Applied *-un-lft-identity11.3

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

  8. Applied associate-/r*11.3

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

  9. Simplified10.8

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

  10. Final simplification10.8

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

Reproduce

herbie shell --seed 2020046 +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))))