Average Error: 10.7 → 3.6
Time: 6.9s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{\frac{y}{\sqrt[3]{\frac{t}{z} - a} \cdot \sqrt[3]{\frac{t}{z} - a}}}{\sqrt[3]{\frac{t}{z} - a}}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{\frac{y}{\sqrt[3]{\frac{t}{z} - a} \cdot \sqrt[3]{\frac{t}{z} - a}}}{\sqrt[3]{\frac{t}{z} - a}}
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x - ((double) (y * z)))) / ((double) (t - ((double) (a * z))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x / ((double) (t - ((double) (a * z)))))) - ((double) (((double) (y / ((double) (((double) cbrt(((double) (((double) (t / z)) - a)))) * ((double) cbrt(((double) (((double) (t / z)) - a)))))))) / ((double) cbrt(((double) (((double) (t / z)) - a))))))));
}

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

Original10.7
Target1.9
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.7

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

  3. Applied div-sub10.7

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

  5. Applied associate-/l*8.1

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

  7. Applied div-sub8.1

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

  8. Simplified3.1

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

  10. Applied add-cube-cbrt3.6

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

  11. Applied associate-/r*3.6

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

  12. Final simplification3.6

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{\frac{y}{\sqrt[3]{\frac{t}{z} - a} \cdot \sqrt[3]{\frac{t}{z} - a}}}{\sqrt[3]{\frac{t}{z} - a}}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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