Average Error: 16.5 → 15.1
Time: 6.5s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \left(\sqrt[3]{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{t}}\right)}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \left(\sqrt[3]{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{t}}\right)}
double code(double x, double y, double z, double t, double a, double b) {
	return ((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t)));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return (fma((y / t), z, x) / ((a + 1.0) + ((cbrt((y / (cbrt(t) * cbrt(t)))) * cbrt((y / (cbrt(t) * cbrt(t))))) * (cbrt((y / (cbrt(t) * cbrt(t)))) * (b / cbrt(t))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.5
Target13.0
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Initial program 16.5

    \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt16.6

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

  4. Applied times-frac16.1

    \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity16.1

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

  7. Applied associate-/r*16.1

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

  8. Simplified15.0

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt15.1

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

  11. Applied associate-*l*15.1

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

  12. Final simplification15.1

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))