Average Error: 16.5 → 14.7
Time: 12.0s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -6.733104175437884 \cdot 10^{-275}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a - 1\right) \cdot \left(y + \left(a + 1\right) \cdot \frac{t}{b}\right)} \cdot \left(\left(a - 1\right) \cdot \frac{t}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.183144001620433 \cdot 10^{+212}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -6.733104175437884 \cdot 10^{-275}:\\
\;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a - 1\right) \cdot \left(y + \left(a + 1\right) \cdot \frac{t}{b}\right)} \cdot \left(\left(a - 1\right) \cdot \frac{t}{b}\right)\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.183144001620433 \cdot 10^{+212}:\\
\;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\

\end{array}
(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
      -6.733104175437884e-275)
   (/
    (+ x (* (/ y (* (cbrt t) (cbrt t))) (/ z (cbrt t))))
    (+ (+ a 1.0) (/ (* y b) t)))
   (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) 0.0)
     (*
      (/ (+ x (* y (/ z t))) (* (- a 1.0) (+ y (* (+ a 1.0) (/ t b)))))
      (* (- a 1.0) (/ t b)))
     (if (<=
          (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
          1.183144001620433e+212)
       (/ (+ x (* (* y z) (/ 1.0 t))) (+ (+ a 1.0) (/ (* y b) t)))
       (/ (+ x (* y (/ z t))) (+ (+ a 1.0) (* b (/ y t))))))))
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) {
	double tmp;
	if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= -6.733104175437884e-275) {
		tmp = (x + ((y / (cbrt(t) * cbrt(t))) * (z / cbrt(t)))) / ((a + 1.0) + ((y * b) / t));
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= 0.0) {
		tmp = ((x + (y * (z / t))) / ((a - 1.0) * (y + ((a + 1.0) * (t / b))))) * ((a - 1.0) * (t / b));
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= 1.183144001620433e+212) {
		tmp = (x + ((y * z) * (1.0 / t))) / ((a + 1.0) + ((y * b) / t));
	} else {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (b * (y / t)));
	}
	return tmp;
}

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
Target13.2
Herbie14.7
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -6.7331041754378842e-275

    1. Initial program 8.1

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

    3. Applied add-cube-cbrt_binary64_120488.4

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

    4. Applied times-frac_binary64_120197.1

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

    if -6.7331041754378842e-275 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

    1. Initial program 27.2

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

    3. Applied *-un-lft-identity_binary64_1201327.2

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

    4. Applied times-frac_binary64_1201926.8

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

    5. Simplified26.8

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

    7. Applied associate-/l*_binary64_1195818.3

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

    9. Applied flip-+_binary64_1198724.6

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

    10. Applied frac-add_binary64_1202137.2

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

    11. Applied associate-/r/_binary64_1195931.0

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

    12. Simplified26.1

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

    if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.1831440016204329e212

    1. Initial program 0.5

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

    3. Applied div-inv_binary64_120100.5

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

    if 1.1831440016204329e212 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

    1. Initial program 52.4

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

    3. Applied *-un-lft-identity_binary64_1201352.4

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

    4. Applied times-frac_binary64_1201948.2

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

    5. Simplified48.2

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

    7. Applied associate-/l*_binary64_1195844.5

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

    9. Applied associate-/r/_binary64_1195944.6

      \[\leadsto \frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -6.733104175437884 \cdot 10^{-275}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a - 1\right) \cdot \left(y + \left(a + 1\right) \cdot \frac{t}{b}\right)} \cdot \left(\left(a - 1\right) \cdot \frac{t}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.183144001620433 \cdot 10^{+212}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))