Average Error: 16.5 → 7.4
Time: 13.0s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -3.7807306835206855 \cdot 10^{-290}:\\ \;\;\;\;\frac{x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\left(\frac{t}{y} \cdot \left(\frac{x}{b} - \frac{z}{b \cdot b}\right) - \frac{t \cdot \left(z \cdot a\right)}{y \cdot \left(b \cdot b\right)}\right) + \frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.5841205928521936 \cdot 10^{+306}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]
\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 -3.7807306835206855 \cdot 10^{-290}:\\
\;\;\;\;\frac{x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}{\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:\\
\;\;\;\;\left(\frac{t}{y} \cdot \left(\frac{x}{b} - \frac{z}{b \cdot b}\right) - \frac{t \cdot \left(z \cdot a\right)}{y \cdot \left(b \cdot b\right)}\right) + \frac{z}{b}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.5841205928521936 \cdot 10^{+306}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\

\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)))
      -3.7807306835206855e-290)
   (/
    (+
     x
     (*
      (/ (* (cbrt y) (cbrt y)) (cbrt t))
      (* (/ (cbrt y) (cbrt t)) (/ z (cbrt t)))))
    (+ (+ a 1.0) (/ (* y b) t)))
   (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) 0.0)
     (+
      (- (* (/ t y) (- (/ x b) (/ z (* b b)))) (/ (* t (* z a)) (* y (* b b))))
      (/ z b))
     (if (<=
          (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
          1.5841205928521936e+306)
       (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (* (* y b) (/ 1.0 t))))
       (/ z b)))))
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))) <= -3.7807306835206855e-290) {
		tmp = (x + (((cbrt(y) * cbrt(y)) / cbrt(t)) * ((cbrt(y) / 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 = (((t / y) * ((x / b) - (z / (b * b)))) - ((t * (z * a)) / (y * (b * b)))) + (z / b);
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= 1.5841205928521936e+306) {
		tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) * (1.0 / t)));
	} else {
		tmp = z / b;
	}
	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
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

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))) < -3.7807306835206855e-290

    1. Initial program 7.1

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

    3. Applied add-cube-cbrt_binary64_171637.4

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

    4. Applied times-frac_binary64_171346.1

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

    6. Applied add-cube-cbrt_binary64_171636.2

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

    7. Applied times-frac_binary64_171346.2

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

    8. Applied associate-*l*_binary64_170695.6

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

    if -3.7807306835206855e-290 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

    1. Initial program 28.2

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

    3. Applied add-cube-cbrt_binary64_1716328.2

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

    4. Applied times-frac_binary64_1713431.0

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

    6. Applied add-cube-cbrt_binary64_1716331.0

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

    7. Applied times-frac_binary64_1713431.0

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

    8. Applied associate-*l*_binary64_1706927.9

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

    9. Simplified27.9

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

    10. Taylor expanded around -inf 27.9

      \[\leadsto \color{blue}{\left(\frac{t \cdot \left(z \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot a\right)\right)}{y \cdot {b}^{2}} + \left(\frac{t \cdot x}{y \cdot b} + \frac{t \cdot \left(z \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}{y \cdot {b}^{2}}\right)\right) - \frac{z \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{b}}\]

    11. Simplified21.4

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

    if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.5841205928521936e306

    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_171250.5

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

    if 1.5841205928521936e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

    1. Initial program 63.8

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

    2. Taylor expanded around inf 11.8

      \[\leadsto \color{blue}{\frac{z}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -3.7807306835206855 \cdot 10^{-290}:\\ \;\;\;\;\frac{x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\left(\frac{t}{y} \cdot \left(\frac{x}{b} - \frac{z}{b \cdot b}\right) - \frac{t \cdot \left(z \cdot a\right)}{y \cdot \left(b \cdot b\right)}\right) + \frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.5841205928521936 \cdot 10^{+306}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))