Average Error: 16.6 → 7.2

Time: 14.8s

Precision: binary64

Cost: 7813

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2.4068459753558044 \cdot 10^{-28}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1.4761138301972 \cdot 10^{-313}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4.834740908361885 \cdot 10^{+299}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

\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 -\infty:\\
\;\;\;\;\frac{z}{b}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2.4068459753558044 \cdot 10^{-28}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{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 -1.4761138301972 \cdot 10^{-313}:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{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{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4.834740908361885 \cdot 10^{+299}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{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))) (- INFINITY))
   (/ z b)
   (if (<=
        (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
        -2.4068459753558044e-28)
     (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
     (if (<=
          (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
          -1.4761138301972e-313)
       (/ (+ x (* y (/ z t))) (+ (+ a 1.0) (/ (* y b) t)))
       (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) 0.0)
         (/ (* y z) (+ (* y b) (* t (+ a 1.0))))
         (if (<=
              (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))
              4.834740908361885e+299)
           (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) 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))) <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= -2.4068459753558044e-28) {
		tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= -1.4761138301972e-313) {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + ((y * b) / t));
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= 0.0) {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	} else if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= 4.834740908361885e+299) {
		tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	16.6
Target	13.1
Herbie	7.2

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

Alternatives

Alternative 1
Error	8.0
Cost	5251

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1.4761138301972 \cdot 10^{-313}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{\frac{t}{y}}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4.834740908361885 \cdot 10^{+299}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Alternative 2
Error	8.0
Cost	5123

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1.4761138301972 \cdot 10^{-313}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{\frac{t}{y}}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4.834740908361885 \cdot 10^{+299}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Alternative 3
Error	14.7
Cost	1416

\[\begin{array}{l} \mathbf{if}\;t \leq -5.169302189694269 \cdot 10^{-164} \lor \neg \left(t \leq 2.9792555516960034 \cdot 10^{-256}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Alternative 4
Error	23.4
Cost	3079

\[\begin{array}{l} \mathbf{if}\;t \leq -2.354572144667007 \cdot 10^{+107}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq -2.736218936347212 \cdot 10^{+95}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -1.0101836680066091 \cdot 10^{-36}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq -6.743397496958266 \cdot 10^{-116}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \leq 8.407808272563171 \cdot 10^{-255}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 5.371074682355507 \cdot 10^{-18}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 4.9909939410880975 \cdot 10^{+115}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \end{array}\]

Alternative 5
Error	22.7
Cost	1802

Alternative 6
Error	23.5
Cost	1297

\[\begin{array}{l} \mathbf{if}\;t \leq -2.354572144667007 \cdot 10^{+107} \lor \neg \left(t \leq -2.736218936347212 \cdot 10^{+95} \lor \neg \left(t \leq -2.547731395923243 \cdot 10^{-36}\right) \land t \leq 5.1518122121940744 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Alternative 7
Error	27.9
Cost	913

\[\begin{array}{l} \mathbf{if}\;t \leq -2.354572144667007 \cdot 10^{+107} \lor \neg \left(t \leq -2.736218936347212 \cdot 10^{+95} \lor \neg \left(t \leq -5.472858635345685 \cdot 10^{-36}\right) \land t \leq 2.390549053795124 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Alternative 8
Error	42.9
Cost	192

\[\frac{z}{b}\]

Alternative 9
Error	55.8
Cost	64

\[0\]

Alternative 10
Error	61.7
Cost	64

\[1\]

Derivation

Split input into 4 regimes
if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 4.83474090836188466e299 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))
1. Initial program 63.4
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Taylor expanded around inf 17.9
  \[\leadsto \color{blue}{\frac{z}{b}}\]
3. Simplified17.9
  \[\leadsto \color{blue}{\frac{z}{b}}\]
if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -2.40684597535580442e-28 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 4.83474090836188466e299
1. Initial program 0.3
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Simplified0.3
  \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\]
if -2.40684597535580442e-28 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.4761138302e-313
1. Initial program 0.7
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Using strategy rm
3. Applied *-un-lft-identity_binary64_195150.7
  \[\leadsto \frac{x + \frac{y \cdot z}{\color{blue}{1 \cdot t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
4. Applied times-frac_binary64_195212.3
  \[\leadsto \frac{x + \color{blue}{\frac{y}{1} \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
5. Simplified2.3
  \[\leadsto \frac{x + \color{blue}{y} \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
6. Simplified2.3
  \[\leadsto \color{blue}{\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\]
if -1.4761138302e-313 < (/.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 div-inv_binary64_1951227.2
  \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\left(y \cdot b\right) \cdot \frac{1}{t}}}\]
4. Taylor expanded around 0 27.5
  \[\leadsto \color{blue}{\frac{z \cdot y}{t \cdot \left(a + \left(\frac{y \cdot b}{t} + 1\right)\right)}}\]
5. Simplified20.5
  \[\leadsto \color{blue}{\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}}\]
6. Simplified20.5
  \[\leadsto \color{blue}{\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}}\]
Recombined 4 regimes into one program.
Final simplification7.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2.4068459753558044 \cdot 10^{-28}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1.4761138301972 \cdot 10^{-313}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4.834740908361885 \cdot 10^{+299}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))

Error

Try it out

Target

Alternatives

Error

Derivation

`if (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -inf.0 or 4.83474090836188466e299 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t)))`

`if -inf.0 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -2.40684597535580442e-28 or -0.0 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < 4.83474090836188466e299`

`if -2.40684597535580442e-28 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -1.4761138302e-313`

`if -1.4761138302e-313 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -0.0`

Reproduce

Error

Try it out

Target

Alternatives

Error

Derivation

if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 4.83474090836188466e299 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -2.40684597535580442e-28 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 4.83474090836188466e299

if -2.40684597535580442e-28 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.4761138302e-313

if -1.4761138302e-313 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

Reproduce

`if (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -inf.0 or 4.83474090836188466e299 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t)))`

`if -inf.0 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -2.40684597535580442e-28 or -0.0 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < 4.83474090836188466e299`

`if -2.40684597535580442e-28 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -1.4761138302e-313`

`if -1.4761138302e-313 < (/.f64 (+.f64 x (/.f64 (.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (.f64 y b) t))) < -0.0`