Average Error: 6.6 → 1.0

Time: 8.1s

Precision: binary64

Cost: 2242

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -\infty:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 1.446264612679845 \cdot 10^{+303}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \end{array}\]

x + \frac{y \cdot \left(z - x\right)}{t}

↓

\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -\infty:\\
\;\;\;\;x + y \cdot \frac{z - x}{t}\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 1.446264612679845 \cdot 10^{+303}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\

\end{array}

(FPCore (x y z t) :precision binary64 (+ x (/ (* y (- z x)) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x (/ (* y (- z x)) t)) (- INFINITY))
   (+ x (* y (/ (- z x) t)))
   (if (<= (+ x (/ (* y (- z x)) t)) 1.446264612679845e+303)
     (+ x (/ (* y (- z x)) t))
     (+ x (/ y (/ t (- z x)))))))

double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / t);
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + ((y * (z - x)) / t)) <= -((double) INFINITY)) {
		tmp = x + (y * ((z - x) / t));
	} else if ((x + ((y * (z - x)) / t)) <= 1.446264612679845e+303) {
		tmp = x + ((y * (z - x)) / t);
	} else {
		tmp = x + (y / (t / (z - x)));
	}
	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	6.6
Target	2.2
Herbie	1.0

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

Alternatives

Alternative 1
Error	1.0
Cost	1928

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -\infty \lor \neg \left(x + \frac{y \cdot \left(z - x\right)}{t} \leq 1.446264612679845 \cdot 10^{+303}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]

Alternative 2
Error	4.5
Cost	904

\[\begin{array}{l} \mathbf{if}\;y \leq -5.026611182633652 \cdot 10^{-85} \lor \neg \left(y \leq 1.111667110290758 \cdot 10^{-271}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array}\]

Alternative 3
Error	11.0
Cost	2374

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2328847118512386 \cdot 10^{-125}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq -2.2726686167378452 \cdot 10^{-163}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;z \leq -6.699490179410647 \cdot 10^{-213}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.2107636882196295 \cdot 10^{-116}:\\ \;\;\;\;x + \frac{y}{\frac{-t}{x}}\\ \mathbf{elif}\;z \leq 7.565506211054061 \cdot 10^{-79}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;z \leq 9.569566529871754 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array}\]

Alternative 4
Error	10.1
Cost	1732

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4936408961461836 \cdot 10^{-125}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq -9.218321964634582 \cdot 10^{-167}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;z \leq -6.699490179410647 \cdot 10^{-213}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.711007259671276 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array}\]

Alternative 5
Error	10.6
Cost	1732

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2328847118512386 \cdot 10^{-125}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq -3.2300503804616212 \cdot 10^{-164}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;z \leq -6.699490179410647 \cdot 10^{-213}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 5.661579578006523 \cdot 10^{-81}:\\ \;\;\;\;x - \frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array}\]

Alternative 6
Error	10.5
Cost	776

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6498297679654513 \cdot 10^{-112} \lor \neg \left(t \leq 3.4193503751268194 \cdot 10^{-144}\right):\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]

Alternative 7
Error	12.6
Cost	448

\[x + z \cdot \frac{y}{t}\]

Alternative 8
Error	28.7
Cost	1604

\[\begin{array}{l} \mathbf{if}\;t \leq -3.070535798668593 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.747959266083649 \cdot 10^{-137}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.1501813924542043 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.783364133062069 \cdot 10^{-08}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 9
Error	28.9
Cost	1101

\[\begin{array}{l} \mathbf{if}\;t \leq -4.77956212560203 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.8867723340850387 \cdot 10^{-139} \lor \neg \left(t \leq 2.9806613133305926 \cdot 10^{-60}\right) \land t \leq 1.9006050264130434 \cdot 10^{-08}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 10
Error	31.8
Cost	64

\[x\]

Alternative 11
Error	61.8
Cost	64

\[-1\]

Alternative 12
Error	61.8
Cost	64

\[1\]

Derivation

Split input into 3 regimes
if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0
1. Initial program 64.0
  \[x + \frac{y \cdot \left(z - x\right)}{t}\]
2. Using strategy rm
3. Applied *-un-lft-identity_binary64_689864.0
  \[\leadsto x + \frac{y \cdot \left(z - x\right)}{\color{blue}{1 \cdot t}}\]
4. Applied times-frac_binary64_69040.2
  \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - x}{t}}\]
5. Simplified0.2
  \[\leadsto x + \color{blue}{y} \cdot \frac{z - x}{t}\]
6. Simplified0.2
  \[\leadsto \color{blue}{x + y \cdot \frac{z - x}{t}}\]
if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1.44626461267984505e303
1. Initial program 0.9
  \[x + \frac{y \cdot \left(z - x\right)}{t}\]
2. Simplified0.9
  \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - x\right)}{t}}\]
if 1.44626461267984505e303 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))
1. Initial program 57.7
  \[x + \frac{y \cdot \left(z - x\right)}{t}\]
2. Using strategy rm
3. Applied associate-/l*_binary64_68433.2
  \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z - x}}}\]
4. Simplified3.2
  \[\leadsto \color{blue}{x + \frac{y}{\frac{t}{z - x}}}\]
Recombined 3 regimes into one program.
Final simplification1.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -\infty:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 1.446264612679845 \cdot 10^{+303}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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

Error

Try it out

Target

Alternatives

Error

Derivation

`if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0`

`if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1.44626461267984505e303`

`if 1.44626461267984505e303 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))`

Reproduce