Average Error: 2.0 → 0.1

Time: 12.0s

Precision: binary64

Cost: 2754

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -3.974157082765867 \cdot 10^{-28}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{\left(t - z\right) + 1}\\ \mathbf{elif}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 1.2519181265166545 \cdot 10^{-136}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \end{array}\]

x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}

↓

\begin{array}{l}
\mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -3.974157082765867 \cdot 10^{-28}:\\
\;\;\;\;x - \left(y - z\right) \cdot \frac{a}{\left(t - z\right) + 1}\\

\mathbf{elif}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 1.2519181265166545 \cdot 10^{-136}:\\
\;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) + 1}\\

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

\end{array}

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- y z) (/ (+ (- t z) 1.0) a)) -3.974157082765867e-28)
   (- x (* (- y z) (/ a (+ (- t z) 1.0))))
   (if (<= (/ (- y z) (/ (+ (- t z) 1.0) a)) 1.2519181265166545e-136)
     (- x (/ (* (- y z) a) (+ (- t z) 1.0)))
     (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))))

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y - z) / (((t - z) + 1.0) / a)) <= -3.974157082765867e-28) {
		tmp = x - ((y - z) * (a / ((t - z) + 1.0)));
	} else if (((y - z) / (((t - z) + 1.0) / a)) <= 1.2519181265166545e-136) {
		tmp = x - (((y - z) * a) / ((t - z) + 1.0));
	} else {
		tmp = x - ((y - z) / (((t - z) + 1.0) / a));
	}
	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	2.0
Target	0.3
Herbie	0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	832

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

Alternative 2
Error	0.1
Cost	1160

\[\begin{array}{l} \mathbf{if}\;a \leq -3.2787870391267375 \cdot 10^{-28} \lor \neg \left(a \leq 7.330490594671804 \cdot 10^{+27}\right):\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) + 1}\\ \end{array}\]

Alternative 3
Error	1.8
Cost	832

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

Alternative 4
Error	6.5
Cost	1032

\[\begin{array}{l} \mathbf{if}\;t \leq -3.0008996469707045 \cdot 10^{+53} \lor \neg \left(t \leq 8.940082290143948 \cdot 10^{+53}\right):\\ \;\;\;\;x - a \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{1 - z}{a}}\\ \end{array}\]

Alternative 5
Error	4.6
Cost	1032

\[\begin{array}{l} \mathbf{if}\;z \leq -2.307721591156877 \lor \neg \left(z \leq 1.580893112046276 \cdot 10^{+43}\right):\\ \;\;\;\;x + a \cdot \frac{z}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{t + 1}{a}}\\ \end{array}\]

Alternative 6
Error	6.6
Cost	1032

\[\begin{array}{l} \mathbf{if}\;y \leq -7.122020901505912 \cdot 10^{+41} \lor \neg \left(y \leq 9.41766290371948 \cdot 10^{+34}\right):\\ \;\;\;\;x - a \cdot \frac{y}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \frac{z}{\left(t - z\right) + 1}\\ \end{array}\]

Alternative 7
Error	8.2
Cost	1032

\[\begin{array}{l} \mathbf{if}\;z \leq -2.690221667062591 \lor \neg \left(z \leq 1.1992523885593028 \cdot 10^{+46}\right):\\ \;\;\;\;x - \frac{y - z}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{\left(t - z\right) + 1}\\ \end{array}\]

Alternative 8
Error	8.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2.690221667062591 \lor \neg \left(z \leq 1539779437.6810768\right):\\ \;\;\;\;x - \frac{y - z}{\frac{-z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array}\]

Alternative 9
Error	8.3
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -2.690221667062591 \lor \neg \left(z \leq 1367964318.0174818\right):\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{a}{z}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array}\]

Alternative 10
Error	9.5
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -2.690221667062591 \lor \neg \left(z \leq 4.2220703348072915 \cdot 10^{+42}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array}\]

Alternative 11
Error	23.3
Cost	192

\[x - a\]

Alternative 12
Error	61.8
Cost	64

\[-1\]

Alternative 13
Error	61.8
Cost	64

\[1\]

Derivation

Split input into 3 regimes
if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)) < -3.9741570827658671e-28
1. Initial program 0.2
  \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
2. Using strategy rm
3. Applied associate-/r/_binary64_167330.1
  \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\]
4. Using strategy rm
5. Applied div-inv_binary64_167840.2
  \[\leadsto x - \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{\left(t - z\right) + 1}\right)} \cdot a\]
6. Applied associate-*l*_binary64_167280.2
  \[\leadsto x - \color{blue}{\left(y - z\right) \cdot \left(\frac{1}{\left(t - z\right) + 1} \cdot a\right)}\]
7. Simplified0.1
  \[\leadsto x - \left(y - z\right) \cdot \color{blue}{\frac{a}{\left(t - z\right) + 1}}\]
8. Simplified0.1
  \[\leadsto \color{blue}{x - \left(y - z\right) \cdot \frac{a}{\left(t - z\right) + 1}}\]
if -3.9741570827658671e-28 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)) < 1.25191812651665447e-136
1. Initial program 4.2
  \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
2. Using strategy rm
3. Applied associate-/r/_binary64_167330.3
  \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a}\]
4. Using strategy rm
5. Applied associate-*l/_binary64_167300.0
  \[\leadsto x - \color{blue}{\frac{\left(y - z\right) \cdot a}{\left(t - z\right) + 1}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{x - \frac{a \cdot \left(y - z\right)}{\left(t - z\right) + 1}}\]
if 1.25191812651665447e-136 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a))
1. Initial program 0.1
  \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
2. Simplified0.1
  \[\leadsto \color{blue}{x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}}\]
Recombined 3 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -3.974157082765867 \cdot 10^{-28}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{\left(t - z\right) + 1}\\ \mathbf{elif}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 1.2519181265166545 \cdot 10^{-136}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

Error

Try it out

Target

Alternatives

Error

Derivation

`if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)) < -3.9741570827658671e-28`

`if -3.9741570827658671e-28 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)) < 1.25191812651665447e-136`

`if 1.25191812651665447e-136 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a))`

Reproduce