Average Error: 4.4 → 2.8

Time: 10.4s

Precision: binary64

Cost: 3587

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.5913527548956093 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot x + \frac{x}{z} \cdot \left(t + \frac{t}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.9748042716300287 \cdot 10^{+255}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.5913527548956093 \cdot 10^{-145}:\\
\;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\

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

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.9748042716300287 \cdot 10^{+255}:\\
\;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\

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

\end{array}

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (/ y z) (/ t (- 1.0 z))) -2.5913527548956093e-145)
   (* (- (/ y z) (/ t (- 1.0 z))) x)
   (if (<= (- (/ y z) (/ t (- 1.0 z))) 0.0)
     (+ (* (/ y z) x) (* (/ x z) (+ t (/ t z))))
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 1.9748042716300287e+255)
       (* (- (/ y z) (/ t (- 1.0 z))) x)
       (/ (* x (- (* y (- 1.0 z)) (* z t))) (* z (- 1.0 z)))))))

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y / z) - (t / (1.0 - z))) <= -2.5913527548956093e-145) {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	} else if (((y / z) - (t / (1.0 - z))) <= 0.0) {
		tmp = ((y / z) * x) + ((x / z) * (t + (t / z)));
	} else if (((y / z) - (t / (1.0 - z))) <= 1.9748042716300287e+255) {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	} else {
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z));
	}
	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	4.4
Target	4.2
Herbie	2.8

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Alternatives

Alternative 1
Error	4.3
Cost	2312

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.5913527548956093 \cdot 10^{-145} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \leq 0\right):\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x + \frac{x}{z} \cdot \left(t + \frac{t}{z}\right)\\ \end{array}\]

Alternative 2
Error	4.4
Cost	704

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

Alternative 3
Error	5.1
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -0.987684137996394 \lor \neg \left(z \leq 0.006930783671939834\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array}\]

Alternative 4
Error	13.3
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2658667193930415 \cdot 10^{-35} \lor \neg \left(z \leq 0.006930783671939834\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Alternative 5
Error	13.6
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -4.1058533444088976 \cdot 10^{-74} \lor \neg \left(z \leq 13708614.872476278\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Alternative 6
Error	27.7
Cost	320

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

Alternative 7
Error	55.4
Cost	64

\[0\]

Alternative 8
Error	61.7
Cost	64

\[1\]

Derivation

Split input into 3 regimes
if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -2.59135275489560931e-145 or 0.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.9748042716300287e255
1. Initial program 2.2
  \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
2. Simplified2.2
  \[\leadsto \color{blue}{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\]
if -2.59135275489560931e-145 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 0.0
1. Initial program 8.1
  \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt_binary64_134128.3
  \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\left(\sqrt[3]{\frac{t}{1 - z}} \cdot \sqrt[3]{\frac{t}{1 - z}}\right) \cdot \sqrt[3]{\frac{t}{1 - z}}}\right)\]
4. Taylor expanded around inf 3.4
  \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(\frac{t \cdot x}{z} + \frac{t \cdot x}{{z}^{2}}\right)}\]
5. Simplified7.0
  \[\leadsto \color{blue}{x \cdot \frac{y}{z} + \frac{x}{z} \cdot \left(t + \frac{t}{z}\right)}\]
6. Simplified7.0
  \[\leadsto \color{blue}{x \cdot \frac{y}{z} + \frac{x}{z} \cdot \left(t + \frac{t}{z}\right)}\]
if 1.9748042716300287e255 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))
1. Initial program 32.0
  \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
2. Using strategy rm
3. Applied frac-sub_binary64_1338634.5
  \[\leadsto x \cdot \color{blue}{\frac{y \cdot \left(1 - z\right) - z \cdot t}{z \cdot \left(1 - z\right)}}\]
4. Applied associate-*r/_binary64_133192.9
  \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}}\]
5. Simplified2.9
  \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}}\]
Recombined 3 regimes into one program.
Final simplification2.8
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.5913527548956093 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot x + \frac{x}{z} \cdot \left(t + \frac{t}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.9748042716300287 \cdot 10^{+255}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))

Error

Try it out

Target

Alternatives

Error

Derivation

`if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -2.59135275489560931e-145 or 0.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.9748042716300287e255`

`if -2.59135275489560931e-145 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 0.0`

`if 1.9748042716300287e255 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))`

Reproduce