Average Error: 1.9 → 1.9

Time: 31.2s

Precision: binary64

Cost: 20160

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

↓

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

\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}

↓

\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

↓

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

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

↓

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

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	1.9
Target	11.0
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Alternatives

Alternative 1
Error	10.0
Cost	85376

\[\frac{x \cdot {\left(e^{\sqrt[3]{\log \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right) - b} \cdot \sqrt[3]{\log \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right) - b}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)}}{y}\]

Alternative 2
Error	23.3
Cost	79424

\[\sqrt[3]{\frac{x}{y} \cdot \left(\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot {z}^{y}\right)} \cdot \left(\sqrt[3]{\frac{x}{y} \cdot \left(\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot {z}^{y}\right)} \cdot \sqrt[3]{\frac{x}{y} \cdot \left(\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot {z}^{y}\right)}\right)\]

Alternative 3
Error	18.3
Cost	52672

\[\frac{x \cdot \left(\sqrt{\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot {z}^{y}} \cdot \sqrt{\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot {z}^{y}}\right)}{y}\]

Alternative 4
Error	1.9
Cost	20224

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

Alternative 5
Error	20.1
Cost	20160

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

Alternative 6
Error	22.5
Cost	20096

\[\frac{x \cdot \left(e^{-b} \cdot \left({z}^{y} \cdot {a}^{t}\right)\right)}{a \cdot y}\]

Alternative 7
Error	18.2
Cost	20096

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

Alternative 8
Error	18.2
Cost	20032

\[\frac{x \cdot \frac{{z}^{y} \cdot {a}^{t}}{a \cdot e^{b}}}{y}\]

Alternative 9
Error	19.1
Cost	20032

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

Alternative 10
Error	20.3
Cost	20032

\[\frac{\frac{x}{\frac{y}{{z}^{y} \cdot {a}^{t}}}}{a \cdot e^{b}}\]

Alternative 11
Error	20.4
Cost	20032

\[\frac{{z}^{y} \cdot {a}^{t}}{y} \cdot \frac{x}{a \cdot e^{b}}\]

Alternative 12
Error	10.1
Cost	20032

\[\frac{x \cdot e^{\left(y \cdot \log z + t \cdot \log a\right) - b}}{y}\]

Alternative 13
Error	18.4
Cost	20032

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

Alternative 14
Error	18.5
Cost	20032

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

Alternative 15
Error	23.2
Cost	20032

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

Alternative 16
Error	18.2
Cost	20032

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

Alternative 17
Error	13.9
Cost	19904

\[\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\]

Alternative 18
Error	20.5
Cost	13632

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

Alternative 19
Error	19.9
Cost	13568

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

Alternative 20
Error	19.9
Cost	13568

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

Alternative 21
Error	18.8
Cost	13504

\[\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot e^{-b}\right)}{y}\]

Alternative 22
Error	20.6
Cost	13504

\[\frac{x}{a \cdot \frac{y}{{z}^{y} \cdot {a}^{t}}}\]

Alternative 23
Error	19.7
Cost	13504

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

Alternative 24
Error	23.1
Cost	13440

\[\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot \frac{x}{y}\]

Alternative 25
Error	23.7
Cost	13440

\[\frac{\frac{x \cdot {z}^{y}}{e^{b}}}{a \cdot y}\]

Alternative 26
Error	20.4
Cost	13440

\[\frac{{a}^{\left(t - 1\right)} \cdot x}{e^{b} \cdot y}\]

Alternative 27
Error	23.5
Cost	13440

\[\frac{{a}^{t}}{\left(a \cdot e^{b}\right) \cdot \frac{y}{x}}\]

Alternative 28
Error	18.3
Cost	13440

\[\frac{x}{\left(a \cdot e^{b}\right) \cdot \frac{y}{{a}^{t}}}\]

Alternative 29
Error	21.2
Cost	13440

\[{a}^{t} \cdot \frac{x}{y \cdot \left(a \cdot e^{b}\right)}\]

Alternative 30
Error	21.0
Cost	13440

\[x \cdot \frac{\frac{{z}^{y}}{a}}{e^{b} \cdot y}\]

Alternative 31
Error	20.1
Cost	13440

\[\frac{x \cdot \frac{{z}^{y}}{a \cdot e^{b}}}{y}\]

Alternative 32
Error	18.8
Cost	13440

\[\frac{\frac{{a}^{\left(t - 1\right)}}{e^{b}} \cdot x}{y}\]

Alternative 33
Error	62.0
Cost	64

\[1\]

Alternative 34
Error	9.7
Cost	64

\[0\]

Alternative 35
Error	62.0
Cost	64

\[-1\]

Error

Derivation

Initial program 1.9
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
Simplified1.9
\[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\]
Final simplification1.9
\[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))