Average Error: 2.0 → 0.5

Time: 19.6s

Precision: binary64

Cost: 13632

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

↓

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

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

↓

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

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

↓

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

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

↓

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

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	1.2
Cost	65280

\[\left(\sqrt[3]{x} \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\]

Alternative 2
Error	1.1
Cost	58880

\[\left(\sqrt[3]{x} \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}\right)\]

Alternative 3
Error	1.2
Cost	52352

\[\left(\sqrt[3]{x} \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5}\right)\]

Alternative 4
Error	25.2
Cost	45568

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

Alternative 5
Error	10.0
Cost	40128

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

Alternative 6
Error	1.9
Cost	39616

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

Alternative 7
Error	28.3
Cost	39296

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

Alternative 8
Error	0.9
Cost	33088

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

Alternative 9
Error	0.9
Cost	33024

\[{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(\sqrt[3]{x} \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}\right)\]

Alternative 10
Error	34.0
Cost	26560

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

Alternative 11
Error	23.2
Cost	26432

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

Alternative 12
Error	25.0
Cost	20288

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

Alternative 13
Error	2.0
Cost	20160

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

Alternative 14
Error	10.2
Cost	13504

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

Alternative 15
Error	2.3
Cost	13504

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

Alternative 16
Error	29.7
Cost	13312

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

Alternative 17
Error	19.9
Cost	13184

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

Alternative 18
Error	7.0
Cost	7168

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

Alternative 19
Error	20.7
Cost	6912

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

Alternative 20
Error	61.9
Cost	64

\[1\]

Alternative 21
Error	18.4
Cost	64

\[0\]

Alternative 22
Error	61.9
Cost	64

\[-1\]

Error

Derivation

Initial program 2.0
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
Taylor expanded around 0 0.5
\[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{-1 \cdot z} - b\right)}\]
Simplified0.5
\[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(-z\right)} - b\right)}\]
Using strategy rm
Applied add-exp-log_binary64_31060.5
\[\leadsto x \cdot \color{blue}{e^{\log \left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(-z\right) - b\right)}\right)}}\]
Simplified0.5
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}}\]
Simplified0.5
\[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}}\]
Final simplification0.5
\[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))