Average Error: 0.1 → 0.1

Time: 16.5s

Precision: binary64

Cost: 26816

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]

↓

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \log \left(\sqrt{t}\right) + z \cdot \log \left(\sqrt{t}\right)\right)\right)\]

\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b

↓

\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \log \left(\sqrt{t}\right) + z \cdot \log \left(\sqrt{t}\right)\right)\right)

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (+
  (* (- a 0.5) b)
  (- (+ (+ x y) z) (+ (* z (log (sqrt t))) (* z (log (sqrt t)))))))

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return ((a - 0.5) * b) + (((x + y) + z) - ((z * log(sqrt(t))) + (z * log(sqrt(t)))));
}

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	0.1
Target	0.4
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Alternatives

Alternative 1
Error	0.1
Cost	65728

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(\log \left(\sqrt[3]{t}\right) + 2 \cdot \log \left(\sqrt[3]{\sqrt[3]{t}}\right)\right) + z \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right)\right)\right)\]

Alternative 2
Error	0.1
Cost	59328

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right) + z \cdot \left(\log \left(\sqrt[3]{t}\right) + 2 \cdot \log \left(\sqrt[3]{\sqrt[3]{t}}\right)\right)\right)\right)\]

Alternative 3
Error	1.0
Cost	40640

\[\left(a - 0.5\right) \cdot b + \sqrt[3]{\left(\left(x + y\right) + z\right) - z \cdot \log t} \cdot \left(\sqrt[3]{\left(\left(x + y\right) + z\right) - z \cdot \log t} \cdot \sqrt[3]{\left(\left(x + y\right) + z\right) - z \cdot \log t}\right)\]

Alternative 4
Error	0.3
Cost	39872

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \sqrt[3]{z \cdot \log t} \cdot \left(\sqrt[3]{z \cdot \log t} \cdot \sqrt[3]{z \cdot \log t}\right)\right)\]

Alternative 5
Error	0.3
Cost	39616

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \sqrt[3]{\log t} \cdot \left(z \cdot \left(\sqrt[3]{\log t} \cdot \sqrt[3]{\log t}\right)\right)\right)\]

Alternative 6
Error	0.5
Cost	27328

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \sqrt[3]{\left(a - 0.5\right) \cdot b} \cdot \left(\sqrt[3]{\left(a - 0.5\right) \cdot b} \cdot \sqrt[3]{\left(a - 0.5\right) \cdot b}\right)\]

Alternative 7
Error	31.8
Cost	27200

\[\left(a - 0.5\right) \cdot b + \sqrt{\left(\left(x + y\right) + z\right) - z \cdot \log t} \cdot \sqrt{\left(\left(x + y\right) + z\right) - z \cdot \log t}\]

Alternative 8
Error	0.1
Cost	26944

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(\log t \cdot 0.8333333333333334\right) + z \cdot \log \left(\sqrt{\sqrt[3]{t}}\right)\right)\right)\]

Alternative 9
Error	0.1
Cost	26944

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \log \left(\sqrt[3]{t}\right) + z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right)\right)\right)\]

Alternative 10
Error	0.3
Cost	26816

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\log t \cdot \sqrt[3]{z}\right)\right)\]

Alternative 11
Error	0.5
Cost	26816

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \sqrt[3]{b} \cdot \left(\left(a - 0.5\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\]

Alternative 12
Error	32.5
Cost	26816

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \sqrt{z \cdot \log t} \cdot \sqrt{z \cdot \log t}\right)\]

Alternative 13
Error	32.7
Cost	26688

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \sqrt{\log t} \cdot \left(z \cdot \sqrt{\log t}\right)\right)\]

Alternative 14
Error	0.1
Cost	20672

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(\log t \cdot 0.8333333333333334\right) + 0.5 \cdot \left(z \cdot \log \left(\sqrt[3]{t}\right)\right)\right)\right)\]

Alternative 15
Error	0.1
Cost	20672

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \log t \cdot \left(z \cdot 0.3333333333333333\right)\right)\right)\]

Alternative 16
Error	32.3
Cost	20544

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \sqrt{\left(a - 0.5\right) \cdot b} \cdot \sqrt{\left(a - 0.5\right) \cdot b}\]

Alternative 17
Error	0.1
Cost	20480

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - 3 \cdot \left(z \cdot \log \left(\sqrt[3]{-t} \cdot \sqrt[3]{-1}\right)\right)\right)\]

Alternative 18
Error	32.2
Cost	20288

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \sqrt{z} \cdot \left(\log t \cdot \sqrt{z}\right)\right)\]

Alternative 19
Error	45.5
Cost	20224

\[\left(a - 0.5\right) \cdot b + \sqrt[3]{{\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)}^{3}}\]

Alternative 20
Error	18.6
Cost	14464

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \frac{b \cdot \left({a}^{3} - 0.125\right)}{a \cdot a + \left(0.25 + a \cdot 0.5\right)}\]

Alternative 21
Error	0.1
Cost	14272

\[\left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(\log t \cdot 0.8333333333333334\right) + z \cdot \left(\log t \cdot 0.16666666666666666\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

Alternative 22
Error	0.1
Cost	14272

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(\log t \cdot 0.8333333333333334\right) + 0.16666666666666666 \cdot \left(z \cdot \log t\right)\right)\right)\]

Alternative 23
Error	13.9
Cost	7744

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \frac{b \cdot \left(a \cdot a - 0.25\right)}{a + 0.5}\]

Alternative 24
Error	0.1
Cost	7360

\[\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)\]

Alternative 25
Error	9.1
Cost	7232

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + b \cdot -0.5\]

Alternative 26
Error	15.6
Cost	7232

\[\left(a - 0.5\right) \cdot b + \left(x + \left(z - z \cdot \log t\right)\right)\]

Alternative 27
Error	15.6
Cost	7232

\[\left(a - 0.5\right) \cdot b + \left(y + \left(z - z \cdot \log t\right)\right)\]

Alternative 28
Error	30.9
Cost	7104

\[\left(a - 0.5\right) \cdot b + \left(z - z \cdot \log t\right)\]

Alternative 29
Error	18.5
Cost	6976

\[\left(\left(x + y\right) + z\right) - z \cdot \log t\]

Alternative 30
Error	48.5
Cost	6720

\[z - z \cdot \log t\]

Alternative 31
Error	14.9
Cost	576

\[\left(x + y\right) + \left(a - 0.5\right) \cdot b\]

Alternative 32
Error	30.1
Cost	448

\[y + \left(a - 0.5\right) \cdot b\]

Alternative 33
Error	30.1
Cost	448

\[x + \left(a - 0.5\right) \cdot b\]

Alternative 34
Error	45.1
Cost	320

\[\left(a - 0.5\right) \cdot b\]

Alternative 35
Error	54.2
Cost	192

\[a \cdot b\]

Alternative 36
Error	47.8
Cost	64

\[y\]

Alternative 37
Error	47.8
Cost	64

\[x\]

Alternative 38
Error	62.2
Cost	64

\[1\]

Alternative 39
Error	62.7
Cost	64

\[0\]

Alternative 40
Error	62.2
Cost	64

\[-1\]

Error

Derivation

Initial program 0.1
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
Using strategy rm
Applied add-sqr-sqrt_binary64_124140.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - z \cdot \log \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right) + \left(a - 0.5\right) \cdot b\]
Applied log-prod_binary64_124780.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - z \cdot \color{blue}{\left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{t}\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]
Applied distribute-rgt-in_binary64_123420.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(\log \left(\sqrt{t}\right) \cdot z + \log \left(\sqrt{t}\right) \cdot z\right)}\right) + \left(a - 0.5\right) \cdot b\]
Simplified0.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \left(\color{blue}{z \cdot \log \left(\sqrt{t}\right)} + \log \left(\sqrt{t}\right) \cdot z\right)\right) + \left(a - 0.5\right) \cdot b\]
Simplified0.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \log \left(\sqrt{t}\right) + \color{blue}{z \cdot \log \left(\sqrt{t}\right)}\right)\right) + \left(a - 0.5\right) \cdot b\]
Simplified0.1
\[\leadsto \color{blue}{\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \log \left(\sqrt{t}\right) + z \cdot \log \left(\sqrt{t}\right)\right)\right)}\]
Final simplification0.1
\[\leadsto \left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \log \left(\sqrt{t}\right) + z \cdot \log \left(\sqrt{t}\right)\right)\right)\]

Reproduce

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

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))