Average Error: 0.1 → 0.1

Time: 17.7s

Precision: binary64

Cost: 27008

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

↓

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

\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 \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + z \cdot \log \left({t}^{0.3333333333333333}\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 (* 2.0 (log (cbrt t)))) (* z (log (pow t 0.3333333333333333)))))))

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 * (2.0 * log(cbrt(t)))) + (z * log(pow(t, 0.3333333333333333)))));
}

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	27008

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

Alternative 2
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 3
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 4
Error	0.1
Cost	7360

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

Alternative 5
Error	0.1
Cost	7360

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

Alternative 6
Error	5.3
Cost	8386

\[\begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -3.837332717043009 \cdot 10^{+75}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + \left(x + \left(z - z \cdot \log t\right)\right)\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 1.1645457480621066 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + b \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y + \left(z + \left(\left(a - 0.5\right) \cdot b - z \cdot \log t\right)\right)\\ \end{array}\]

Alternative 7
Error	4.9
Cost	7560

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2566158354924457 \cdot 10^{-58} \lor \neg \left(z \leq 1.0108056647527016 \cdot 10^{+110}\right):\\ \;\;\;\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + b \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + \left(x + y\right)\\ \end{array}\]

Alternative 8
Error	6.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1828078958522263 \cdot 10^{+23} \lor \neg \left(z \leq 2.858469858985963 \cdot 10^{+131}\right):\\ \;\;\;\;\left(\left(x + y\right) + z\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + \left(x + y\right)\\ \end{array}\]

Alternative 9
Error	11.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6333961189459529 \cdot 10^{+74} \lor \neg \left(z \leq 2.8183139587770886 \cdot 10^{+137}\right):\\ \;\;\;\;z - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + \left(x + y\right)\\ \end{array}\]

Alternative 10
Error	15.7
Cost	576

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

Alternative 11
Error	21.5
Cost	776

\[\begin{array}{l} \mathbf{if}\;x \leq -2.0494752822550678 \cdot 10^{+134} \lor \neg \left(x \leq 8.363689555113998 \cdot 10^{+79}\right):\\ \;\;\;\;\left(a - 0.5\right) \cdot b + x\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + y\\ \end{array}\]

Alternative 12
Error	24.1
Cost	1090

\[\begin{array}{l} \mathbf{if}\;x \leq -1.224134889511494 \cdot 10^{+192}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8625377596714915 \cdot 10^{+147}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 13
Error	38.6
Cost	3851

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1781902181087815 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.44107789962921 \cdot 10^{+17}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -7.049875901991692 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.8227182944494455 \cdot 10^{-209}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{elif}\;x \leq 1.730650257365584 \cdot 10^{-171}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 6.812370483895836 \cdot 10^{-50}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{elif}\;x \leq 307979719320171:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 2.2205578951413834 \cdot 10^{+59}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{elif}\;x \leq 1.6710165652471078 \cdot 10^{+71}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 2.1575185153870931 \cdot 10^{+89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8656966078373864 \cdot 10^{+150}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 14
Error	39.2
Cost	706

\[\begin{array}{l} \mathbf{if}\;x \leq -4.594492640159438 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8962071903446619 \cdot 10^{+80}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 15
Error	48.2
Cost	64

\[y\]

Alternative 16
Error	62.2
Cost	64

\[-1\]

Alternative 17
Error	62.2
Cost	64

\[1\]

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-cube-cbrt_binary64_120480.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - z \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}\right) + \left(a - 0.5\right) \cdot b\]
Applied log-prod_binary64_120990.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - z \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]
Applied distribute-rgt-in_binary64_119630.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot z + \log \left(\sqrt[3]{t}\right) \cdot z\right)}\right) + \left(a - 0.5\right) \cdot b\]
Simplified0.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \left(\color{blue}{z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right)} + \log \left(\sqrt[3]{t}\right) \cdot z\right)\right) + \left(a - 0.5\right) \cdot b\]
Simplified0.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \color{blue}{z \cdot \log \left(\sqrt[3]{t}\right)}\right)\right) + \left(a - 0.5\right) \cdot b\]
Using strategy rm
Applied pow1/3_binary64_120950.1
\[\leadsto \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + z \cdot \log \color{blue}{\left({t}^{0.3333333333333333}\right)}\right)\right) + \left(a - 0.5\right) \cdot b\]
Simplified0.1
\[\leadsto \color{blue}{\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + z \cdot \log \left({t}^{0.3333333333333333}\right)\right)\right)}\]
Final simplification0.1
\[\leadsto \left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + z \cdot \log \left({t}^{0.3333333333333333}\right)\right)\right)\]

Reproduce

herbie shell --seed 2021044 
(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)))