Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 14.4s

Precision: binary64

Cost: 26560

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	20032

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

Alternative 2
Error	1.1
Cost	20488

\[\begin{array}{l} \mathbf{if}\;-0.5 + a \leq -26940019.480830804 \lor \neg \left(-0.5 + a \leq -0.4999974342373137\right):\\ \;\;\;\;\log t \cdot \left(-0.5 + a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot -0.5 + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)\\ \end{array}\]

Alternative 3
Error	4.2
Cost	20225

\[\begin{array}{l} \mathbf{if}\;y \leq 1.0535220162098499 \cdot 10^{-105}:\\ \;\;\;\;\left(\log z + \left(\log t \cdot \left(-0.5 + a\right) - t\right)\right) + \log x\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log t \cdot \left(-0.5 + a\right) - t\right)\right) + \log y\\ \end{array}\]

Alternative 4
Error	9.2
Cost	20225

\[\begin{array}{l} \mathbf{if}\;y \leq 4.319119741426114 \cdot 10^{-90}:\\ \;\;\;\;\left(\log z + \left(\log t \cdot \left(-0.5 + a\right) - t\right)\right) + \log x\\ \mathbf{elif}\;y \leq 8.75993593636454 \cdot 10^{+166}:\\ \;\;\;\;\log t \cdot \left(-0.5 + a\right) + \left(\log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(-0.5 + a\right) - t\\ \end{array}\]

Alternative 5
Error	9.3
Cost	14225

\[\begin{array}{l} \mathbf{if}\;a \leq -9.708605631387478 \cdot 10^{-17} \lor \neg \left(a \leq 3.8285228051860155 \cdot 10^{-137} \lor \neg \left(a \leq 5.709540516196257 \cdot 10^{-42}\right) \land a \leq 506673596.9651076\right):\\ \;\;\;\;\log t \cdot \left(-0.5 + a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(-0.5 + a\right) + \left(\log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \end{array}\]

Alternative 6
Error	14.7
Cost	6848

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

Alternative 7
Error	23.8
Cost	6920

\[\begin{array}{l} \mathbf{if}\;a \leq -2.7477081485001602 \cdot 10^{+25} \lor \neg \left(a \leq 66650640802147256\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array}\]

Alternative 8
Error	38.3
Cost	449

\[\begin{array}{l} \mathbf{if}\;t \leq 3335.9441908706367:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array}\]

Alternative 9
Error	61.2
Cost	64

\[1\]

Derivation

1. Initial program 0.3

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

2. Taylor expanded around inf 0.3

   \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z + 0.5 \cdot \log \left(\frac{1}{t}\right)\right)\right) - \left(t + \log \left(\frac{1}{t}\right) \cdot a\right)}\]

3. Simplified 0.3

   \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z + \log t \cdot -0.5\right)\right) - \left(t - a \cdot \log t\right)}\]

4. Simplified 0.3

   \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z + \log t \cdot -0.5\right)\right) - \left(t - a \cdot \log t\right)}\]

5. Final simplification 0.3

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

Reproduce

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

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

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