Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.2

Time: 21.7s

Precision: binary64

Cost: 20032

Math

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

↓

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

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

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) + (((a - 0.5) * log(t)) + log(z))) - t;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.2
Target	0.3
Herbie	0.2

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

Alternatives

Alternative 1
Error	36.4
Cost	93952

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

Alternative 2
Error	33.4
Cost	87104

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

Alternative 3
Error	0.5
Cost	85184

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

Alternative 4
Error	0.5
Cost	84672

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

Alternative 5
Error	1.3
Cost	79424

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

Alternative 6
Error	0.3
Cost	65856

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

Alternative 7
Error	0.3
Cost	65728

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

Alternative 8
Error	0.3
Cost	65600

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

Alternative 9
Error	31.9
Cost	65344

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

Alternative 10
Error	0.5
Cost	59328

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

Alternative 11
Error	18.5
Cost	53184

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

Alternative 12
Error	39.9
Cost	52928

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

Alternative 13
Error	56.2
Cost	52416

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

Alternative 14
Error	29.3
Cost	47296

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

Alternative 15
Error	0.3
Cost	46400

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

Alternative 16
Error	0.3
Cost	46144

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

Alternative 17
Error	22.0
Cost	46016

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

Alternative 18
Error	0.3
Cost	39744

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

Alternative 19
Error	32.1
Cost	39616

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

Alternative 20
Error	0.3
Cost	39616

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

Alternative 21
Error	0.3
Cost	39488

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

Alternative 22
Error	0.3
Cost	39488

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

Alternative 23
Error	0.3
Cost	39488

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

Alternative 24
Error	31.9
Cost	39360

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

Alternative 25
Error	0.3
Cost	39360

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

Alternative 26
Error	15.3
Cost	33216

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

Alternative 27
Error	48.8
Cost	33088

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

Alternative 28
Error	0.4
Cost	32960

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

Alternative 29
Error	0.3
Cost	32896

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

Alternative 30
Error	35.8
Cost	32896

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

Alternative 31
Error	33.2
Cost	32832

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

Alternative 32
Error	22.0
Cost	32832

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

Alternative 33
Error	41.0
Cost	32832

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

Alternative 34
Error	21.1
Cost	27136

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

Alternative 35
Error	49.8
Cost	26368

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

Alternative 36
Error	15.9
Cost	20416

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

Alternative 37
Error	0.2
Cost	20032

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

Alternative 38
Error	0.3
Cost	20032

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

Alternative 39
Error	52.9
Cost	19968

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

Alternative 40
Error	23.3
Cost	19904

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

Alternative 41
Error	19.8
Cost	19904

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

Alternative 42
Error	20.3
Cost	19904

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

Alternative 43
Error	19.8
Cost	19904

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

Alternative 44
Error	20.3
Cost	19904

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

Alternative 45
Error	15.2
Cost	13632

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

Alternative 46
Error	14.9
Cost	6848

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

Alternative 47
Error	40.1
Cost	6592

\[a \cdot \log t\]

Alternative 48
Error	39.2
Cost	128

\[-t\]

Alternative 49
Error	61.3
Cost	64

\[1\]

Alternative 50
Error	62.5
Cost	64

\[0\]

Alternative 51
Error	61.4
Cost	64

\[-1\]

Derivation

1. Initial program 0.2

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
2. Using strategy rm

3. Applied associate--l+_binary64_7667 0.2

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

4. Applied associate-+l+_binary64_7663 0.3

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

5. Simplified 0.3

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

7. Applied associate-+r-_binary64_7664 0.3

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

8. Applied associate-+r-_binary64_7664 0.2

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

9. Simplified 0.2

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

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2021042 
(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))))