Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Accuracy: 99.6% → 99.6%

Time: 32.5s

Precision: binary64

Cost: 26304

Math

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

↓

\[\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right) \]

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) (fma (log t) (- 0.5 a) 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)) + (log(z) - fma(log(t), (0.5 - a), t));
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

↓

function code(x, y, z, t, a)
	return Float64(log(Float64(x + y)) + Float64(log(z) - fma(log(t), Float64(0.5 - a), t)))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.6%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 99.6%

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

2. Simplified 99.6%

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

   [Start] 99.6	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   associate-+l- [=>] 99.6	\[ \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]
   associate--l+ [=>] 99.6	\[ \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]
   sub-neg [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\left(t + \left(-\left(a - 0.5\right) \cdot \log t\right)\right)}\right) \]
   +-commutative [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\left(\left(-\left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]
   associate--r+ [=>] 99.6	\[ \log \left(x + y\right) + \color{blue}{\left(\left(\log z - \left(-\left(a - 0.5\right) \cdot \log t\right)\right) - t\right)} \]
   sub-neg [=>] 99.6	\[ \log \left(x + y\right) + \color{blue}{\left(\left(\log z - \left(-\left(a - 0.5\right) \cdot \log t\right)\right) + \left(-t\right)\right)} \]
   associate-+l- [=>] 99.6	\[ \log \left(x + y\right) + \color{blue}{\left(\log z - \left(\left(-\left(a - 0.5\right) \cdot \log t\right) - \left(-t\right)\right)\right)} \]
   neg-sub0 [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \left(\color{blue}{\left(0 - \left(a - 0.5\right) \cdot \log t\right)} - \left(-t\right)\right)\right) \]
   associate--r+ [<=] 99.6	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\left(0 - \left(\left(a - 0.5\right) \cdot \log t + \left(-t\right)\right)\right)}\right) \]
   unsub-neg [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \left(0 - \color{blue}{\left(\left(a - 0.5\right) \cdot \log t - t\right)}\right)\right) \]
   associate-+l- [<=] 99.6	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\left(\left(0 - \left(a - 0.5\right) \cdot \log t\right) + t\right)}\right) \]
   neg-sub0 [<=] 99.6	\[ \log \left(x + y\right) + \left(\log z - \left(\color{blue}{\left(-\left(a - 0.5\right) \cdot \log t\right)} + t\right)\right) \]
   *-commutative [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \left(\left(-\color{blue}{\log t \cdot \left(a - 0.5\right)}\right) + t\right)\right) \]
   distribute-rgt-neg-in [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]
   fma-def [=>] 99.6	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

3. Final simplification 99.6%

   \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right) \]

Alternatives

Alternative 1
Accuracy	74.2%
Cost	20296

\[\begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -10000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 2
Accuracy	74.2%
Cost	20296

\[\begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -10000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 3
Accuracy	74.2%
Cost	20296

\[\begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -10000:\\ \;\;\;\;\left(\log y + \left(\log z + t_1\right)\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 4
Accuracy	64.3%
Cost	20233

\[\begin{array}{l} \mathbf{if}\;a - 0.5 \leq -10000 \lor \neg \left(a - 0.5 \leq -0.5\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 5
Accuracy	85.9%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;t \leq 0.42:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot a\right)\right) - t\\ \end{array} \]

Alternative 6
Accuracy	99.6%
Cost	20032

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

Alternative 7
Accuracy	68.7%
Cost	19904

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

Alternative 8
Accuracy	68.7%
Cost	19904

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

Alternative 9
Accuracy	69.8%
Cost	13904

\[\begin{array}{l} t_1 := \log \left(\left(x + y\right) \cdot \frac{z}{\sqrt{t}}\right) - t\\ t_2 := \left(\log y + \log t \cdot a\right) - t\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{-272}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	69.7%
Cost	13904

\[\begin{array}{l} t_1 := \left(\log y + \log t \cdot a\right) - t\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-303}:\\ \;\;\;\;\log \left(\frac{x + y}{\frac{\sqrt{t}}{z}}\right) - t\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-272}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-20}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \frac{z}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	56.5%
Cost	13840

\[\begin{array}{l} t_1 := \log \left(\left(y \cdot z\right) \cdot {t}^{\left(a - 0.5\right)}\right)\\ t_2 := \left(\log y + \log t \cdot a\right) - t\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-269}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Accuracy	59.3%
Cost	13840

\[\begin{array}{l} t_1 := \log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ t_2 := \left(\log y + \log t \cdot a\right) - t\\ \mathbf{if}\;a \leq -1.82 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.06 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-271}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Accuracy	60.3%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0048 \lor \neg \left(a \leq 1.85 \cdot 10^{-20}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 14
Accuracy	85.9%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a + \left(\log z - t\right)\\ \end{array} \]

Alternative 15
Accuracy	73.1%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;t \leq 19500:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a + \left(\log z - t\right)\\ \end{array} \]

Alternative 16
Accuracy	78.2%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;a \leq -0.31 \lor \neg \left(a \leq 2.25\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \]

Alternative 17
Accuracy	76.7%
Cost	13248

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

Alternative 18
Accuracy	57.2%
Cost	13248

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

Alternative 19
Accuracy	77.2%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -0.49 \lor \neg \left(a \leq 1\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 20
Accuracy	61.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -2.45 \cdot 10^{+39} \lor \neg \left(a \leq 3.2 \cdot 10^{+73}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 21
Accuracy	74.2%
Cost	6720

\[\log t \cdot a - t \]

Alternative 22
Accuracy	38.0%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023147 
(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))))