Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 29.5s

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	0.3
Target	0.3
Herbie	0.3

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

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. Simplified 0.3

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

   [Start] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   sub-neg [=>] 0.3	\[ \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) + \left(-t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   associate-+l+ [=>] 0.3	\[ \color{blue}{\left(\log \left(x + y\right) + \log z\right) + \left(\left(-t\right) + \left(a - 0.5\right) \cdot \log t\right)} \]
   associate-+l+ [=>] 0.3	\[ \color{blue}{\log \left(x + y\right) + \left(\log z + \left(\left(-t\right) + \left(a - 0.5\right) \cdot \log t\right)\right)} \]
   associate-+l+ [<=] 0.3	\[ \log \left(x + y\right) + \color{blue}{\left(\left(\log z + \left(-t\right)\right) + \left(a - 0.5\right) \cdot \log t\right)} \]
   sub-neg [<=] 0.3	\[ \log \left(x + y\right) + \left(\color{blue}{\left(\log z - t\right)} + \left(a - 0.5\right) \cdot \log t\right) \]
   associate-+l- [=>] 0.3	\[ \log \left(x + y\right) + \color{blue}{\left(\log z - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]
   cancel-sign-sub-inv [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\left(t + \left(-\left(a - 0.5\right)\right) \cdot \log t\right)}\right) \]
   +-commutative [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\left(\left(-\left(a - 0.5\right)\right) \cdot \log t + t\right)}\right) \]
   *-commutative [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]
   fma-def [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]
   neg-sub0 [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{0 - \left(a - 0.5\right)}, t\right)\right) \]
   associate--r- [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{\left(0 - a\right) + 0.5}, t\right)\right) \]
   neg-sub0 [<=] 0.3	\[ \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{\left(-a\right)} + 0.5, t\right)\right) \]
   +-commutative [<=] 0.3	\[ \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{0.5 + \left(-a\right)}, t\right)\right) \]
   unsub-neg [=>] 0.3	\[ \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	12.5
Cost	20296

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a + -0.5 \leq -0.49998:\\ \;\;\;\;\log y + \left(\log z + \left(\log t \cdot -0.5 - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 2
Error	0.9
Cost	20036

\[\begin{array}{l} \mathbf{if}\;t \leq 350:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 3
Error	0.3
Cost	20032

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

Alternative 4
Error	12.4
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 310:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 5
Error	16.0
Cost	13904

\[\begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-194}:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-179}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 124000:\\ \;\;\;\;\log \left(\frac{z}{\sqrt{t}} \cdot \left(x + y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 6
Error	9.8
Cost	13904

\[\begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-194}:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-183}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 124000:\\ \;\;\;\;\log \left(\frac{z}{\sqrt{t}} \cdot \left(x + y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 7
Error	17.9
Cost	13777

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-260} \lor \neg \left(a \leq 3.3 \cdot 10^{-108}\right) \land a \leq 2.7 \cdot 10^{-18}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]

Alternative 8
Error	17.8
Cost	13776

\[\begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{-108}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]

Alternative 9
Error	17.6
Cost	13776

\[\begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot -0.5\\ t_2 := \log z - t\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.35 \cdot 10^{-108}:\\ \;\;\;\;\log \left(x + y\right) + t_2\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 + \log t \cdot a\\ \end{array} \]

Alternative 10
Error	17.9
Cost	13776

\[\begin{array}{l} t_1 := \log z - t\\ t_2 := \log \left(y \cdot \frac{z}{\sqrt{t}}\right) - t\\ \mathbf{if}\;a \leq -5.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-183}:\\ \;\;\;\;\log \left(x + y\right) + t_1\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 + \log t \cdot a\\ \end{array} \]

Alternative 11
Error	15.5
Cost	13772

\[\begin{array}{l} \mathbf{if}\;a \leq -110000:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 1.76 \cdot 10^{-202}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \mathbf{elif}\;a \leq 3.05 \cdot 10^{-18}:\\ \;\;\;\;\log \left(\frac{z}{\sqrt{t}} \cdot \left(x + y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 12
Error	9.7
Cost	13772

\[\begin{array}{l} \mathbf{if}\;a \leq -110000:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-202}:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot -0.5\right) - t\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-18}:\\ \;\;\;\;\log \left(\frac{z}{\sqrt{t}} \cdot \left(x + y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 13
Error	17.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 14
Error	14.9
Cost	13184

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

Alternative 15
Error	14.9
Cost	6848

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

Alternative 16
Error	24.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 17
Error	16.6
Cost	6720

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

Alternative 18
Error	40.0
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023018 
(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))))