Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 26.2s

Precision: binary64

Cost: 26304

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

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)) (fma (+ 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)) + fma((a + -0.5), log(t), (log(z) - 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)) + fma(Float64(a + -0.5), log(t), Float64(log(z) - 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[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[z], $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%

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

   [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%	\[ \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   associate-+l+ [=>] 99.6%	\[ \color{blue}{\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)} \]
   +-commutative [=>] 99.6%	\[ \log \left(x + y\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\right)} \]
   fma-def [=>] 99.6%	\[ \log \left(x + y\right) + \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log z - t\right)} \]
   remove-double-neg [<=] 99.6%	\[ \log \left(x + y\right) + \mathsf{fma}\left(\color{blue}{-\left(-\left(a - 0.5\right)\right)}, \log t, \log z - t\right) \]
   remove-double-neg [=>] 99.6%	\[ \log \left(x + y\right) + \mathsf{fma}\left(\color{blue}{a - 0.5}, \log t, \log z - t\right) \]
   sub-neg [=>] 99.6%	\[ \log \left(x + y\right) + \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, \log t, \log z - t\right) \]
   metadata-eval [=>] 99.6%	\[ \log \left(x + y\right) + \mathsf{fma}\left(a + \color{blue}{-0.5}, \log t, \log z - t\right) \]

3. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	26304

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

Alternative 2
Accuracy	75.4%
Cost	20108

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{if}\;a \leq -1.86 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-125}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	20032

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

Alternative 4
Accuracy	80.3%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 480:\\ \;\;\;\;\log z + \left(\log t \cdot \left(a - 0.5\right) + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \end{array} \]

Alternative 5
Accuracy	80.4%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 220:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \end{array} \]

Alternative 6
Accuracy	68.8%
Cost	19904

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

Alternative 7
Accuracy	84.2%
Cost	13900

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t\\ t_2 := t_1 + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{if}\;t \leq 1.16 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-109}:\\ \;\;\;\;t_1 + \log y\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \end{array} \]

Alternative 8
Accuracy	73.6%
Cost	13708

\[\begin{array}{l} t_1 := \log \left(\left(y \cdot z\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \mathbf{if}\;t \leq 1.25 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-45}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \end{array} \]

Alternative 9
Accuracy	73.1%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-8} \lor \neg \left(a \leq 4.8 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \]

Alternative 10
Accuracy	66.2%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \end{array} \]

Alternative 11
Accuracy	73.0%
Cost	13577

\[\begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-160} \lor \neg \left(a \leq 4 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 12
Accuracy	77.2%
Cost	13184

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

Alternative 13
Accuracy	56.6%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+53}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{+72}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{t}\right) \cdot \left(-a\right)\\ \end{array} \]

Alternative 14
Accuracy	56.6%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{+54} \lor \neg \left(a \leq 7.8 \cdot 10^{+72}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 15
Accuracy	40.6%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 60000000:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16
Accuracy	62.0%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+42}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 17
Accuracy	37.4%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023171 
(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))))