Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.41% → 0.44%

Time: 26.1s

Precision: binary64

Cost: 26560

Math

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

↓

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

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 (+ y x)) t) (* 3.0 (log (cbrt z)))) (* (- a 0.5) (log 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((y + x)) - t) + (3.0 * log(cbrt(z)))) + ((a - 0.5) * log(t));
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((y + x)) - t) + (3.0 * Math.log(Math.cbrt(z)))) + ((a - 0.5) * Math.log(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(Float64(Float64(log(Float64(y + x)) - t) + Float64(3.0 * log(cbrt(z)))) + Float64(Float64(a - 0.5) * log(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[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(3.0 * N[Log[N[Power[z, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.41%
Target	0.41%
Herbie	0.44%

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

Derivation

1. Initial program 0.41

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

2. Applied egg-rr 0.43

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

3. Simplified 0.44

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

   [Start] 0.43	\[ \left(\log \left({\left(\sqrt[3]{z}\right)}^{2}\right) + \left(\log \left(\sqrt[3]{z}\right) + \left(\log \left(x + y\right) - t\right)\right)\right) + \left(a - 0.5\right) \cdot \log t \]
   associate-+r+ [=>] 0.44	\[ \color{blue}{\left(\left(\log \left({\left(\sqrt[3]{z}\right)}^{2}\right) + \log \left(\sqrt[3]{z}\right)\right) + \left(\log \left(x + y\right) - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   +-commutative [=>] 0.44	\[ \color{blue}{\left(\left(\log \left(x + y\right) - t\right) + \left(\log \left({\left(\sqrt[3]{z}\right)}^{2}\right) + \log \left(\sqrt[3]{z}\right)\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   +-commutative [=>] 0.44	\[ \left(\left(\log \color{blue}{\left(y + x\right)} - t\right) + \left(\log \left({\left(\sqrt[3]{z}\right)}^{2}\right) + \log \left(\sqrt[3]{z}\right)\right)\right) + \left(a - 0.5\right) \cdot \log t \]
   log-pow [=>] 0.44	\[ \left(\left(\log \left(y + x\right) - t\right) + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{z}\right)} + \log \left(\sqrt[3]{z}\right)\right)\right) + \left(a - 0.5\right) \cdot \log t \]
   distribute-lft1-in [=>] 0.44	\[ \left(\left(\log \left(y + x\right) - t\right) + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{z}\right)}\right) + \left(a - 0.5\right) \cdot \log t \]
   metadata-eval [=>] 0.44	\[ \left(\left(\log \left(y + x\right) - t\right) + \color{blue}{3} \cdot \log \left(\sqrt[3]{z}\right)\right) + \left(a - 0.5\right) \cdot \log t \]

4. Final simplification 0.44

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

Alternatives

Alternative 1
Error	0.41%
Cost	26304

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

Alternative 2
Error	7.79%
Cost	20424

\[\begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\log \left(y + x\right) + \left(\left(\log z + \log t \cdot -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 3
Error	26.27%
Cost	20296

\[\begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\left(\log z + \log y\right) + \left(\log t \cdot -0.5 - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 4
Error	0.41%
Cost	20032

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

Alternative 5
Error	31.49%
Cost	19904

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

Alternative 6
Error	27.16%
Cost	13768

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{-81}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;a \leq 12600:\\ \;\;\;\;\left(t_1 + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 7
Error	27.21%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-80} \lor \neg \left(a \leq 3.35 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 - t\right) + \log \left(y \cdot z\right)\\ \end{array} \]

Alternative 8
Error	25.57%
Cost	13577

\[\begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{-75} \lor \neg \left(a \leq 3.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a - 0.5\right)}\right)\right)\\ \end{array} \]

Alternative 9
Error	35.92%
Cost	13576

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{-80}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \mathbf{elif}\;a \leq 0.06:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \end{array} \]

Alternative 10
Error	25.17%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-75} \lor \neg \left(a \leq 1.55 \cdot 10^{-21}\right):\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot -0.5 + \log \left(y \cdot z\right)\\ \end{array} \]

Alternative 11
Error	25.27%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-76} \lor \neg \left(a \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot -0.5 + \log \left(y \cdot z\right)\\ \end{array} \]

Alternative 12
Error	38.3%
Cost	6989

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{+19} \lor \neg \left(t \leq 9 \cdot 10^{+91}\right) \land t \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13
Error	23.65%
Cost	6848

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

Alternative 14
Error	62.67%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023121 
(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))))