Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.2

Time: 22.7s

Precision: binary64

Cost: 20032

Math

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

↓

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

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = log((x + y)) + ((log(z) - t) + ((a - 0.5d0) * log(t)))
end function

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((x + y)) + ((Math.log(z) - t) + ((a - 0.5) * Math.log(t)));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

↓

def code(x, y, z, t, a):
	return math.log((x + y)) + ((math.log(z) - t) + ((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(log(Float64(x + y)) + Float64(Float64(log(z) - t) + Float64(Float64(a - 0.5) * log(t))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

↓

function tmp = code(x, y, z, t, a)
	tmp = log((x + y)) + ((log(z) - t) + ((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[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.2
Herbie	0.2

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

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 0.2

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

   [Start] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   rational.json-simplify-1 [=>] 0.3	\[ \color{blue}{\left(a - 0.5\right) \cdot \log t + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]
   rational.json-simplify-1 [=>] 0.3	\[ \left(a - 0.5\right) \cdot \log t + \left(\color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) \]
   rational.json-simplify-48 [=>] 0.3	\[ \left(a - 0.5\right) \cdot \log t + \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} \]
   rational.json-simplify-41 [=>] 0.2	\[ \color{blue}{\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	12.5
Cost	20296

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -4000:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a - 0.5 \leq 0.2:\\ \;\;\;\;\log y + \left(\log z + \left(\log t \cdot -0.5 - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + \left(t_1 + -1\right)\\ \end{array} \]

Alternative 2
Error	16.5
Cost	20296

\[\begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4000:\\ \;\;\;\;\left(\log z + \left(\log y + \log t \cdot a\right)\right) - t\\ \mathbf{elif}\;a - 0.5 \leq 0.2:\\ \;\;\;\;\log y + \left(\log z + \left(\log t \cdot -0.5 - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + \left(a \cdot \log t + -1\right)\\ \end{array} \]

Alternative 3
Error	19.3
Cost	20176

\[\begin{array}{l} t_1 := \log y + \left(\log z + -0.5 \cdot \log t\right)\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	19.9
Cost	19904

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

Alternative 5
Error	18.8
Cost	13384

\[\begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -90:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.9:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.9
Cost	6988

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;t \leq 510000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 7
Error	16.3
Cost	6720

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

Alternative 8
Error	39.9
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023074 
(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))))