Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.39% → 0.39%

Time: 25.1s

Precision: binary64

Cost: 20032

Math

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

↓

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

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)) (* (log t) (+ a -0.5))))

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)) + (log(t) * (a + -0.5));
}

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)) + (log(t) * (a + (-0.5d0)))
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)) + (Math.log(t) * (a + -0.5));
}

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)) + (math.log(t) * (a + -0.5))

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(log(Float64(x + y)) + Float64(log(z) - t)) + Float64(log(t) * Float64(a + -0.5)))
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)) + (log(t) * (a + -0.5));
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[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.39%
Target	0.38%
Herbie	0.39%

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

Derivation

1. Initial program 0.39

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

2. Simplified 0.39

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

   [Start] 0.39	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   associate--l+ [=>] 0.39	\[ \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   remove-double-neg [<=] 0.39	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-\left(-\left(a - 0.5\right)\right)\right)} \cdot \log t \]
   remove-double-neg [=>] 0.39	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(a - 0.5\right)} \cdot \log t \]
   sub-neg [=>] 0.39	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t \]
   metadata-eval [=>] 0.39	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + \color{blue}{-0.5}\right) \cdot \log t \]

3. Final simplification 0.39

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

Alternatives

Alternative 1
Error	1.73%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -1000000000 \lor \neg \left(a + -0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 2
Error	19.69%
Cost	20297

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -1000000000 \lor \neg \left(a + -0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 3
Error	19.69%
Cost	20297

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -1000000000 \lor \neg \left(a + -0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \]

Alternative 4
Error	31.58%
Cost	19904

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

Alternative 5
Error	15.09%
Cost	14152

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -8200000000:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a \leq -2.55 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{\log t \cdot \left(0.25 - a \cdot a\right)}{-0.5 - a} + \log \left(y \cdot z\right)\right) - t\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(-0.5 \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 6
Error	14.51%
Cost	13768

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_1 - t\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(-0.5 \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 7
Error	26%
Cost	13640

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.32 \cdot 10^{-61}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_1 - t\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-25}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 8
Error	14.17%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]

Alternative 9
Error	23.83%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{-70}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{-31}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 10
Error	23.48%
Cost	13576

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq 8.7 \cdot 10^{-72}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_1 - t\right)\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-31}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 11
Error	23.97%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-18} \lor \neg \left(x \leq -6.3 \cdot 10^{-65}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \end{array} \]

Alternative 12
Error	24.01%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-18}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 13
Error	26.87%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]

Alternative 14
Error	22.83%
Cost	6848

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

Alternative 15
Error	38.08%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+48}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16
Error	25.37%
Cost	6720

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

Alternative 17
Error	62.08%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023088 
(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))))