Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.5

Time: 24.4s

Precision: binary64

Cost: 19904

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

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
 (- (+ (* (+ a -0.5) (log t)) (+ (log z) (log y))) 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 (((a + -0.5) * log(t)) + (log(z) + log(y))) - 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 = (((a + (-0.5d0)) * log(t)) + (log(z) + log(y))) - 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 (((a + -0.5) * Math.log(t)) + (Math.log(z) + Math.log(y))) - 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 (((a + -0.5) * math.log(t)) + (math.log(z) + math.log(y))) - 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(Float64(a + -0.5) * log(t)) + Float64(log(z) + log(y))) - 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 = (((a + -0.5) * log(t)) + (log(z) + log(y))) - 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[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Target

Original	0.3
Target	0.3
Herbie	0.5

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

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

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

   [Start] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   associate--l+ [=>] 0.3	\[ \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   remove-double-neg [<=] 0.3	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-\left(-\left(a - 0.5\right)\right)\right)} \cdot \log t \]
   remove-double-neg [=>] 0.3	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(a - 0.5\right)} \cdot \log t \]
   sub-neg [=>] 0.3	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t \]
   metadata-eval [=>] 0.3	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + \color{blue}{-0.5}\right) \cdot \log t \]

3. Taylor expanded in x around 0 0.5

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

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.1
Cost	20297

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -5 \lor \neg \left(a + -0.5 \leq -0.4\right):\\ \;\;\;\;\log \left(y + x\right) + \left(a \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 2
Error	8.7
Cost	20036

\[\begin{array}{l} \mathbf{if}\;\log z \leq 176:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 3
Error	1.2
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 17.5:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 4
Error	0.5
Cost	19904

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

Alternative 5
Error	15.2
Cost	13776

\[\begin{array}{l} t_1 := \log t \cdot -0.5 + \log \left(z \cdot y\right)\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{-206}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 6
Error	8.2
Cost	13640

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\log \left(y + x\right) + \left(t_1 - t\right)\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot -0.5 - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 7
Error	9.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 8
Error	24.3
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -23000000 \lor \neg \left(a \leq 2.9 \cdot 10^{+52}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9
Error	14.7
Cost	6848

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

Alternative 10
Error	38.1
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 150:\\ \;\;\;\;\log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11
Error	16.4
Cost	6720

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

Alternative 12
Error	39.8
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2022364 
(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))))