Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 21.3s

Precision: binary64

Cost: 20544

Math

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

↓

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

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

Target

Original	0.3
Target	0.3
Herbie	0.3

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

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

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

   [Start] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 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_best_oopsla_all_46_json_45_simplify-108 [=>] 0.3	\[ \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) + \left(a - 0.5\right) \cdot \log t\right) - t} \]

3. Applied egg-rr 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) + \left(a \cdot \left(\left(a + -0.5\right) \cdot \frac{\log t}{a + -0.5}\right) - \left(\left(a + -0.5\right) \cdot \frac{\log t}{a + -0.5}\right) \cdot 0.5\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) + \color{blue}{\left(\left(a + -0.5\right) \cdot \frac{\log t}{a + -0.5}\right) \cdot \left(a - 0.5\right)}\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) + \color{blue}{\left(a - 0.5\right) \cdot \left(\left(a + -0.5\right) \cdot \frac{\log t}{a + -0.5}\right)}\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{\left(-0.5 + a\right)} \cdot \frac{\log t}{a + -0.5}\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) + \left(a - 0.5\right) \cdot \left(\left(-0.5 + a\right) \cdot \frac{\log t}{\color{blue}{-0.5 + a}}\right)\right) - t \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	12.4
Cost	20040

\[\begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -2.8:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.3
Cost	20040

\[\begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log t \cdot a + \log y\right)\right) - t\\ \mathbf{elif}\;a \leq 1.1:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 3
Error	16.4
Cost	20040

\[\begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log t \cdot a + \log y\right)\right) - t\\ \mathbf{elif}\;a \leq 0.95:\\ \;\;\;\;\left(\left(\log z + \log y\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 4
Error	0.3
Cost	20032

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

Alternative 5
Error	0.3
Cost	20032

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

Alternative 6
Error	0.3
Cost	20032

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

Alternative 7
Error	0.3
Cost	20032

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

Alternative 8
Error	19.7
Cost	19904

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

Alternative 9
Error	19.7
Cost	19904

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

Alternative 10
Error	19.0
Cost	13384

\[\begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -1.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.25:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	14.6
Cost	7048

\[\begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -0.39:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;\log \left(x + y\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	19.3
Cost	6984

\[\begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -0.25:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	37.8
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 0.0065:\\ \;\;\;\;\log \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14
Error	44.6
Cost	6592

\[\log y - t \]

Alternative 15
Error	39.9
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023090 
(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))))