Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Accuracy: 99.6% → 99.6%

Time: 28.1s

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	99.6%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 99.6%

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

2. Simplified 99.6%

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

   [Start] 99.6	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   associate--l+ [=>] 99.6	\[ \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)} + \left(a - 0.5\right) \cdot \log t \]
   remove-double-neg [<=] 99.6	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-\left(-\left(a - 0.5\right)\right)\right)} \cdot \log t \]
   remove-double-neg [=>] 99.6	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(a - 0.5\right)} \cdot \log t \]
   sub-neg [=>] 99.6	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t \]
   metadata-eval [=>] 99.6	\[ \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + \color{blue}{-0.5}\right) \cdot \log t \]

3. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	92.0%
Cost	20424

\[\begin{array}{l} t_1 := \log z - t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;a + -0.5 \leq -20000:\\ \;\;\;\;\left(\log y + t_2\right) - t\\ \mathbf{elif}\;a + -0.5 \leq -0.4:\\ \;\;\;\;\left(\log \left(x + y\right) + t_1\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_2\\ \end{array} \]

Alternative 2
Accuracy	70.5%
Cost	20232

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a + -0.5 \leq -20000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a + -0.5 \leq -0.4:\\ \;\;\;\;\log y + \left(\log \left(z \cdot {t}^{-0.5}\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 3
Accuracy	92.0%
Cost	20168

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.95:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 1.15:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \left(-0.5 \cdot \log t - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 4
Accuracy	70.3%
Cost	20108

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.00024:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{-168}:\\ \;\;\;\;\log y + \left(\log \left(z \cdot {t}^{-0.5}\right) - t\right)\\ \mathbf{elif}\;a \leq 0.95:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 5
Accuracy	74.0%
Cost	20040

\[\begin{array}{l} t_1 := \log z - t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.95:\\ \;\;\;\;\left(\log y + t_2\right) - t\\ \mathbf{elif}\;a \leq 2.3:\\ \;\;\;\;-0.5 \cdot \log t + \left(t_1 + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_2\\ \end{array} \]

Alternative 6
Accuracy	68.2%
Cost	19904

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

Alternative 7
Accuracy	78.5%
Cost	14280

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -5.6 \cdot 10^{-69}:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 14500:\\ \;\;\;\;\left(\frac{\log t}{-0.5 - a} \cdot \left(0.25 - a \cdot a\right) + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 8
Accuracy	64.2%
Cost	14036

\[\begin{array}{l} t_1 := \log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.000145:\\ \;\;\;\;\left(\log y + t_2\right) - t\\ \mathbf{elif}\;a \leq -2.85 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-288}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.7 \cdot 10^{-24}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_2 - t\right)\\ \end{array} \]

Alternative 9
Accuracy	72.5%
Cost	14036

\[\begin{array}{l} t_1 := \log \left(\left(z \cdot {t}^{-0.5}\right) \cdot \left(x + y\right)\right) - t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.00028:\\ \;\;\;\;\left(\log y + t_2\right) - t\\ \mathbf{elif}\;a \leq -2.85 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-288}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-23}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_2 - t\right)\\ \end{array} \]

Alternative 10
Accuracy	64.1%
Cost	13772

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.0056:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-173}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_1 - t\right)\\ \end{array} \]

Alternative 11
Accuracy	78.3%
Cost	13768

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_1 - t\right)\\ \end{array} \]

Alternative 12
Accuracy	57.6%
Cost	13708

\[\begin{array}{l} t_1 := \left(\log y + a \cdot \log t\right) - t\\ \mathbf{if}\;a \leq -0.0039:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 10^{-177}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	64.9%
Cost	13576

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-109}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \]

Alternative 14
Accuracy	56.4%
Cost	13248

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

Alternative 15
Accuracy	59.9%
Cost	7122

\[\begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+90} \lor \neg \left(a \leq 290000000000 \lor \neg \left(a \leq 6.2 \cdot 10^{+51}\right) \land a \leq 3.6 \cdot 10^{+98}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16
Accuracy	63.3%
Cost	7121

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1650000000000:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+52} \lor \neg \left(a \leq 6 \cdot 10^{+101}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 17
Accuracy	76.6%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5 \lor \neg \left(a \leq 3.3\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 18
Accuracy	76.2%
Cost	6848

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

Alternative 19
Accuracy	36.6%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023138 
(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))))