Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Accuracy: 99.6% → 99.6%

Time: 27.5s

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	98.3%
Cost	20424

\[\begin{array}{l} t_1 := \log z - t\\ \mathbf{if}\;a + -0.5 \leq -200000000000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, t_1\right)\\ \mathbf{elif}\;a + -0.5 \leq -0.4995:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 + a \cdot \log t\\ \end{array} \]

Alternative 2
Accuracy	80.6%
Cost	20296

\[\begin{array}{l} t_1 := \log z - t\\ \mathbf{if}\;a + -0.5 \leq -200000000000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, t_1\right)\\ \mathbf{elif}\;a + -0.5 \leq -0.4995:\\ \;\;\;\;\left(\log z + \log y\right) + \left(-0.5 \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + a \cdot \log t\\ \end{array} \]

Alternative 3
Accuracy	86.8%
Cost	19652

\[\begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right)\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{elif}\;a \leq 25000:\\ \;\;\;\;t_1 + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 4
Accuracy	86.8%
Cost	13896

\[\begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right)\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{-31}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;a \leq 27000:\\ \;\;\;\;t_1 + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 5
Accuracy	73.9%
Cost	13776

\[\begin{array}{l} t_1 := -0.5 \cdot \log t + \log \left(y \cdot z\right)\\ t_2 := \log t \cdot \left(a + -0.5\right) - t\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.22 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	86.5%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-30} \lor \neg \left(a \leq 7.4 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \end{array} \]

Alternative 7
Accuracy	73.8%
Cost	13768

\[\begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right) - t\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;a \leq 29000:\\ \;\;\;\;t_1 + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	73.8%
Cost	13768

\[\begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right)\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;a \leq 25000:\\ \;\;\;\;\left(t_1 + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 9
Accuracy	73.9%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-30} \lor \neg \left(a \leq 8 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log t + \left(\log \left(y \cdot z\right) - t\right)\\ \end{array} \]

Alternative 10
Accuracy	76.7%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-227} \lor \neg \left(a \leq -2.3 \cdot 10^{-257}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ \end{array} \]

Alternative 11
Accuracy	60.9%
Cost	6989

\[\begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{+16} \lor \neg \left(t \leq 6.2 \cdot 10^{+68}\right) \land t \leq 3.8 \cdot 10^{+117}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12
Accuracy	65.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+16} \lor \neg \left(a \leq 1.6 \cdot 10^{+64}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 13
Accuracy	64.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+14} \lor \neg \left(a \leq 1.3 \cdot 10^{+66}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \]

Alternative 14
Accuracy	77.4%
Cost	6848

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

Alternative 15
Accuracy	41.5%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;t \leq 0.00079:\\ \;\;\;\;\log z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16
Accuracy	38.8%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023137 
(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))))