Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Accuracy: 99.6% → 99.6%

Time: 31.6s

Precision: binary64

Cost: 20032

Math

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

↓

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

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)) t) (* (- a 0.5) (log 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)) - t) + ((a - 0.5) * log(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)) - t) + ((a - 0.5d0) * log(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)) - t) + ((a - 0.5) * Math.log(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)) - t) + ((a - 0.5) * math.log(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)) - t) + Float64(Float64(a - 0.5) * log(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)) - t) + ((a - 0.5) * log(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] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	80.3%
Cost	20305

\[\begin{array}{l} t_1 := \log z + \log y\\ \mathbf{if}\;t \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + t_1\\ \mathbf{elif}\;t \leq 3300000000:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{elif}\;t \leq 180000000000 \lor \neg \left(t \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \log t \cdot -0.5\right) - t\\ \end{array} \]

Alternative 2
Accuracy	98.3%
Cost	20305

\[\begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{elif}\;t \leq 3300000000:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{elif}\;t \leq 190000000000 \lor \neg \left(t \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	20032

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

Alternative 4
Accuracy	76.3%
Cost	19976

\[\begin{array}{l} t_1 := \log z - t\\ \mathbf{if}\;a \leq -2 \cdot 10^{-19}:\\ \;\;\;\;t_1 + a \cdot \log t\\ \mathbf{elif}\;a \leq 0.0008:\\ \;\;\;\;\log \left(z \cdot {t}^{-0.5}\right) + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, t_1\right)\\ \end{array} \]

Alternative 5
Accuracy	80.0%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 85:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \end{array} \]

Alternative 6
Accuracy	68.8%
Cost	19904

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

Alternative 7
Accuracy	86.7%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;t \leq 3300000000:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \end{array} \]

Alternative 8
Accuracy	71.1%
Cost	13776

\[\begin{array}{l} t_1 := \log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right)\\ \mathbf{if}\;a \leq -1.22 \cdot 10^{-172}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-111}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 9
Accuracy	86.7%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;t \leq 3300000000:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 10
Accuracy	71.0%
Cost	13712

\[\begin{array}{l} t_1 := \log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right)\\ t_2 := \log t \cdot \left(a + -0.5\right) - t\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-111}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Accuracy	72.9%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-17}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-39}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]

Alternative 12
Accuracy	86.3%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;t \leq 25:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 13
Accuracy	72.5%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;t \leq 3300000000:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 14
Accuracy	71.2%
Cost	13577

\[\begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-93} \lor \neg \left(a \leq 5.5 \cdot 10^{-40}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 15
Accuracy	69.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -15500000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 0.01:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]

Alternative 16
Accuracy	69.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -15500000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 0.01:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]

Alternative 17
Accuracy	76.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -15500000 \lor \neg \left(a \leq 0.01\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \]

Alternative 18
Accuracy	76.4%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -15500000 \lor \neg \left(a \leq 0.01\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 19
Accuracy	61.8%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+21} \lor \neg \left(a \leq 4500000000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 20
Accuracy	64.6%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+21} \lor \neg \left(a \leq 3700000000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \]

Alternative 21
Accuracy	76.1%
Cost	6848

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

Alternative 22
Accuracy	38.6%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 180:\\ \;\;\;\;\log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 23
Accuracy	37.3%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023151 
(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))))