Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 24.0s

Precision: binary64

Cost: 20032

Math

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

↓

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

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(Float64(log(Float64(x + y)) + 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[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $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. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	26304

\[\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right) \]

Alternative 2
Error	0.9
Cost	20036

\[\begin{array}{l} \mathbf{if}\;t \leq 260:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 3
Error	12.7
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 240:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log t \cdot a\\ \end{array} \]

Alternative 4
Error	20.5
Cost	19904

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

Alternative 5
Error	28.4
Cost	14044

\[\begin{array}{l} t_1 := \log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;a \leq -12600000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{-169}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-133}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 0.029:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{+121}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	29.4
Cost	14044

\[\begin{array}{l} t_1 := \log \left(\frac{y \cdot z}{\sqrt{t}}\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;a \leq -1500000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-169}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-281}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-131}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 0.029:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{+121}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	28.6
Cost	14044

\[\begin{array}{l} t_1 := \log \left(\frac{z}{\frac{\sqrt{t}}{x + y}}\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;a \leq -12000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.18 \cdot 10^{-169}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-281}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-131}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 0.029:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+121}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	28.3
Cost	14044

\[\begin{array}{l} t_1 := \log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;a \leq -115000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{-169}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-281}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-131}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 0.029:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+121}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	15.7
Cost	14040

\[\begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot -0.5\\ t_2 := \log z - t\\ t_3 := t_2 + \log t \cdot a\\ t_4 := \log \left(x + y\right) + t_2\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{-169}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-200}:\\ \;\;\;\;\log \left(\frac{z}{\frac{\sqrt{t}}{x + y}}\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-281}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	8.8
Cost	14028

\[\begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right)\\ t_2 := \left(\log z - t\right) + \log t \cdot a\\ t_3 := \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{if}\;t \leq 3 \cdot 10^{-67}:\\ \;\;\;\;t_3 + t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 17500000000:\\ \;\;\;\;\left(t_3 - t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	17.2
Cost	13900

\[\begin{array}{l} t_1 := \left(\log z - t\right) + \log t \cdot a\\ t_2 := \log \left(y \cdot z\right)\\ t_3 := \log t \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t \leq 5.5 \cdot 10^{-68}:\\ \;\;\;\;t_2 + t_3\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 20000000000:\\ \;\;\;\;\left(t_2 - t\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	10.4
Cost	13900

\[\begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right)\\ t_2 := \left(\log z - t\right) + \log t \cdot a\\ \mathbf{if}\;t \leq 2.85 \cdot 10^{-67}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 15000000000:\\ \;\;\;\;\left(\log \left(y \cdot z\right) - t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	23.0
Cost	13776

\[\begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{if}\;a \leq -2500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-24}:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+121}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	16.6
Cost	13776

\[\begin{array}{l} t_1 := \log \left(y \cdot \frac{z}{\sqrt{t}}\right)\\ t_2 := \left(\log z - t\right) + \log t \cdot a\\ \mathbf{if}\;t \leq 1.4 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	17.1
Cost	13772

\[\begin{array}{l} t_1 := \left(\log z - t\right) + \log t \cdot a\\ t_2 := \log \left(y \cdot z\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t \leq 2.5 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 58:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	17.2
Cost	13640

\[\begin{array}{l} t_1 := \left(\log z - t\right) + \log t \cdot a\\ \mathbf{if}\;a \leq -7.4 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.034:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	25.0
Cost	7120

\[\begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -98000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.76 \cdot 10^{+19}:\\ \;\;\;\;-t\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	39.6
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2022330 
(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))))