Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 31.7s

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	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

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

Alternatives

Alternative 1
Error	12.5
Cost	20296

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -10000000000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a + -0.5 \leq -0.4:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 2
Error	12.3
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 8.204331088690704 \cdot 10^{-5}:\\ \;\;\;\;\log z + \left(\left(a + -0.5\right) \cdot \log t + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 3
Error	19.6
Cost	19904

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

Alternative 4
Error	8.6
Cost	14280

\[\begin{array}{l} \mathbf{if}\;a \leq -3578721763033670700:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 2.9953795478634104 \cdot 10^{-36}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \frac{\log t \cdot \left(a \cdot a + -0.25\right)}{a + 0.5}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 5
Error	8.6
Cost	13896

\[\begin{array}{l} \mathbf{if}\;a \leq -3578721763033670700:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 2.9953795478634104 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 6
Error	8.7
Cost	13768

\[\begin{array}{l} \mathbf{if}\;a \leq -14113584122.741764:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 2.9953795478634104 \cdot 10^{-36}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 7
Error	14.2
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a \leq -14113584122.741764:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 2.9953795478634104 \cdot 10^{-36}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 8
Error	16.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;t \leq 1.114784771892651 \cdot 10^{-93}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 9
Error	14.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;t \leq 2.490427083310917 \cdot 10^{-13}:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 10
Error	16.5
Cost	13056

\[\mathsf{fma}\left(\log t, a, -t\right) \]

Alternative 11
Error	24.8
Cost	6856

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -14113584122.741764:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.2277541476288655 \cdot 10^{+91}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	16.5
Cost	6720

\[a \cdot \log t - t \]

Alternative 13
Error	39.8
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2022316 
(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))))