Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 23.1s

Precision: binary64

Cost: 20032

Math

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

↓

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

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	4.5
Cost	40008

\[\begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;t_1 \leq -750:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 688:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a + -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	20.9
Cost	39880

\[\begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;t_1 \leq -750:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 688:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a + -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	1.0
Cost	20036

\[\begin{array}{l} \mathbf{if}\;t \leq 8.217575593061328:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 4
Error	12.1
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 8.217575593061328:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 5
Error	19.6
Cost	19904

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

Alternative 6
Error	16.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5301997294794886 \cdot 10^{-33}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 5.899865342903392 \cdot 10^{-61}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 7
Error	18.1
Cost	13576

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5301997294794886 \cdot 10^{-33}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 5.899865342903392 \cdot 10^{-61}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 8
Error	14.2
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a \leq -510.2186519132732:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 2.5727053258482297 \cdot 10^{-78}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 9
Error	14.8
Cost	13252

\[\begin{array}{l} \mathbf{if}\;t \leq 8.217575593061328:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 10
Error	16.4
Cost	13056

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

Alternative 11
Error	24.7
Cost	6856

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -35784061596.46999:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.215882395629914 \cdot 10^{+45}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	16.4
Cost	6720

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

Alternative 13
Error	39.0
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2022300 
(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))))