Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 20.2s

Precision: binary64

Math

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

↓

\[\left(\log \left(y + x\right) + a \cdot \log t\right) - \left(\left(t + \log \left(\frac{1}{z}\right)\right) + \log t \cdot 0.5\right) \]

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 (+ y x)) (* a (log t))) (+ (+ t (log (/ 1.0 z))) (* (log t) 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((y + x)) + (a * log(t))) - ((t + log((1.0 / z))) + (log(t) * 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((y + x)) + (a * log(t))) - ((t + log((1.0d0 / z))) + (log(t) * 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((y + x)) + (a * Math.log(t))) - ((t + Math.log((1.0 / z))) + (Math.log(t) * 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((y + x)) + (a * math.log(t))) - ((t + math.log((1.0 / z))) + (math.log(t) * 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(y + x)) + Float64(a * log(t))) - Float64(Float64(t + log(Float64(1.0 / z))) + Float64(log(t) * 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((y + x)) + (a * log(t))) - ((t + log((1.0 / z))) + (log(t) * 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[(y + x), $MachinePrecision]], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t + N[Log[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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. Simplified 0.3

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

3. Taylor expanded in z around inf 0.3

   \[\leadsto \color{blue}{\left(\log \left(y + x\right) + a \cdot \log t\right) - \left(\log \left(\frac{1}{z}\right) + \left(t + 0.5 \cdot \log t\right)\right)} \]

4. Applied associate-+r+_binary64 0.3

   \[\leadsto \left(\log \left(y + x\right) + a \cdot \log t\right) - \color{blue}{\left(\left(\log \left(\frac{1}{z}\right) + t\right) + 0.5 \cdot \log t\right)} \]

5. Final simplification 0.3

   \[\leadsto \left(\log \left(y + x\right) + a \cdot \log t\right) - \left(\left(t + \log \left(\frac{1}{z}\right)\right) + \log t \cdot 0.5\right) \]

Reproduce

herbie shell --seed 2022129 
(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))))