Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 11.1s

Precision: binary64

Math

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

↓

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

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 t) (+ a -0.5)) (log (+ y x))) t) (log z)))

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(t) * (a + -0.5)) + log((y + x))) - t) + log(z);
}

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(t) * (a + (-0.5d0))) + log((y + x))) - t) + log(z)
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(t) * (a + -0.5)) + Math.log((y + x))) - t) + Math.log(z);
}

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(t) * (a + -0.5)) + math.log((y + x))) - t) + math.log(z)

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

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

3. Taylor expanded in z around 0 0.3

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

4. Simplified 0.2

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

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022153 
(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))))