Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 22.5s

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	0.9
Cost	20036

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

Alternative 2
Error	12.3
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 230:\\ \;\;\;\;\left(\log z + \log y\right) + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 3
Error	20.3
Cost	19904

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

Alternative 4
Error	8.3
Cost	14024

\[\begin{array}{l} \mathbf{if}\;a \leq -23000000000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{elif}\;a \leq 1150000:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} + \left(\log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 5
Error	16.5
Cost	13776

\[\begin{array}{l} t_1 := \log \left(\frac{y \cdot z}{\sqrt{t}}\right)\\ t_2 := \log z - t\\ t_3 := t_2 + a \cdot \log t\\ \mathbf{if}\;a \leq -1.82 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-259}:\\ \;\;\;\;\log \left(x + y\right) + t_2\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	16.6
Cost	13776

\[\begin{array}{l} t_1 := \log z - t\\ t_2 := t_1 + a \cdot \log t\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-259}:\\ \;\;\;\;\log \left(x + y\right) + t_1\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-248}:\\ \;\;\;\;\log \left(\frac{y \cdot z}{\sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	8.8
Cost	13768

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \mathbf{elif}\;a \leq 0.0006:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) - t\right) + \log t \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 8
Error	16.8
Cost	13648

\[\begin{array}{l} t_1 := \log \left(\frac{y \cdot z}{\sqrt{t}}\right)\\ t_2 := \log \left(x + y\right) - t\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.035:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 9
Error	17.5
Cost	13640

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \mathbf{elif}\;a \leq 0.044:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 10
Error	14.0
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a \leq -100000000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{elif}\;a \leq 0.066:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 11
Error	17.1
Cost	13508

\[\begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 12
Error	14.6
Cost	13188

\[\begin{array}{l} \mathbf{if}\;a \leq -100000000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{elif}\;a \leq 0.066:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 13
Error	24.5
Cost	6989

\[\begin{array}{l} \mathbf{if}\;t \leq 9.3 \cdot 10^{+21} \lor \neg \left(t \leq 1.15 \cdot 10^{+59}\right) \land t \leq 1.56 \cdot 10^{+77}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14
Error	14.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -100000000 \lor \neg \left(a \leq 0.066\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 15
Error	16.4
Cost	6720

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

Alternative 16
Error	39.7
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023016 
(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))))