Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Average Accuracy: 99.8% → 99.8%

Time: 31.2s

Precision: binary64

Cost: 14144

\[ \begin{array}{c}[z, t, a] = \mathsf{sort}([z, t, a])\\ \end{array} \]

Math

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

↓

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (- z (* (log (/ 1.0 y)) x)) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((z - (log((1.0 / y)) * x)) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

↓

real(8) function code(x, y, z, t, a, b, c, i)
    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
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((z - (log((1.0d0 / y)) * x)) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((z - (Math.log((1.0 / y)) * x)) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

↓

def code(x, y, z, t, a, b, c, i):
	return ((((z - (math.log((1.0 / y)) * x)) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

↓

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(z - Float64(log(Float64(1.0 / y)) * x)) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

↓

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((z - (log((1.0 / y)) * x)) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(z - N[(N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around inf 99.8%

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

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	90.4%
Cost	14024

\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;b - 0.5 \leq -2.5 \cdot 10^{+101}:\\ \;\;\;\;t_1 + \left(z + \left(t + a\right)\right)\\ \mathbf{elif}\;b - 0.5 \leq 0.5:\\ \;\;\;\;y \cdot i + \left(\left(t + a\right) + \mathsf{fma}\left(x, \log y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(t_1 + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	95.3%
Cost	14020

\[\begin{array}{l} t_1 := a + \left(z + t\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;x \leq -5 \cdot 10^{+25}:\\ \;\;\;\;t_2 + \left(t_1 - \log \left(\frac{1}{y}\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+27}:\\ \;\;\;\;y \cdot i + \left(t_2 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(x \cdot \log y + \left(z + a\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	14016

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

Alternative 4
Accuracy	95.2%
Cost	13897

\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+24} \lor \neg \left(x \leq 4.1 \cdot 10^{+27}\right):\\ \;\;\;\;t_1 + \left(x \cdot \log y + \left(z + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(t_1 + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	89.1%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+201} \lor \neg \left(x \leq 2.8 \cdot 10^{+215}\right):\\ \;\;\;\;x \cdot \log y + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	89.2%
Cost	13512

\[\begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+202}:\\ \;\;\;\;t_1 - \log \left(\frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+212}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + t_1\\ \end{array} \]

Alternative 7
Accuracy	88.2%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+238} \lor \neg \left(x \leq 1.72 \cdot 10^{+218}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\left(b - 0.5\right) \cdot \log c + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	50.6%
Cost	7376

\[\begin{array}{l} t_1 := a + \left(b - 0.5\right) \cdot \log c\\ t_2 := x \cdot \log y\\ \mathbf{if}\;x \leq -7.7 \cdot 10^{+163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-35}:\\ \;\;\;\;a + y \cdot i\\ \mathbf{elif}\;x \leq 1.88 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	76.4%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+239} \lor \neg \left(x \leq 5.2 \cdot 10^{+217}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(b - 0.5\right) \cdot \log c + \left(z + \left(t + a\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	61.5%
Cost	7245

\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+94}:\\ \;\;\;\;z + t_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-52} \lor \neg \left(z \leq -1.15 \cdot 10^{-90}\right):\\ \;\;\;\;a + t_1\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 11
Accuracy	75.8%
Cost	7236

\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+17}:\\ \;\;\;\;t_1 + \left(z + \left(t + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + t_1\right)\\ \end{array} \]

Alternative 12
Accuracy	69.2%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+97}:\\ \;\;\;\;z + \left(b - 0.5\right) \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + b \cdot \log c\right)\\ \end{array} \]

Alternative 13
Accuracy	46.1%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+165} \lor \neg \left(b \leq 1.22 \cdot 10^{+117}\right):\\ \;\;\;\;b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 14
Accuracy	31.8%
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq 30000000000:\\ \;\;\;\;y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 15
Accuracy	37.1%
Cost	320

\[a + y \cdot i \]

Alternative 16
Accuracy	25.4%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))