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

Average Error: 0.1 → 0.1

Time: 18.6s

Precision: binary64

Cost: 20288

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

Math

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

↓

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

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
 (+ (+ (fma x (log y) z) (+ t a)) (+ (* y i) (* (log c) (+ b -0.5)))))

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 (fma(x, log(y), z) + (t + a)) + ((y * i) + (log(c) * (b + -0.5)));
}

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right)} \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{\left(\left(x \cdot \log y + z\right) + \left(t + a\right)\right)} + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) \]
   fma-def [=>] 0.1	\[ \left(\color{blue}{\mathsf{fma}\left(x, \log y, z\right)} + \left(t + a\right)\right) + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) \]
   sub-neg [=>] 0.1	\[ \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right) + \left(\color{blue}{\left(b + \left(-0.5\right)\right)} \cdot \log c + y \cdot i\right) \]
   metadata-eval [=>] 0.1	\[ \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right) + \left(\left(b + \color{blue}{-0.5}\right) \cdot \log c + y \cdot i\right) \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	3.6
Cost	14024

\[\begin{array}{l} t_1 := a + \left(z + t\right)\\ t_2 := \log c \cdot \left(b + -0.5\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+59}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right) + y \cdot i\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{+153}:\\ \;\;\;\;y \cdot i + \left(t_2 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(t_1 + x \cdot \log y\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	14016

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

Alternative 3
Error	3.8
Cost	13769

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+59} \lor \neg \left(x \leq 1.25 \cdot 10^{+152}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b + -0.5\right) + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	3.8
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+59} \lor \neg \left(x \leq 3.3 \cdot 10^{+148}\right):\\ \;\;\;\;y \cdot i + \left(\left(t + a\right) + \left(z + x \cdot \log y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b + -0.5\right) + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	17.0
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+122} \lor \neg \left(x \leq 9.5 \cdot 10^{+185}\right):\\ \;\;\;\;y \cdot i + \left(\left(t + a\right) + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(z + \left(t + a\right)\right)\\ \end{array} \]

Alternative 6
Error	12.2
Cost	7232

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

Alternative 7
Error	19.4
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+120} \lor \neg \left(x \leq 2.4 \cdot 10^{+219}\right):\\ \;\;\;\;y \cdot i + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(z + \left(t + a\right)\right)\\ \end{array} \]

Alternative 8
Error	23.7
Cost	576

\[y \cdot i + \left(z + \left(t + a\right)\right) \]

Alternative 9
Error	30.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+173}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 10
Error	29.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+48}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 11
Error	36.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+48}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12
Error	47.7
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022364 
(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)))