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

Average Error: 0.1 → 0.1

Time: 25.8s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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

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

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

   \[\leadsto \color{blue}{z + \left(t + \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(b + -0.5, \log c, \mathsf{fma}\left(y, i, a\right)\right)\right)\right)} \]
   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	\[ \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right)} \]
   +-commutative [=>] 0.1	\[ \color{blue}{\left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} \]
   +-commutative [=>] 0.1	\[ \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \color{blue}{\left(a + \left(\left(x \cdot \log y + z\right) + t\right)\right)} \]
   associate-+l+ [=>] 0.1	\[ \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \left(a + \color{blue}{\left(x \cdot \log y + \left(z + t\right)\right)}\right) \]
   associate-+r+ [=>] 0.1	\[ \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \color{blue}{\left(\left(a + x \cdot \log y\right) + \left(z + t\right)\right)} \]
   associate-+r+ [=>] 0.1	\[ \color{blue}{\left(\left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \left(a + x \cdot \log y\right)\right) + \left(z + t\right)} \]
   +-commutative [=>] 0.1	\[ \color{blue}{\left(z + t\right) + \left(\left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \left(a + x \cdot \log y\right)\right)} \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{z + \left(t + \left(\left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + \left(a + x \cdot \log y\right)\right)\right)} \]
   associate-+r+ [=>] 0.1	\[ z + \left(t + \color{blue}{\left(\left(\left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right) + a\right) + x \cdot \log y\right)}\right) \]
   +-commutative [<=] 0.1	\[ z + \left(t + \left(\color{blue}{\left(a + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right)\right)} + x \cdot \log y\right)\right) \]
   +-commutative [<=] 0.1	\[ z + \left(t + \color{blue}{\left(x \cdot \log y + \left(a + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right)\right)\right)}\right) \]
   fma-def [=>] 0.1	\[ z + \left(t + \color{blue}{\mathsf{fma}\left(x, \log y, a + \left(\left(b - 0.5\right) \cdot \log c + y \cdot i\right)\right)}\right) \]
   associate-+r+ [=>] 0.1	\[ z + \left(t + \mathsf{fma}\left(x, \log y, \color{blue}{\left(a + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i}\right)\right) \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	3.4
Cost	14025

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+175} \lor \neg \left(x \leq 8.5 \cdot 10^{+28}\right):\\ \;\;\;\;\log c \cdot \left(b + -0.5\right) + \left(x \cdot \log y + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z + t\right)\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	14016

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

Alternative 3
Error	5.6
Cost	13897

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+175} \lor \neg \left(x \leq 1.45 \cdot 10^{+42}\right):\\ \;\;\;\;z + \left(t + \mathsf{fma}\left(x, \log y, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z + t\right)\right)\\ \end{array} \]

Alternative 4
Error	5.6
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+175} \lor \neg \left(x \leq 2.3 \cdot 10^{+42}\right):\\ \;\;\;\;z + \left(t + \mathsf{fma}\left(x, \log y, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + \left(\log c \cdot \left(b + -0.5\right) + \left(a + y \cdot i\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	29.0
Cost	8044

\[\begin{array}{l} t_1 := z + \left(t + a\right)\\ t_2 := z + \left(t + b \cdot \log c\right)\\ t_3 := a + y \cdot i\\ t_4 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+199}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.15 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-240}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 6
Error	29.3
Cost	7904

\[\begin{array}{l} t_1 := z + \left(t + x \cdot \log y\right)\\ t_2 := z + \left(t + b \cdot \log c\right)\\ t_3 := a + y \cdot i\\ t_4 := z + \left(t + a\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-169}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-243}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-167}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+65}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	6.4
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+202} \lor \neg \left(x \leq 2.2 \cdot 10^{+183}\right):\\ \;\;\;\;z + \left(t + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + \left(\log c \cdot \left(b + -0.5\right) + \left(a + y \cdot i\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	28.2
Cost	7384

\[\begin{array}{l} t_1 := z + \left(t + a\right)\\ t_2 := a + y \cdot i\\ t_3 := x \cdot \log y\\ \mathbf{if}\;x \leq -1 \cdot 10^{+200}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.42 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	14.7
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+199} \lor \neg \left(x \leq 3.8 \cdot 10^{+98}\right):\\ \;\;\;\;z + \left(t + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + \left(a + \log c \cdot \left(b + -0.5\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	14.9
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+186} \lor \neg \left(x \leq 3.8 \cdot 10^{+98}\right):\\ \;\;\;\;z + \left(t + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(a + \log c \cdot \left(b + -0.5\right)\right)\\ \end{array} \]

Alternative 11
Error	29.4
Cost	6856

\[\begin{array}{l} t_1 := z + \left(t + a\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+71}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 12
Error	30.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 13
Error	28.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 14
Error	27.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+129}:\\ \;\;\;\;z + \left(t + a\right)\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 15
Error	36.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 16
Error	36.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+79}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 17
Error	47.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023016 
(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)))