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

Average Error: 0.1 → 0.1

Time: 26.1s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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

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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	32832

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

Alternative 2
Error	15.3
Cost	20425

\[\begin{array}{l} t_1 := \log c \cdot \left(b + -0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+172} \lor \neg \left(t_1 \leq 10^{+230}\right):\\ \;\;\;\;a + t_1\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \end{array} \]

Alternative 3
Error	15.4
Cost	20424

\[\begin{array}{l} t_1 := \log c \cdot \left(b + -0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;a + t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+229}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + b \cdot \log c\\ \end{array} \]

Alternative 4
Error	3.1
Cost	14025

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+59} \lor \neg \left(x \leq 1.32 \cdot 10^{+172}\right):\\ \;\;\;\;\log c \cdot \left(b + -0.5\right) + \left(x \cdot \log y + \left(a + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b + -0.5, \log c, \left(t + a\right) + \left(z + y \cdot i\right)\right)\\ \end{array} \]

Alternative 5
Error	0.1
Cost	14016

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

Alternative 6
Error	4.8
Cost	13896

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(b + -0.5, \log c, \left(t + a\right) + \left(z + y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]

Alternative 7
Error	4.8
Cost	13384

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+134}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+172}:\\ \;\;\;\;z + \left(y \cdot i + \left(\log c \cdot \left(b + -0.5\right) + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]

Alternative 8
Error	4.8
Cost	13384

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+172}:\\ \;\;\;\;z + \left(y \cdot i + \left(\log c \cdot \left(b + -0.5\right) + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]

Alternative 9
Error	16.4
Cost	7888

\[\begin{array}{l} t_1 := z + \log c \cdot \left(b + -0.5\right)\\ \mathbf{if}\;b + -0.5 \leq -1 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b + -0.5 \leq 10^{+158}:\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{elif}\;b + -0.5 \leq 5 \cdot 10^{+199}:\\ \;\;\;\;y \cdot i + b \cdot \log c\\ \mathbf{elif}\;b + -0.5 \leq 5 \cdot 10^{+247}:\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	4.8
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+134} \lor \neg \left(x \leq 1.38 \cdot 10^{+172}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b + -0.5\right) + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	4.8
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+134} \lor \neg \left(x \leq 1.95 \cdot 10^{+172}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(y \cdot i + \left(\log c \cdot \left(b + -0.5\right) + \left(t + a\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	5.2
Cost	7497

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+134} \lor \neg \left(x \leq 1.4 \cdot 10^{+172}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(z + \log c \cdot \left(b + -0.5\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	11.7
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+133} \lor \neg \left(x \leq 2.8 \cdot 10^{+141}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\log c \cdot \left(b + -0.5\right) + \left(a + \left(z + t\right)\right)\\ \end{array} \]

Alternative 14
Error	24.8
Cost	7245

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+148}:\\ \;\;\;\;z + \left(t + a\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+101} \lor \neg \left(z \leq -2.4 \cdot 10^{+79}\right):\\ \;\;\;\;a + \log c \cdot \left(b + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot i\\ \end{array} \]

Alternative 15
Error	11.9
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+134} \lor \neg \left(x \leq 1.3 \cdot 10^{+130}\right):\\ \;\;\;\;a + \left(z + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + \log c \cdot \left(b + -0.5\right)\right)\\ \end{array} \]

Alternative 16
Error	27.7
Cost	6988

\[\begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;b \leq -2.26 \cdot 10^{+226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+96}:\\ \;\;\;\;z + \left(t + a\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+247}:\\ \;\;\;\;a + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	30.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+148}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 18
Error	29.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+148}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 19
Error	27.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + a\right)\\ \end{array} \]

Alternative 20
Error	36.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+148}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 21
Error	47.6
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023018 
(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)))