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

Percentage Accurate: 99.8% → 99.8%

Time: 14.6s

Precision: binary64

Cost: 32832

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

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

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

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

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9%	\[ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   associate-+l+ [=>] 99.9%	\[ \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + \left(a + \left(b - 0.5\right) \cdot \log c\right)\right)} + y \cdot i \]
   +-commutative [=>] 99.9%	\[ \color{blue}{\left(\left(a + \left(b - 0.5\right) \cdot \log c\right) + \left(\left(x \cdot \log y + z\right) + t\right)\right)} + y \cdot i \]
   associate-+l+ [=>] 99.9%	\[ \color{blue}{\left(a + \left(b - 0.5\right) \cdot \log c\right) + \left(\left(\left(x \cdot \log y + z\right) + t\right) + y \cdot i\right)} \]
   +-commutative [=>] 99.9%	\[ \color{blue}{\left(\left(b - 0.5\right) \cdot \log c + a\right)} + \left(\left(\left(x \cdot \log y + z\right) + t\right) + y \cdot i\right) \]
   fma-def [=>] 99.9%	\[ \color{blue}{\mathsf{fma}\left(b - 0.5, \log c, a\right)} + \left(\left(\left(x \cdot \log y + z\right) + t\right) + y \cdot i\right) \]
   sub-neg [=>] 99.9%	\[ \mathsf{fma}\left(\color{blue}{b + \left(-0.5\right)}, \log c, a\right) + \left(\left(\left(x \cdot \log y + z\right) + t\right) + y \cdot i\right) \]
   metadata-eval [=>] 99.9%	\[ \mathsf{fma}\left(b + \color{blue}{-0.5}, \log c, a\right) + \left(\left(\left(x \cdot \log y + z\right) + t\right) + y \cdot i\right) \]
   +-commutative [<=] 99.9%	\[ \mathsf{fma}\left(b + -0.5, \log c, a\right) + \color{blue}{\left(y \cdot i + \left(\left(x \cdot \log y + z\right) + t\right)\right)} \]
   fma-def [=>] 99.9%	\[ \mathsf{fma}\left(b + -0.5, \log c, a\right) + \color{blue}{\mathsf{fma}\left(y, i, \left(x \cdot \log y + z\right) + t\right)} \]
   +-commutative [=>] 99.9%	\[ \mathsf{fma}\left(b + -0.5, \log c, a\right) + \mathsf{fma}\left(y, i, \color{blue}{\left(z + x \cdot \log y\right)} + t\right) \]
   associate-+l+ [=>] 99.9%	\[ \mathsf{fma}\left(b + -0.5, \log c, a\right) + \mathsf{fma}\left(y, i, \color{blue}{z + \left(x \cdot \log y + t\right)}\right) \]
   fma-def [=>] 99.9%	\[ \mathsf{fma}\left(b + -0.5, \log c, a\right) + \mathsf{fma}\left(y, i, z + \color{blue}{\mathsf{fma}\left(x, \log y, t\right)}\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	32832

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

Alternative 2
Accuracy	92.5%
Cost	20809

\[\begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+113} \lor \neg \left(t_1 \leq 5 \cdot 10^{+41}\right):\\ \;\;\;\;y \cdot i + \left(a + \left(z + t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(x \cdot \log y + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	75.2%
Cost	20681

\[\begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+127} \lor \neg \left(t_1 \leq 10^{+64}\right):\\ \;\;\;\;y \cdot i + \left(z + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(z + t\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.8%
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 5
Accuracy	99.1%
Cost	13888

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

Alternative 6
Accuracy	94.8%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+84} \lor \neg \left(x \leq 3.7 \cdot 10^{+113}\right):\\ \;\;\;\;y \cdot i + \left(a + \left(x \cdot \log y + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	88.3%
Cost	7496

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+203}:\\ \;\;\;\;y \cdot i + \left(a + \left(z + \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + t_1\\ \end{array} \]

Alternative 8
Accuracy	72.8%
Cost	7112

\[\begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+183}:\\ \;\;\;\;y \cdot i + t_1\\ \mathbf{elif}\;b - 0.5 \leq 10^{+218}:\\ \;\;\;\;y \cdot i + \left(a + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	71.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+248} \lor \neg \left(b \leq 1.6 \cdot 10^{+218}\right):\\ \;\;\;\;b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(z + t\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	66.9%
Cost	576

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

Alternative 11
Accuracy	59.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+32}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 12
Accuracy	44.5%
Cost	320

\[a + y \cdot i \]

Alternative 13
Accuracy	23.5%
Cost	192

\[y \cdot i \]

Reproduce

herbie shell --seed 2023167 
(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)))