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

Percentage Accurate: 99.7% → 99.8%

Time: 33.9s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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

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

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

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	32832

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

Alternative 2
Accuracy	93.0%
Cost	27464

\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{+180}:\\ \;\;\;\;\left(z + a\right) + \left(t_1 + \left(t + b \cdot \log c\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+190}:\\ \;\;\;\;\left(z + a\right) + \left(y \cdot i + \left(t_1 + \left(t + -0.5 \cdot \log c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + \left(y \cdot i + \left(t + t_2\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	91.3%
Cost	14024

\[\begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ t_2 := t + t_1\\ \mathbf{if}\;i \leq -4.5 \cdot 10^{+47}:\\ \;\;\;\;\left(z + a\right) + \left(y \cdot i + t_2\right)\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-34}:\\ \;\;\;\;\left(z + a\right) + \left(x \cdot \log y + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(t_1 + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.7%
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	92.3%
Cost	13897

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+111} \lor \neg \left(x \leq 2.5 \cdot 10^{+92}\right):\\ \;\;\;\;\left(z + a\right) + \left(x \cdot \log y + \left(t + b \cdot \log c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + \left(y \cdot i + \left(t + \log c \cdot \left(b - 0.5\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	92.3%
Cost	13897

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+112} \lor \neg \left(x \leq 2.5 \cdot 10^{+92}\right):\\ \;\;\;\;\left(z + a\right) + \left(x \cdot \log y + \left(t + b \cdot \log c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a + \left(\log c \cdot \left(b - 0.5\right) + \left(z + t\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	90.4%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+180} \lor \neg \left(x \leq 1.3 \cdot 10^{+141}\right):\\ \;\;\;\;x \cdot \log y + \left(a + \left(z + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + \left(y \cdot i + \left(t + \log c \cdot \left(b - 0.5\right)\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	60.3%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;b - 0.5 \leq -2 \cdot 10^{+209} \lor \neg \left(b - 0.5 \leq 5 \cdot 10^{+168}\right):\\ \;\;\;\;\left(z + a\right) + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + y \cdot i\\ \end{array} \]

Alternative 9
Accuracy	71.3%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;\log c \cdot \left(b - 0.5\right) + \left(y \cdot i + \left(z + t\right)\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+142} \lor \neg \left(a \leq 4.8 \cdot 10^{+222}\right):\\ \;\;\;\;\left(z + a\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + b \cdot \log c\\ \end{array} \]

Alternative 10
Accuracy	57.7%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+237} \lor \neg \left(b \leq 1.1 \cdot 10^{+187}\right):\\ \;\;\;\;b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + y \cdot i\\ \end{array} \]

Alternative 11
Accuracy	57.8%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+171} \lor \neg \left(x \leq 6.8 \cdot 10^{+207}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) + y \cdot i\\ \end{array} \]

Alternative 12
Accuracy	39.9%
Cost	589

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+47} \lor \neg \left(y \leq 9.8 \cdot 10^{+86}\right) \land y \leq 8.8 \cdot 10^{+135}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]

Alternative 13
Accuracy	23.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+153}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-129}:\\ \;\;\;\;y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 14
Accuracy	45.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+45}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 15
Accuracy	43.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.66 \cdot 10^{+83}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 16
Accuracy	53.0%
Cost	448

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

Alternative 17
Accuracy	21.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+82}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 18
Accuracy	16.1%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023272 
(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)))