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

Percentage Accurate: 99.9% → 99.9%

Time: 18.4s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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

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

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

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

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

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	32832

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

Alternative 2
Accuracy	99.9%
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
Accuracy	91.9%
Cost	13897

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+145} \lor \neg \left(x \leq 9.5 \cdot 10^{+230}\right):\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \mathsf{fma}\left(\log c, b + -0.5, t + z\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	92.0%
Cost	13896

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+145}:\\ \;\;\;\;t + \left(\log c \cdot \left(b - 0.5\right) + \left(a + \left(z + t_1\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+230}:\\ \;\;\;\;y \cdot i + \left(a + \mathsf{fma}\left(\log c, b + -0.5, t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + t_1\right)\\ \end{array} \]

Alternative 5
Accuracy	91.9%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+145} \lor \neg \left(x \leq 1.95 \cdot 10^{+232}\right):\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(z + \left(t + a\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	90.4%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+146} \lor \neg \left(x \leq 4.8 \cdot 10^{+234}\right):\\ \;\;\;\;t + \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(z + \left(t + a\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	74.8%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+151} \lor \neg \left(b - 0.5 \leq 10^{+206}\right):\\ \;\;\;\;t + \left(\log c \cdot \left(b - 0.5\right) + \left(a + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + z\right)\\ \end{array} \]

Alternative 8
Accuracy	88.5%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+207} \lor \neg \left(x \leq 1.1 \cdot 10^{+243}\right):\\ \;\;\;\;t + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(\log c \cdot \left(b - 0.5\right) + \left(z + \left(t + a\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	26.4%
Cost	7512

\[\begin{array}{l} t_1 := t + y \cdot i\\ \mathbf{if}\;a \leq 6.2 \cdot 10^{-277}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;t + x \cdot \log y\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-41}:\\ \;\;\;\;t + b \cdot \log c\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]

Alternative 10
Accuracy	73.2%
Cost	7496

\[\begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;t + \left(a + t_1\right)\\ \mathbf{elif}\;b - 0.5 \leq 10^{+206}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + z\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z + t_1\right)\\ \end{array} \]

Alternative 11
Accuracy	55.9%
Cost	7376

\[\begin{array}{l} t_1 := a + \log c \cdot \left(b - 0.5\right)\\ t_2 := t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{if}\;a \leq 4 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a\right)\\ \end{array} \]

Alternative 12
Accuracy	71.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+151} \lor \neg \left(b - 0.5 \leq 10^{+206}\right):\\ \;\;\;\;a + \log c \cdot \left(b - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + z\right)\\ \end{array} \]

Alternative 13
Accuracy	72.8%
Cost	7368

\[\begin{array}{l} t_1 := a + \log c \cdot \left(b - 0.5\right)\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;t + t_1\\ \mathbf{elif}\;b - 0.5 \leq 10^{+206}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Accuracy	58.0%
Cost	7248

\[\begin{array}{l} t_1 := t + \mathsf{fma}\left(y, i, a\right)\\ t_2 := t + b \cdot \log c\\ \mathbf{if}\;b \leq -1.95 \cdot 10^{+202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{+68}:\\ \;\;\;\;z\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Accuracy	58.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+137}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, a\right)\\ \end{array} \]

Alternative 16
Accuracy	23.1%
Cost	852

\[\begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{-277}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-181}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-24}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{+135}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]

Alternative 17
Accuracy	26.5%
Cost	852

\[\begin{array}{l} t_1 := t + y \cdot i\\ \mathbf{if}\;a \leq 1.8 \cdot 10^{-276}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{-74}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+135}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]

Alternative 18
Accuracy	21.9%
Cost	724

\[\begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{-277}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-180}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 3.05 \cdot 10^{-74}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 10^{-25}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 19
Accuracy	21.1%
Cost	196

\[\begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{+136}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 20
Accuracy	15.8%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023165 
(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)))