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

Average Error: 0.17% → 0.16%

Time: 27.4s

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 0.17

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

2. Simplified 0.16

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

   [Start] 0.17	\[ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   +-commutative [=>] 0.17	\[ \color{blue}{y \cdot i + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right)} \]
   fma-def [=>] 0.16	\[ \color{blue}{\mathsf{fma}\left(y, i, \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right)} \]
   +-commutative [=>] 0.16	\[ \mathsf{fma}\left(y, i, \color{blue}{\left(b - 0.5\right) \cdot \log c + \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)}\right) \]
   fma-def [=>] 0.16	\[ \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(b - 0.5, \log c, \left(\left(x \cdot \log y + z\right) + t\right) + a\right)}\right) \]
   sub-neg [=>] 0.16	\[ \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{b + \left(-0.5\right)}, \log c, \left(\left(x \cdot \log y + z\right) + t\right) + a\right)\right) \]
   metadata-eval [=>] 0.16	\[ \mathsf{fma}\left(y, i, \mathsf{fma}\left(b + \color{blue}{-0.5}, \log c, \left(\left(x \cdot \log y + z\right) + t\right) + a\right)\right) \]
   associate-+l+ [=>] 0.16	\[ \mathsf{fma}\left(y, i, \mathsf{fma}\left(b + -0.5, \log c, \color{blue}{\left(x \cdot \log y + z\right) + \left(t + a\right)}\right)\right) \]
   fma-def [=>] 0.16	\[ \mathsf{fma}\left(y, i, \mathsf{fma}\left(b + -0.5, \log c, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)} + \left(t + a\right)\right)\right) \]

3. Final simplification 0.16

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

Alternatives

Alternative 1
Error	0.17%
Cost	20288

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

Alternative 2
Error	4.59%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+132} \lor \neg \left(x \leq 7.2 \cdot 10^{+118}\right):\\ \;\;\;\;\left(x \cdot \log y + \left(a + \left(z + t\right)\right)\right) + \left(b + -0.5\right) \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b + -0.5, \log c, \left(t + a\right) + \left(z + y \cdot i\right)\right)\\ \end{array} \]

Alternative 3
Error	0.17%
Cost	14016

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

Alternative 4
Error	9.64%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \log y + \left(z + a\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(b + -0.5, \log c, \left(t + a\right) + \left(z + y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot \left(3 \cdot \log \left(\sqrt[3]{y}\right)\right)\\ \end{array} \]

Alternative 5
Error	9.77%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \log y + \left(z + a\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+242}:\\ \;\;\;\;y \cdot i + \left(\left(b + -0.5\right) \cdot \log c + \left(z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot \left(3 \cdot \log \left(\sqrt[3]{y}\right)\right)\\ \end{array} \]

Alternative 6
Error	9.68%
Cost	13256

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \log y + \left(z + a\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+240}:\\ \;\;\;\;y \cdot i + \left(\left(b + -0.5\right) \cdot \log c + \left(z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, z\right)\\ \end{array} \]

Alternative 7
Error	19.4%
Cost	7632

\[\begin{array}{l} t_1 := \left(b + -0.5\right) \cdot \log c\\ t_2 := x \cdot \log y + \left(z + a\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-263}:\\ \;\;\;\;a + \left(z + t_1\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-220}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;a + \left(\left(z + t\right) + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	9.64%
Cost	7624

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+207}:\\ \;\;\;\;t_1 + \left(z + a\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+240}:\\ \;\;\;\;y \cdot i + \left(\left(b + -0.5\right) \cdot \log c + \left(z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z + t_1\\ \end{array} \]

Alternative 9
Error	19.55%
Cost	7504

\[\begin{array}{l} t_1 := a + \left(z + \left(b + -0.5\right) \cdot \log c\right)\\ t_2 := x \cdot \log y + \left(z + a\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-220}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	20.74%
Cost	7504

\[\begin{array}{l} t_1 := z + \left(t + \left(a + x \cdot \log y\right)\right)\\ t_2 := a + \left(z + \left(b + -0.5\right) \cdot \log c\right)\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{-243}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	27.4%
Cost	7376

\[\begin{array}{l} t_1 := x \cdot \log y + \left(z + a\right)\\ t_2 := b \cdot \log c\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+230}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-240}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	23.41%
Cost	7372

\[\begin{array}{l} t_1 := \left(b + -0.5\right) \cdot \log c\\ t_2 := x \cdot \log y\\ \mathbf{if}\;a \leq 1.32 \cdot 10^{+125}:\\ \;\;\;\;\left(z + y \cdot i\right) + t_1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+154}:\\ \;\;\;\;t_2 + \left(z + a\right)\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{+196}:\\ \;\;\;\;a + \left(z + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + \left(a + t_2\right)\right)\\ \end{array} \]

Alternative 13
Error	23.15%
Cost	7372

\[\begin{array}{l} t_1 := \left(b + -0.5\right) \cdot \log c\\ t_2 := x \cdot \log y\\ \mathbf{if}\;a \leq 2.65 \cdot 10^{+125}:\\ \;\;\;\;t_1 + \left(t + \left(z + y \cdot i\right)\right)\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;t_2 + \left(z + a\right)\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{+196}:\\ \;\;\;\;a + \left(z + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + \left(a + t_2\right)\right)\\ \end{array} \]

Alternative 14
Error	41.92%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;a \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+154}:\\ \;\;\;\;z + \left(t + x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + a\right)\\ \end{array} \]

Alternative 15
Error	42.04%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+19}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;z + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + a\right)\\ \end{array} \]

Alternative 16
Error	57.48%
Cost	724

\[\begin{array}{l} \mathbf{if}\;a \leq 1.52 \cdot 10^{-133}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-71}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 325:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{+41}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 17
Error	57.28%
Cost	724

\[\begin{array}{l} \mathbf{if}\;a \leq 3.25 \cdot 10^{-133}:\\ \;\;\;\;z + t\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 340:\\ \;\;\;\;z\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 18
Error	47.21%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+143}:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 19
Error	44.88%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{+147}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]

Alternative 20
Error	41.95%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;z + \left(t + a\right)\\ \end{array} \]

Alternative 21
Error	56.01%
Cost	196

\[\begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 22
Error	73.09%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023088 
(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)))