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

Average Error: 0.15% → 0.12%

Time: 19.2s

Precision: binary64

Cost: 19904

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (+ x (fma z (- 1.0 (log t)) (fma (+ a -0.5) b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return x + fma(z, (1.0 - log(t)), fma((a + -0.5), b, y));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
end

↓

function code(x, y, z, t, a, b)
	return Float64(x + fma(z, Float64(1.0 - log(t)), fma(Float64(a + -0.5), b, y)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.15%
Target	0.61%
Herbie	0.12%

\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \]

Derivation

1. Initial program 0.15

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

2. Simplified 0.12

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right)} \]
   Proof

   [Start] 0.15	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   sub-neg [=>] 0.15	\[ \color{blue}{\left(\left(\left(x + y\right) + z\right) + \left(-z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 0.15	\[ \left(\color{blue}{\left(x + \left(y + z\right)\right)} + \left(-z \cdot \log t\right)\right) + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 0.15	\[ \color{blue}{\left(x + \left(\left(y + z\right) + \left(-z \cdot \log t\right)\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 0.15	\[ \color{blue}{x + \left(\left(\left(y + z\right) + \left(-z \cdot \log t\right)\right) + \left(a - 0.5\right) \cdot b\right)} \]
   sub-neg [<=] 0.15	\[ x + \left(\color{blue}{\left(\left(y + z\right) - z \cdot \log t\right)} + \left(a - 0.5\right) \cdot b\right) \]
   associate-+r- [<=] 0.15	\[ x + \left(\color{blue}{\left(y + \left(z - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b\right) \]
   +-commutative [=>] 0.15	\[ x + \left(\color{blue}{\left(\left(z - z \cdot \log t\right) + y\right)} + \left(a - 0.5\right) \cdot b\right) \]
   associate-+l+ [=>] 0.15	\[ x + \color{blue}{\left(\left(z - z \cdot \log t\right) + \left(y + \left(a - 0.5\right) \cdot b\right)\right)} \]
   +-commutative [<=] 0.15	\[ x + \left(\left(z - z \cdot \log t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot b + y\right)}\right) \]
   sub-neg [=>] 0.15	\[ x + \left(\color{blue}{\left(z + \left(-z \cdot \log t\right)\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   +-commutative [=>] 0.15	\[ x + \left(\color{blue}{\left(\left(-z \cdot \log t\right) + z\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   neg-sub0 [=>] 0.15	\[ x + \left(\left(\color{blue}{\left(0 - z \cdot \log t\right)} + z\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   associate-+l- [=>] 0.15	\[ x + \left(\color{blue}{\left(0 - \left(z \cdot \log t - z\right)\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   associate-+l- [=>] 0.15	\[ x + \color{blue}{\left(0 - \left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]
   sub0-neg [=>] 0.15	\[ x + \color{blue}{\left(-\left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]

3. Final simplification 0.12

   \[\leadsto x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]

Alternatives

Alternative 1
Error	11.7%
Cost	7753

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+138} \lor \neg \left(t_1 \leq 4 \cdot 10^{+18}\right):\\ \;\;\;\;\left(z + \left(x + y\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \end{array} \]

Alternative 2
Error	14.98%
Cost	7633

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ t_2 := z \cdot \left(1 - \log t\right)\\ t_3 := \left(z + \left(x + y\right)\right) + t_1\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+18}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 60000000:\\ \;\;\;\;x + \left(y + t_2\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+80} \lor \neg \left(b \leq 5.2 \cdot 10^{+192}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_2\\ \end{array} \]

Alternative 3
Error	13.53%
Cost	7378

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+200} \lor \neg \left(z \leq -3.15 \cdot 10^{+175} \lor \neg \left(z \leq -8.5 \cdot 10^{+133}\right) \land z \leq 2.2 \cdot 10^{+148}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 4
Error	13.52%
Cost	7376

\[\begin{array}{l} t_1 := \left(z + \left(x + y\right)\right) + \left(a + -0.5\right) \cdot b\\ t_2 := x + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+130}:\\ \;\;\;\;x + \left(z - z \cdot \log t\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	9.9%
Cost	7364

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(z + y\right)\right) - z \cdot \log t\\ \end{array} \]

Alternative 6
Error	0.15%
Cost	7360

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

Alternative 7
Error	15.04%
Cost	7112

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+130}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+52}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + \left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 8
Error	16.62%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+200} \lor \neg \left(z \leq 9.5 \cdot 10^{+259}\right):\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 9
Error	48.81%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{+47}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 10^{-217}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x + y \leq 1000000000000:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10
Error	41.01%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{+47}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11
Error	37.48%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-17}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Error	35%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-17}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 1000000000000:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \]

Alternative 13
Error	34.88%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-17}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 10^{-59}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 14
Error	58.37%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x \leq -2.02 \cdot 10^{-252}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 15
Error	29.99%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x + y \leq 10^{-59}:\\ \;\;\;\;x + \left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 16
Error	24.93%
Cost	708

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -1 \cdot 10^{-136}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 17
Error	23.3%
Cost	704

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

Alternative 18
Error	57.7%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 19
Error	74.87%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))