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

Average Error: 0.1 → 0.1

Time: 20.0s

Precision: binary64

Cost: 19904

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	0.1
Target	0.4
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   sub-neg [=>] 0.1	\[ \color{blue}{\left(\left(\left(x + y\right) + z\right) + \left(-z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 0.1	\[ \left(\color{blue}{\left(x + \left(y + z\right)\right)} + \left(-z \cdot \log t\right)\right) + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{\left(x + \left(\left(y + z\right) + \left(-z \cdot \log t\right)\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{x + \left(\left(\left(y + z\right) + \left(-z \cdot \log t\right)\right) + \left(a - 0.5\right) \cdot b\right)} \]
   sub-neg [<=] 0.1	\[ x + \left(\color{blue}{\left(\left(y + z\right) - z \cdot \log t\right)} + \left(a - 0.5\right) \cdot b\right) \]
   associate-+r- [<=] 0.1	\[ x + \left(\color{blue}{\left(y + \left(z - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b\right) \]
   +-commutative [=>] 0.1	\[ x + \left(\color{blue}{\left(\left(z - z \cdot \log t\right) + y\right)} + \left(a - 0.5\right) \cdot b\right) \]
   associate-+l+ [=>] 0.1	\[ x + \color{blue}{\left(\left(z - z \cdot \log t\right) + \left(y + \left(a - 0.5\right) \cdot b\right)\right)} \]
   +-commutative [<=] 0.1	\[ x + \left(\left(z - z \cdot \log t\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot b + y\right)}\right) \]
   sub-neg [=>] 0.1	\[ x + \left(\color{blue}{\left(z + \left(-z \cdot \log t\right)\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   +-commutative [=>] 0.1	\[ x + \left(\color{blue}{\left(\left(-z \cdot \log t\right) + z\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   neg-sub0 [=>] 0.1	\[ x + \left(\left(\color{blue}{\left(0 - z \cdot \log t\right)} + z\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   associate-+l- [=>] 0.1	\[ x + \left(\color{blue}{\left(0 - \left(z \cdot \log t - z\right)\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   associate-+l- [=>] 0.1	\[ x + \color{blue}{\left(0 - \left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]
   sub0-neg [=>] 0.1	\[ x + \color{blue}{\left(-\left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	6.0
Cost	7497

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{+99} \lor \neg \left(z \leq 2 \cdot 10^{+152}\right):\\ \;\;\;\;t_1 + \left(\left(z + y\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(z + \left(x + y\right)\right)\\ \end{array} \]

Alternative 2
Error	15.3
Cost	7492

\[\begin{array}{l} t_1 := z \cdot \log t\\ t_2 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{-141}:\\ \;\;\;\;t_2 + \left(\left(x + z\right) - t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\left(z + y\right) - t_1\right)\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7360

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

Alternative 4
Error	6.5
Cost	7241

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+178} \lor \neg \left(z \leq 5.2 \cdot 10^{+116}\right):\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(z + \left(x + y\right)\right)\\ \end{array} \]

Alternative 5
Error	8.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+178}:\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+165}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(z + \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) - z \cdot \log t\\ \end{array} \]

Alternative 6
Error	8.3
Cost	7112

\[\begin{array}{l} t_1 := z \cdot \log t\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+178}:\\ \;\;\;\;\left(x + z\right) - t_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+165}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(z + \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) - t_1\\ \end{array} \]

Alternative 7
Error	9.5
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+178} \lor \neg \left(z \leq 3.4 \cdot 10^{+208}\right):\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(z + \left(x + y\right)\right)\\ \end{array} \]

Alternative 8
Error	9.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+178}:\\ \;\;\;\;z - z \cdot \log t\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+211}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(z + \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \end{array} \]

Alternative 9
Error	8.9
Cost	6984

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+178}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+209}:\\ \;\;\;\;b \cdot \left(a + -0.5\right) + \left(z + \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	25.1
Cost	1097

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+137} \lor \neg \left(t_1 \leq 2 \cdot 10^{+151}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11
Error	15.7
Cost	1097

\[\begin{array}{l} t_1 := z + \left(x + y\right)\\ \mathbf{if}\;a + -0.5 \leq -5 \cdot 10^{+38} \lor \neg \left(a + -0.5 \leq -0.499\right):\\ \;\;\;\;t_1 + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1 + -0.5 \cdot b\\ \end{array} \]

Alternative 12
Error	31.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-42}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+96}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13
Error	36.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-42}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 10^{-20}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 14
Error	45.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+99}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \leq -1.06 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 15
Error	33.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;x + y \leq 10^{-20}:\\ \;\;\;\;x + b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 16
Error	30.3
Cost	708

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;x + y \leq 10^{-38}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 17
Error	14.9
Cost	704

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

Alternative 18
Error	42.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+123}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 19
Error	30.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+129}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+135}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 20
Error	47.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023055 
(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)))