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

Average Error: 0.1 → 0.1

Time: 23.1s

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	0.1
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	\[ x + \color{blue}{\left(-\left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]
   sub0-neg [<=] 0.1	\[ x + \color{blue}{\left(0 - \left(\left(z \cdot \log t - z\right) - \left(\left(a - 0.5\right) \cdot b + y\right)\right)\right)} \]
   associate-+l- [<=] 0.1	\[ x + \color{blue}{\left(\left(0 - \left(z \cdot \log t - z\right)\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right)} \]
   associate-+l- [<=] 0.1	\[ x + \left(\color{blue}{\left(\left(0 - z \cdot \log t\right) + z\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   neg-sub0 [<=] 0.1	\[ x + \left(\left(\color{blue}{\left(-z \cdot \log t\right)} + z\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   *-commutative [=>] 0.1	\[ x + \left(\left(\left(-\color{blue}{\log t \cdot z}\right) + z\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   distribute-lft-neg-in [=>] 0.1	\[ x + \left(\left(\color{blue}{\left(-\log t\right) \cdot z} + z\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   distribute-lft1-in [=>] 0.1	\[ x + \left(\color{blue}{\left(\left(-\log t\right) + 1\right) \cdot z} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   *-commutative [=>] 0.1	\[ x + \left(\color{blue}{z \cdot \left(\left(-\log t\right) + 1\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   fma-def [=>] 0.1	\[ x + \color{blue}{\mathsf{fma}\left(z, \left(-\log t\right) + 1, \left(a - 0.5\right) \cdot b + y\right)} \]
   +-commutative [=>] 0.1	\[ x + \mathsf{fma}\left(z, \color{blue}{1 + \left(-\log t\right)}, \left(a - 0.5\right) \cdot b + y\right) \]
   unsub-neg [=>] 0.1	\[ x + \mathsf{fma}\left(z, \color{blue}{1 - \log t}, \left(a - 0.5\right) \cdot b + y\right) \]
   fma-def [=>] 0.1	\[ x + \mathsf{fma}\left(z, 1 - \log t, \color{blue}{\mathsf{fma}\left(a - 0.5, b, y\right)}\right) \]
   sub-neg [=>] 0.1	\[ x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, b, y\right)\right) \]
   metadata-eval [=>] 0.1	\[ x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + \color{blue}{-0.5}, b, y\right)\right) \]

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	10.9
Cost	8528

\[\begin{array}{l} t_1 := \left(x + y\right) + z \cdot \left(1 - \log t\right)\\ t_2 := \left(a + -0.5\right) \cdot b\\ t_3 := \mathsf{fma}\left(a + -0.5, b, y\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+190}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -4000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(y + -0.5 \cdot b\right)\\ \end{array} \]

Alternative 2
Error	11.1
Cost	8528

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ t_2 := \left(a + -0.5\right) \cdot b\\ t_3 := \left(x + y\right) + t_1\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+208}:\\ \;\;\;\;t_2 + t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+49}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -4000000000:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \mathbf{elif}\;t_2 \leq 10^{+128}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(y + -0.5 \cdot b\right)\\ \end{array} \]

Alternative 3
Error	10.5
Cost	8528

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ t_2 := \left(x + y\right) + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+208}:\\ \;\;\;\;\left(\left(x + z\right) + t_1\right) - z \cdot \log t\\ \mathbf{elif}\;t_1 \leq -2.2 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -4000000000:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \mathbf{elif}\;t_1 \leq 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(y + -0.5 \cdot b\right)\\ \end{array} \]

Alternative 4
Error	19.9
Cost	7760

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ t_2 := x + a \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 5
Error	19.9
Cost	7760

\[\begin{array}{l} t_1 := z - z \cdot \log t\\ t_2 := x + a \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 6
Error	19.9
Cost	7760

\[\begin{array}{l} t_1 := z - z \cdot \log t\\ t_2 := x + a \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \end{array} \]

Alternative 7
Error	17.1
Cost	7629

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;x + y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+106} \lor \neg \left(x + y \leq 4.5 \cdot 10^{+160}\right):\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;y + t_1\\ \end{array} \]

Alternative 8
Error	0.1
Cost	7488

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

Alternative 9
Error	0.1
Cost	7360

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

Alternative 10
Error	17.0
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x + y \leq 2 \cdot 10^{-169}:\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, b, y\right)\\ \end{array} \]

Alternative 11
Error	31.0
Cost	1492

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{+34}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-46}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-11}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + y \leq 200:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+39}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Error	25.1
Cost	1097

\[\begin{array}{l} t_1 := \left(a + -0.5\right) \cdot b\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+208} \lor \neg \left(t_1 \leq 10^{+172}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13
Error	22.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-18}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\left(a + -0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \]

Alternative 14
Error	18.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-18}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(y + -0.5 \cdot b\right)\\ \end{array} \]

Alternative 15
Error	18.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-18}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + \left(a + -0.5\right) \cdot b\\ \end{array} \]

Alternative 16
Error	36.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 17
Error	48.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(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)))