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

Percentage Accurate: 99.8% → 99.9%

Time: 25.2s

Precision: binary64

Cost: 33024

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

Math

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

↓

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

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 (pow (log t) 2.0)) (+ 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 - pow(log(t), 2.0)) / (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(Float64(1.0 - (log(t) ^ 2.0)) / 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[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.8%
Target	99.5%
Herbie	99.9%

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

Derivation

1. Initial program 99.8%

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

2. Simplified 99.9%

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

   [Start] 99.8%	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   associate--l+ [=>] 99.9%	\[ \color{blue}{\left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]
   associate-+l+ [=>] 99.9%	\[ \color{blue}{\left(x + y\right) + \left(\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\right)} \]
   associate-+l+ [=>] 99.9%	\[ \color{blue}{x + \left(y + \left(\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\right)\right)} \]
   +-commutative [=>] 99.9%	\[ x + \left(y + \color{blue}{\left(\left(a - 0.5\right) \cdot b + \left(z - z \cdot \log t\right)\right)}\right) \]
   associate-+r+ [=>] 99.9%	\[ x + \color{blue}{\left(\left(y + \left(a - 0.5\right) \cdot b\right) + \left(z - z \cdot \log t\right)\right)} \]
   +-commutative [<=] 99.9%	\[ x + \left(\color{blue}{\left(\left(a - 0.5\right) \cdot b + y\right)} + \left(z - z \cdot \log t\right)\right) \]
   +-commutative [<=] 99.9%	\[ x + \color{blue}{\left(\left(z - z \cdot \log t\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right)} \]
   *-commutative [=>] 99.9%	\[ x + \left(\left(z - \color{blue}{\log t \cdot z}\right) + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   cancel-sign-sub-inv [=>] 99.9%	\[ x + \left(\color{blue}{\left(z + \left(-\log t\right) \cdot z\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   distribute-rgt1-in [=>] 99.9%	\[ x + \left(\color{blue}{\left(\left(-\log t\right) + 1\right) \cdot z} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   *-commutative [=>] 99.9%	\[ x + \left(\color{blue}{z \cdot \left(\left(-\log t\right) + 1\right)} + \left(\left(a - 0.5\right) \cdot b + y\right)\right) \]
   fma-def [=>] 99.9%	\[ x + \color{blue}{\mathsf{fma}\left(z, \left(-\log t\right) + 1, \left(a - 0.5\right) \cdot b + y\right)} \]
   +-commutative [=>] 99.9%	\[ x + \mathsf{fma}\left(z, \color{blue}{1 + \left(-\log t\right)}, \left(a - 0.5\right) \cdot b + y\right) \]
   unsub-neg [=>] 99.9%	\[ x + \mathsf{fma}\left(z, \color{blue}{1 - \log t}, \left(a - 0.5\right) \cdot b + y\right) \]
   fma-def [=>] 99.9%	\[ x + \mathsf{fma}\left(z, 1 - \log t, \color{blue}{\mathsf{fma}\left(a - 0.5, b, y\right)}\right) \]
   sub-neg [=>] 99.9%	\[ x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, b, y\right)\right) \]
   metadata-eval [=>] 99.9%	\[ x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + \color{blue}{-0.5}, b, y\right)\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto x + \mathsf{fma}\left(z, \color{blue}{\left(1 - {\log t}^{2}\right) \cdot \frac{1}{1 + \log t}}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   Step-by-step derivation

   [Start] 99.9%	\[ x + \mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   flip-- [=>] 99.9%	\[ x + \mathsf{fma}\left(z, \color{blue}{\frac{1 \cdot 1 - \log t \cdot \log t}{1 + \log t}}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   div-inv [=>] 99.8%	\[ x + \mathsf{fma}\left(z, \color{blue}{\left(1 \cdot 1 - \log t \cdot \log t\right) \cdot \frac{1}{1 + \log t}}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   metadata-eval [=>] 99.8%	\[ x + \mathsf{fma}\left(z, \left(\color{blue}{1} - \log t \cdot \log t\right) \cdot \frac{1}{1 + \log t}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   pow2 [=>] 99.8%	\[ x + \mathsf{fma}\left(z, \left(1 - \color{blue}{{\log t}^{2}}\right) \cdot \frac{1}{1 + \log t}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]

4. Simplified 99.9%

   \[\leadsto x + \mathsf{fma}\left(z, \color{blue}{\frac{1 - {\log t}^{2}}{1 + \log t}}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   Step-by-step derivation

   [Start] 99.8%	\[ x + \mathsf{fma}\left(z, \left(1 - {\log t}^{2}\right) \cdot \frac{1}{1 + \log t}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   associate-*r/ [=>] 99.9%	\[ x + \mathsf{fma}\left(z, \color{blue}{\frac{\left(1 - {\log t}^{2}\right) \cdot 1}{1 + \log t}}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]
   *-rgt-identity [=>] 99.9%	\[ x + \mathsf{fma}\left(z, \frac{\color{blue}{1 - {\log t}^{2}}}{1 + \log t}, \mathsf{fma}\left(a + -0.5, b, y\right)\right) \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	33024

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

Alternative 2
Accuracy	99.9%
Cost	19904

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

Alternative 3
Accuracy	89.8%
Cost	8140

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+110}:\\ \;\;\;\;x + \left(t_1 + z \cdot \left(1 - \log t\right)\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(x + y\right) + -0.5 \cdot b\right) + a \cdot b\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\left(y + \left(x + z\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + t_1\\ \end{array} \]

Alternative 4
Accuracy	89.8%
Cost	7752

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(x + y\right) + -0.5 \cdot b\right) + a \cdot b\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\left(y + \left(x + z\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + t_1\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	7360

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

Alternative 6
Accuracy	82.2%
Cost	7248

\[\begin{array}{l} t_1 := x + z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(x + y\right) + -0.5 \cdot b\right) + a \cdot b\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+283}:\\ \;\;\;\;\left(x + y\right) + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot \log t\\ \end{array} \]

Alternative 7
Accuracy	82.6%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+214} \lor \neg \left(z \leq 4.8 \cdot 10^{+283}\right):\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y\right) + -0.5 \cdot b\right) + a \cdot b\\ \end{array} \]

Alternative 8
Accuracy	82.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -3.55 \cdot 10^{+213}:\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+285}:\\ \;\;\;\;\left(\left(x + y\right) + -0.5 \cdot b\right) + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot \log t\\ \end{array} \]

Alternative 9
Accuracy	71.0%
Cost	1225

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+79} \lor \neg \left(t_1 \leq 10^{+241}\right):\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + -0.5 \cdot b\\ \end{array} \]

Alternative 10
Accuracy	65.2%
Cost	1097

\[\begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+79} \lor \neg \left(t_1 \leq 5 \cdot 10^{+182}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11
Accuracy	58.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-54}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -0.5 \cdot b\right) + a \cdot b\\ \end{array} \]

Alternative 12
Accuracy	49.7%
Cost	721

\[\begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+138}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{+71} \lor \neg \left(a \leq -2.4 \cdot 10^{+19}\right) \land a \leq 6 \cdot 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 13
Accuracy	78.6%
Cost	704

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

Alternative 14
Accuracy	61.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+165}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 15
Accuracy	78.6%
Cost	576

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

Alternative 16
Accuracy	29.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-295}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 17
Accuracy	28.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+70}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 18
Accuracy	21.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(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)))