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

Average Error: 0.15% → 0.18%

Time: 19.8s

Precision: binary64

Cost: 33280

Math

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

↓

\[\left(\left(x + y\right) + \left(z - \left(z \cdot \log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + z \cdot \log \left(\sqrt[3]{t}\right)\right)\right)\right) + b \cdot \left(a + -0.5\right) \]

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 y) (- z (+ (* z (log (pow (cbrt t) 2.0))) (* z (log (cbrt t))))))
  (* b (+ a -0.5))))

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 + y) + (z - ((z * log(pow(cbrt(t), 2.0))) + (z * log(cbrt(t)))))) + (b * (a + -0.5));
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (z - ((z * Math.log(Math.pow(Math.cbrt(t), 2.0))) + (z * Math.log(Math.cbrt(t)))))) + (b * (a + -0.5));
}

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(Float64(Float64(x + y) + Float64(z - Float64(Float64(z * log((cbrt(t) ^ 2.0))) + Float64(z * log(cbrt(t)))))) + Float64(b * Float64(a + -0.5)))
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[(N[(N[(x + y), $MachinePrecision] + N[(z - N[(N[(z * N[Log[N[Power[N[Power[t, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.15%
Target	0.68%
Herbie	0.18%

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

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 0.15

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

   [Start] 0.15	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   associate--l+ [=>] 0.15	\[ \color{blue}{\left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right)} + \left(a - 0.5\right) \cdot b \]
   remove-double-neg [<=] 0.15	\[ \left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right) + \color{blue}{\left(-\left(-\left(a - 0.5\right)\right)\right)} \cdot b \]
   remove-double-neg [=>] 0.15	\[ \left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right) + \color{blue}{\left(a - 0.5\right)} \cdot b \]
   sub-neg [=>] 0.15	\[ \left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right) + \color{blue}{\left(a + \left(-0.5\right)\right)} \cdot b \]
   metadata-eval [=>] 0.15	\[ \left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right) + \left(a + \color{blue}{-0.5}\right) \cdot b \]

3. Applied egg-rr 33.88

   \[\leadsto \left(\left(x + y\right) + \left(z - \color{blue}{\sqrt[3]{{\left(z \cdot \log t\right)}^{3}}}\right)\right) + \left(a + -0.5\right) \cdot b \]

4. Applied egg-rr 0.18

   \[\leadsto \left(\left(x + y\right) + \left(z - \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) \cdot z + \log \left(\sqrt[3]{t}\right) \cdot z\right)}\right)\right) + \left(a + -0.5\right) \cdot b \]

5. Final simplification 0.18

   \[\leadsto \left(\left(x + y\right) + \left(z - \left(z \cdot \log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + z \cdot \log \left(\sqrt[3]{t}\right)\right)\right)\right) + b \cdot \left(a + -0.5\right) \]

Alternatives

Alternative 1
Error	0.18%
Cost	13888

\[\left(\left(x + y\right) + \left(z + \left(z \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot -3\right)\right) + b \cdot \left(a + -0.5\right) \]

Alternative 2
Error	10.81%
Cost	8008

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\left(x + y\right) + \left(-0.5 \cdot b + a \cdot b\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\left(y + \left(x + z\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot \left(1 - \log t\right) + t_1\right)\\ \end{array} \]

Alternative 3
Error	0.15%
Cost	7616

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

Alternative 4
Error	23.68%
Cost	7492

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ t_2 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{-83}:\\ \;\;\;\;x + \left(t_2 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(y + t_1\right)\\ \end{array} \]

Alternative 5
Error	0.15%
Cost	7360

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

Alternative 6
Error	10.2%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+99} \lor \neg \left(z \leq 5.1 \cdot 10^{+47}\right):\\ \;\;\;\;\left(y + \left(x + z\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(-0.5 \cdot b + a \cdot b\right)\\ \end{array} \]

Alternative 7
Error	14.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+146} \lor \neg \left(z \leq 1.75 \cdot 10^{+199}\right):\\ \;\;\;\;\left(y + z\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(-0.5 \cdot b + a \cdot b\right)\\ \end{array} \]

Alternative 8
Error	16.09%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+154} \lor \neg \left(z \leq 1.05 \cdot 10^{+209}\right):\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + \left(-0.5 \cdot b + a \cdot b\right)\\ \end{array} \]

Alternative 9
Error	15.09%
Cost	6984

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+103}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+208}:\\ \;\;\;\;\left(x + y\right) + \left(-0.5 \cdot b + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	67.73%
Cost	1644

\[\begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+133}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-21}:\\ \;\;\;\;y\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-60}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-91}:\\ \;\;\;\;y\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-204}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-247}:\\ \;\;\;\;y\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+166}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 11
Error	32.21%
Cost	1225

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+164} \lor \neg \left(t_1 \leq 2 \cdot 10^{+54}\right):\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + -0.5 \cdot b\\ \end{array} \]

Alternative 12
Error	39.3%
Cost	1097

\[\begin{array}{l} t_1 := b \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+125} \lor \neg \left(t_1 \leq 10^{+140}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13
Error	25.24%
Cost	969

\[\begin{array}{l} \mathbf{if}\;a + -0.5 \leq -1 \cdot 10^{+23} \lor \neg \left(a + -0.5 \leq -0.5\right):\\ \;\;\;\;\left(x + y\right) + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + -0.5 \cdot b\\ \end{array} \]

Alternative 14
Error	56.19%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-28}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \]

Alternative 15
Error	50.36%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x + y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;x + b \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \]

Alternative 16
Error	23.91%
Cost	704

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

Alternative 17
Error	47.12%
Cost	588

\[\begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{+132}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+104}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{+166}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot b\\ \end{array} \]

Alternative 18
Error	23.91%
Cost	576

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

Alternative 19
Error	69.29%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 20
Error	68.08%
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 21
Error	75.05%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))