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

Average Error: 0.1 → 0.1

Time: 18.3s

Precision: binary64

Cost: 7360

Math

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

↓

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

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 (+ (* (- 1.0 (log t)) z) (+ (* (- 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 + (((1.0 - log(t)) * z) + (((a - 0.5) * b) + y));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + (((1.0d0 - log(t)) * z) + (((a - 0.5d0) * b) + y))
end function

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 + (((1.0 - Math.log(t)) * z) + (((a - 0.5) * b) + y));
}

Python

def code(x, y, z, t, a, b):
	return (((x + y) + z) - (z * math.log(t))) + ((a - 0.5) * b)

↓

def code(x, y, z, t, a, b):
	return x + (((1.0 - math.log(t)) * z) + (((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 + Float64(Float64(Float64(1.0 - log(t)) * z) + Float64(Float64(Float64(a - 0.5) * b) + y)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
end

↓

function tmp = code(x, y, z, t, a, b)
	tmp = x + (((1.0 - log(t)) * z) + (((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[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision] + 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

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

   [Start] 0.1	\[ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   rational.json-simplify-1 [=>] 0.1	\[ \color{blue}{\left(a - 0.5\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]
   rational.json-simplify-1 [=>] 0.1	\[ \left(a - 0.5\right) \cdot b + \left(\color{blue}{\left(z + \left(x + y\right)\right)} - z \cdot \log t\right) \]
   rational.json-simplify-48 [=>] 0.1	\[ \left(a - 0.5\right) \cdot b + \color{blue}{\left(\left(x + y\right) + \left(z - z \cdot \log t\right)\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ \color{blue}{\left(x + y\right) + \left(\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\right)} \]
   rational.json-simplify-1 [=>] 0.1	\[ \color{blue}{\left(\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\right) + \left(x + y\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ \color{blue}{x + \left(y + \left(\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\right)\right)} \]
   rational.json-simplify-1 [=>] 0.1	\[ x + \left(y + \color{blue}{\left(\left(a - 0.5\right) \cdot b + \left(z - z \cdot \log t\right)\right)}\right) \]

3. Taylor expanded in z around 0 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.4
Cost	8272

\[\begin{array}{l} t_1 := z \cdot \log t\\ t_2 := x + \left(y + \left(\left(-0.5 \cdot b + z\right) - t_1\right)\right)\\ t_3 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;x + \left(\left(t_3 + z\right) - t_1\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq 10^{+107}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z + t_3\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+181}:\\ \;\;\;\;x + \left(t_3 + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	11.4
Cost	8272

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := x + \left(y + \left(\left(-0.5 \cdot b + z\right) - z \cdot \log t\right)\right)\\ t_3 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;x + y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;t_3 + \left(t_1 + x\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x + y \leq 10^{+107}:\\ \;\;\;\;t_3 + t_1\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+181}:\\ \;\;\;\;x + \left(t_1 + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	4.4
Cost	7496

\[\begin{array}{l} t_1 := x + \left(y + \left(\left(-0.5 \cdot b + z\right) - z \cdot \log t\right)\right)\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+26}:\\ \;\;\;\;x + \left(\left(a - 0.5\right) \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.1
Cost	7360

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

Alternative 5
Error	6.3
Cost	7240

\[\begin{array}{l} t_1 := x + \left(y + \left(z - z \cdot \log t\right)\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;x + \left(\left(a - 0.5\right) \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	6.2
Cost	7240

\[\begin{array}{l} t_1 := x + \left(\left(1 - \log t\right) \cdot z + y\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+132}:\\ \;\;\;\;x + \left(\left(a - 0.5\right) \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	8.2
Cost	7112

\[\begin{array}{l} t_1 := x + \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;x + \left(\left(a - 0.5\right) \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	9.8
Cost	6984

\[\begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+197}:\\ \;\;\;\;x + \left(\left(a - 0.5\right) \cdot b + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	33.3
Cost	1360

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10
Error	32.3
Cost	1360

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{-80}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 11
Error	32.0
Cost	1360

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;-0.5 \cdot b + x\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 12
Error	27.1
Cost	1228

\[\begin{array}{l} t_1 := x + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x + y \leq 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13
Error	37.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x + y \leq 5000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 14
Error	25.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;x + y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;x + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.5 \cdot b + y\right)\\ \end{array} \]

Alternative 15
Error	15.2
Cost	576

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

Alternative 16
Error	48.8
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 17
Error	47.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(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)))