NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 0.2

Time: 14.9s

Precision: binary64

Cost: 7040

Math

\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

↓

\[0.5 \cdot \frac{\frac{\pi}{a + b}}{a \cdot b} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

↓

(FPCore (a b) :precision binary64 (* 0.5 (/ (/ PI (+ a b)) (* a b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

↓

double code(double a, double b) {
	return 0.5 * ((((double) M_PI) / (a + b)) / (a * b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

↓

public static double code(double a, double b) {
	return 0.5 * ((Math.PI / (a + b)) / (a * b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

↓

def code(a, b):
	return 0.5 * ((math.pi / (a + b)) / (a * b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

↓

function code(a, b)
	return Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(a * b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

↓

function tmp = code(a, b)
	tmp = 0.5 * ((pi / (a + b)) / (a * b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.2

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

2. Simplified 14.2

   \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]
   Proof

   [Start] 14.2	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   rational.json-simplify-2 [=>] 14.2	\[ \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]
   rational.json-simplify-43 [=>] 14.2	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

3. Applied egg-rr 0.8

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]

4. Simplified 0.3

   \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{\pi}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]
   Proof

   [Start] 0.8	\[ \frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]
   rational.json-simplify-49 [=>] 0.8	\[ \color{blue}{0.5 \cdot \frac{\pi}{\left(b + a\right) \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]
   rational.json-simplify-46 [=>] 0.3	\[ 0.5 \cdot \color{blue}{\frac{\frac{\pi}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]
   rational.json-simplify-1 [=>] 0.3	\[ 0.5 \cdot \frac{\frac{\pi}{\color{blue}{a + b}}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

5. Taylor expanded in b around 0 0.2

   \[\leadsto 0.5 \cdot \frac{\frac{\pi}{a + b}}{\color{blue}{a \cdot b}} \]

6. Final simplification 0.2

   \[\leadsto 0.5 \cdot \frac{\frac{\pi}{a + b}}{a \cdot b} \]

Alternatives

Alternative 1
Error	7.0
Cost	7176

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \mathbf{if}\;b \leq -1120000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 560:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -520000:\\ \;\;\;\;\frac{0.5}{b \cdot \left(a \cdot b\right)} \cdot \pi\\ \mathbf{elif}\;b \leq 100000:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]

Alternative 3
Error	26.0
Cost	6912

\[0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)} \]

Alternative 4
Error	25.9
Cost	6912

\[0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b} \]

Reproduce

herbie shell --seed 2023075 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))