NMSE Section 6.1 mentioned, B

Average Error: 14.6 → 0.3

Time: 17.0s

Precision: binary64

Cost: 7552

Math

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

↓

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

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
 (* (/ (- (/ 1.0 a) (/ 1.0 b)) (+ a b)) (/ PI (* (- b a) 2.0))))

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 (((1.0 / a) - (1.0 / b)) / (a + b)) * (((double) M_PI) / ((b - a) * 2.0));
}

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 (((1.0 / a) - (1.0 / b)) / (a + b)) * (Math.PI / ((b - a) * 2.0));
}

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 (((1.0 / a) - (1.0 / b)) / (a + b)) * (math.pi / ((b - a) * 2.0))

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(Float64(Float64(Float64(1.0 / a) - Float64(1.0 / b)) / Float64(a + b)) * Float64(pi / Float64(Float64(b - a) * 2.0)))
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 = (((1.0 / a) - (1.0 / b)) / (a + b)) * (pi / ((b - a) * 2.0));
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[(N[(N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6

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

2. Simplified 14.6

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

   [Start] 14.6	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   rational.json-simplify-2 [=>] 14.6	\[ \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]
   rational.json-simplify-43 [=>] 14.6	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

3. Applied egg-rr 0.7

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

4. Simplified 9.7

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

   [Start] 0.7	\[ \frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]
   rational.json-simplify-46 [=>] 0.3	\[ \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]
   rational.json-simplify-61 [=>] 9.7	\[ \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b - a}{\frac{\pi \cdot 0.5}{b + a}}}} \]
   rational.json-simplify-61 [=>] 9.7	\[ \frac{\frac{1}{a} - \frac{1}{b}}{\color{blue}{\frac{b + a}{\frac{\pi \cdot 0.5}{b - a}}}} \]
   rational.json-simplify-1 [=>] 9.7	\[ \frac{\frac{1}{a} - \frac{1}{b}}{\frac{\color{blue}{a + b}}{\frac{\pi \cdot 0.5}{b - a}}} \]
   rational.json-simplify-2 [=>] 9.7	\[ \frac{\frac{1}{a} - \frac{1}{b}}{\frac{a + b}{\frac{\color{blue}{0.5 \cdot \pi}}{b - a}}} \]

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.8
Cost	7304

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.3
Cost	7176

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{if}\;a \leq -880000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -880000000:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \end{array} \]

Alternative 4
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -280000000:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot b}}{2}}{a}\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	7040

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

Alternative 6
Error	0.3
Cost	7040

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

Alternative 7
Error	25.6
Cost	6912

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

Reproduce

herbie shell --seed 2023074 
(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))))