NMSE Section 6.1 mentioned, B

Average Error: 14.5 → 0.7

Time: 17.4s

Precision: binary64

Cost: 7104

Math

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

↓

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

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) (* (+ b a) (* b (- a)))))

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)) / ((b + a) * (b * -a));
}

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) / ((b + a) * (b * -a));
}

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) / ((b + a) * (b * -a))

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(-0.5 * pi) / Float64(Float64(b + a) * Float64(b * Float64(-a))))
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) / ((b + a) * (b * -a));
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[(-0.5 * Pi), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.5

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

2. Simplified 14.5

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

   [Start] 14.5	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   rational_best-simplify-2 [=>] 14.5	\[ \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]
   rational_best-simplify-2 [=>] 14.5	\[ \left(\frac{1}{a} - \frac{1}{b}\right) \cdot \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \]
   rational_best-simplify-44 [=>] 14.5	\[ \color{blue}{\frac{1}{b \cdot b - a \cdot a} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\pi}{2}\right)} \]
   rational_best-simplify-2 [=>] 14.5	\[ \frac{1}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{\pi}{2} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

3. Taylor expanded in a around inf 28.2

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

4. Applied egg-rr 18.8

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

5. Taylor expanded in b around 0 0.7

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

6. Simplified 0.7

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

   [Start] 0.7	\[ \frac{-0.5 \cdot \pi}{\left(b + a\right) \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]
   rational_best-simplify-2 [=>] 0.7	\[ \frac{-0.5 \cdot \pi}{\left(b + a\right) \cdot \left(-1 \cdot \color{blue}{\left(b \cdot a\right)}\right)} \]
   rational_best-simplify-44 [=>] 0.7	\[ \frac{-0.5 \cdot \pi}{\left(b + a\right) \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a\right)\right)}} \]
   rational_best-simplify-2 [=>] 0.7	\[ \frac{-0.5 \cdot \pi}{\left(b + a\right) \cdot \left(b \cdot \color{blue}{\left(a \cdot -1\right)}\right)} \]
   rational_best-simplify-12 [=>] 0.7	\[ \frac{-0.5 \cdot \pi}{\left(b + a\right) \cdot \left(b \cdot \color{blue}{\left(-a\right)}\right)} \]

7. Final simplification 0.7

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

Alternatives

Alternative 1
Error	19.1
Cost	7040

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

Reproduce

herbie shell --seed 2023096 
(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))))