NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.3

Time: 3.7s

Precision: binary64

Math

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

↓

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

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
 (/ (* (/ (- (/ PI a) (/ PI b)) (+ a b)) -0.5) (- 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 ((((((double) M_PI) / a) - (((double) M_PI) / b)) / (a + b)) * -0.5) / (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 ((((Math.PI / a) - (Math.PI / b)) / (a + b)) * -0.5) / (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 ((((math.pi / a) - (math.pi / b)) / (a + b)) * -0.5) / (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(Float64(Float64(Float64(Float64(pi / a) - Float64(pi / b)) / Float64(a + b)) * -0.5) / 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 = ((((pi / a) - (pi / b)) / (a + b)) * -0.5) / (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[(N[(N[(N[(N[(Pi / a), $MachinePrecision] - N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.3

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

2. Simplified 14.3

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

3. Applied egg-rr 0.3

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

4. Taylor expanded in b around 0 0.3

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

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022150 
(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))))