NMSE Section 6.1 mentioned, B

Average Error: 14.5 → 0.3

Time: 11.7s

Precision: binary64

Cost: 7040

Math

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

↓

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

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

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

   [Start] 14.5	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 14.5	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 14.5	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 14.5	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 14.5	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 14.5	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 29.9

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]

4. Simplified 0.3

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

   [Start] 29.9	\[ e^{\mathsf{log1p}\left(\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b \cdot a\right)}\right)} - 1 \]
   expm1-def [=>] 9.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b \cdot a\right)}\right)\right)} \]
   expm1-log1p [=>] 0.8	\[ \color{blue}{\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]
   times-frac [=>] 0.3	\[ \color{blue}{\frac{\pi}{b + a} \cdot \frac{0.5}{b \cdot a}} \]
   +-commutative [=>] 0.3	\[ \frac{\pi}{\color{blue}{a + b}} \cdot \frac{0.5}{b \cdot a} \]
   *-commutative [=>] 0.3	\[ \frac{\pi}{a + b} \cdot \frac{0.5}{\color{blue}{a \cdot b}} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+66}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\frac{\frac{\pi}{a}}{b}}}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot \frac{a}{\pi}\right)}\\ \end{array} \]

Alternative 2
Error	6.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1400000 \lor \neg \left(a \leq 7 \cdot 10^{-38}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \]

Alternative 3
Error	7.2
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-46} \lor \neg \left(b \leq 24500000\right):\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.72 \cdot 10^{-43}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 11500:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot b}\\ \end{array} \]

Alternative 5
Error	25.6
Cost	6912

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

Reproduce

herbie shell --seed 2022364 
(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))))