NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.1%

Time: 7.2s

Precision: binary64

Cost: 7040

Math

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

↓

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

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

Derivation

1. Initial program 79.7%

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

2. Simplified 79.6%

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

   [Start] 79.7%	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 79.6%	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 79.6%	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 79.6%

   \[\leadsto \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   Step-by-step derivation

   [Start] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   frac-add [=>] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]
   *-un-lft-identity [<=] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

4. Simplified 79.6%

   \[\leadsto \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]
   Step-by-step derivation

   [Start] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{b + a \cdot -1}{a \cdot b} \]
   *-commutative [=>] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]
   neg-mul-1 [<=] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]
   sub-neg [<=] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

5. Applied egg-rr 75.1%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(b - a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(b \cdot a\right)}} \]
   Step-by-step derivation

   [Start] 79.6%	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{b - a}{a \cdot b} \]
   frac-times [=>] 75.1%	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot \left(b - a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(a \cdot b\right)}} \]
   div-inv [=>] 75.1%	\[ \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \left(b - a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(a \cdot b\right)} \]
   metadata-eval [=>] 75.1%	\[ \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \left(b - a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(a \cdot b\right)} \]
   *-commutative [=>] 75.1%	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(b - a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(b \cdot a\right)}} \]

6. Simplified 99.6%

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

   [Start] 75.1%	\[ \frac{\left(\pi \cdot 0.5\right) \cdot \left(b - a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(b \cdot a\right)} \]
   times-frac [=>] 79.6%	\[ \color{blue}{\frac{\pi \cdot 0.5}{b \cdot b - a \cdot a} \cdot \frac{b - a}{b \cdot a}} \]
   associate-*l/ [=>] 79.6%	\[ \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \frac{b - a}{b \cdot a}}{b \cdot b - a \cdot a}} \]
   associate-*l* [=>] 79.6%	\[ \frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{b - a}{b \cdot a}\right)}}{b \cdot b - a \cdot a} \]
   difference-of-squares [=>] 89.7%	\[ \frac{\pi \cdot \left(0.5 \cdot \frac{b - a}{b \cdot a}\right)}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \]
   times-frac [=>] 99.6%	\[ \color{blue}{\frac{\pi}{b + a} \cdot \frac{0.5 \cdot \frac{b - a}{b \cdot a}}{b - a}} \]
   +-commutative [=>] 99.6%	\[ \frac{\pi}{\color{blue}{a + b}} \cdot \frac{0.5 \cdot \frac{b - a}{b \cdot a}}{b - a} \]
   associate-*r/ [=>] 99.6%	\[ \frac{\pi}{a + b} \cdot \frac{\color{blue}{\frac{0.5 \cdot \left(b - a\right)}{b \cdot a}}}{b - a} \]
   *-commutative [=>] 99.6%	\[ \frac{\pi}{a + b} \cdot \frac{\frac{0.5 \cdot \left(b - a\right)}{\color{blue}{a \cdot b}}}{b - a} \]

7. Taylor expanded in b around 0 99.6%

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

8. Simplified 99.7%

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

   [Start] 99.6%	\[ \frac{\pi}{a + b} \cdot \frac{0.5}{a \cdot b} \]
   associate-/l/ [<=] 99.7%	\[ \frac{\pi}{a + b} \cdot \color{blue}{\frac{\frac{0.5}{b}}{a}} \]

9. Applied egg-rr 99.7%

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

   [Start] 99.7%	\[ \frac{\pi}{a + b} \cdot \frac{\frac{0.5}{b}}{a} \]
   associate-/l/ [=>] 99.6%	\[ \frac{\pi}{a + b} \cdot \color{blue}{\frac{0.5}{a \cdot b}} \]
   frac-times [=>] 99.7%	\[ \color{blue}{\frac{\pi \cdot 0.5}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]

10. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	7040

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

Alternative 2
Accuracy	80.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-53} \lor \neg \left(b \leq 4.5 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \end{array} \]

Alternative 3
Accuracy	86.2%
Cost	7177

\[\begin{array}{l} t_0 := \frac{\frac{0.5}{b}}{a}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-52} \lor \neg \left(b \leq 2.4 \cdot 10^{-40}\right):\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{a}\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	7040

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

Alternative 5
Accuracy	99.6%
Cost	7040

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

Alternative 6
Accuracy	99.6%
Cost	7040

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

Alternative 7
Accuracy	56.9%
Cost	6912

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

Reproduce

herbie shell --seed 2023178 
(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))))