NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 15.3s

Precision: binary64

Cost: 7424

Math

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

↓

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

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

Derivation

1. Initial program 79.5%

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

2. Simplified 87.4%

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

   [Start] 79.5%	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   times-frac [<=] 79.6%	\[ \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-commutative [<=] 79.6%	\[ \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   times-frac [=>] 79.6%	\[ \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   difference-of-squares [=>] 87.0%	\[ \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-/r* [=>] 87.4%	\[ \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   metadata-eval [=>] 87.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 87.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 87.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 87.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 87.4%

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

   [Start] 87.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   clear-num [=>] 87.4%	\[ \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   inv-pow [=>] 87.4%	\[ \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

4. Simplified 87.4%

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

   [Start] 87.4%	\[ \left(\frac{{\left(\frac{b + a}{\pi}\right)}^{-1}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   unpow-1 [=>] 87.4%	\[ \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   +-commutative [=>] 87.4%	\[ \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

5. Applied egg-rr 83.4%

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

   [Start] 87.4%	\[ \left(\frac{\frac{1}{\frac{a + b}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   distribute-lft-in [=>] 83.8%	\[ \color{blue}{\left(\frac{\frac{1}{\frac{a + b}{\pi}}}{b - a} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{\frac{1}{\frac{a + b}{\pi}}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b}} \]
   associate-/l/ [=>] 83.6%	\[ \left(\color{blue}{\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}}} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{\frac{1}{\frac{a + b}{\pi}}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b} \]
   associate-/l/ [=>] 83.4%	\[ \left(\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\color{blue}{\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}}} \cdot 0.5\right) \cdot \frac{-1}{b} \]

6. Simplified 87.5%

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

   [Start] 83.4%	\[ \left(\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}} \cdot 0.5\right) \cdot \frac{-1}{b} \]
   distribute-lft-out [=>] 86.9%	\[ \color{blue}{\left(\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   associate-*r* [<=] 86.9%	\[ \color{blue}{\frac{1}{\left(b - a\right) \cdot \frac{a + b}{\pi}} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)} \]
   *-commutative [=>] 86.9%	\[ \frac{1}{\color{blue}{\frac{a + b}{\pi} \cdot \left(b - a\right)}} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right) \]
   associate-/r* [=>] 87.4%	\[ \color{blue}{\frac{\frac{1}{\frac{a + b}{\pi}}}{b - a}} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right) \]
   associate-/l* [<=] 87.5%	\[ \frac{\color{blue}{\frac{1 \cdot \pi}{a + b}}}{b - a} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right) \]
   *-lft-identity [=>] 87.5%	\[ \frac{\frac{\color{blue}{\pi}}{a + b}}{b - a} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right) \]
   distribute-rgt-in [=>] 87.5%	\[ \frac{\frac{\pi}{a + b}}{b - a} \cdot \color{blue}{\left(\frac{1}{a} \cdot 0.5 + \frac{-1}{b} \cdot 0.5\right)} \]
   associate-*l/ [=>] 87.5%	\[ \frac{\frac{\pi}{a + b}}{b - a} \cdot \left(\color{blue}{\frac{1 \cdot 0.5}{a}} + \frac{-1}{b} \cdot 0.5\right) \]
   metadata-eval [=>] 87.5%	\[ \frac{\frac{\pi}{a + b}}{b - a} \cdot \left(\frac{\color{blue}{0.5}}{a} + \frac{-1}{b} \cdot 0.5\right) \]
   associate-*l/ [=>] 87.5%	\[ \frac{\frac{\pi}{a + b}}{b - a} \cdot \left(\frac{0.5}{a} + \color{blue}{\frac{-1 \cdot 0.5}{b}}\right) \]
   metadata-eval [=>] 87.5%	\[ \frac{\frac{\pi}{a + b}}{b - a} \cdot \left(\frac{0.5}{a} + \frac{\color{blue}{-0.5}}{b}\right) \]

7. Applied egg-rr 99.6%

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

   [Start] 87.5%	\[ \frac{\frac{\pi}{a + b}}{b - a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right) \]
   associate-*l/ [=>] 99.6%	\[ \color{blue}{\frac{\frac{\pi}{a + b} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)}{b - a}} \]

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	7424

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

Alternative 2
Accuracy	96.3%
Cost	7952

\[\begin{array}{l} t_0 := \left(\frac{0.5}{a} + \frac{-0.5}{b}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{b \cdot \frac{a}{0.5}}\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\pi}{a + b} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 3
Accuracy	96.6%
Cost	7952

\[\begin{array}{l} t_0 := \frac{0.5}{a} + \frac{-0.5}{b}\\ t_1 := \frac{\pi}{a + b}\\ \mathbf{if}\;b \leq -6 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{b \cdot \frac{a}{0.5}}\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-153}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{t_1 \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+89}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 4
Accuracy	89.7%
Cost	7433

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-50} \lor \neg \left(a \leq 1.02 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\frac{\pi}{a + b} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{b \cdot \frac{a}{0.5}}\\ \end{array} \]

Alternative 5
Accuracy	89.7%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-50} \lor \neg \left(a \leq 3.6 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{b \cdot \frac{a}{0.5}}\\ \end{array} \]

Alternative 6
Accuracy	80.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-63} \lor \neg \left(b \leq 8.5 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \end{array} \]

Alternative 7
Accuracy	80.5%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+20}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 8
Accuracy	80.6%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 9
Accuracy	86.6%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+19}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 10
Accuracy	63.4%
Cost	6912

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

Reproduce

herbie shell --seed 2023263 
(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))))