NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.7%

Time: 11.0s

Alternatives: 11

Speedup: 2.4×

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 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));
}

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));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / 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

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / 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]

Initial Program: 78.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 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));
}

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));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / 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

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / 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]

Alternative 1: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b + a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (/ (* PI 0.5) (* b a)) (+ b a)))

C

double code(double a, double b) {
	return ((((double) M_PI) * 0.5) / (b * a)) / (b + a);
}

Java

public static double code(double a, double b) {
	return ((Math.PI * 0.5) / (b * a)) / (b + a);
}

Python

def code(a, b):
	return ((math.pi * 0.5) / (b * a)) / (b + a)

Julia

function code(a, b)
	return Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / Float64(b + a))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi * 0.5) / (b * a)) / (b + a);
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.6%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. *-lft-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. frac-times N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r/ N/A

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b + a}} \]
7. Add Preprocessing

Alternative 2: 95.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+52)
   (* PI (/ 0.5 (* a (* b (+ b a)))))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+52) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * (b + a))));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+52) {
		tmp = Math.PI * (0.5 / (a * (b * (b + a))));
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e+52:
		tmp = math.pi * (0.5 / (a * (b * (b + a))))
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+52)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * Float64(b + a)))));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+52)
		tmp = pi * (0.5 / (a * (b * (b + a))));
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+52], N[(Pi * N[(0.5 / N[(a * N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e52

   1. Initial program 80.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 95.1%

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

   if 5e52 < b

   1. Initial program 71.5%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. frac-times N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\left(b + a\right) \cdot \color{blue}{\left(b \cdot a\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\left(b + a\right) \cdot \color{blue}{\left(a \cdot b\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot a\right) \cdot b}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot a\right) \cdot b}} \]

      6. lower-*.f64 98.7

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

   8. Applied rewrites 98.7%

      \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(\left(b + a\right) \cdot a\right) \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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))))