NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.6%

Time: 10.0s

Alternatives: 4

Speedup: 1.9×

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: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e+113)
   (* (/ 0.5 a) (/ PI (* b (+ a b))))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8.8e+113) {
		tmp = (0.5 / a) * (((double) M_PI) / (b * (a + b)));
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8.8e+113) {
		tmp = (0.5 / a) * (Math.PI / (b * (a + b)));
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8.8e+113:
		tmp = (0.5 / a) * (math.pi / (b * (a + b)))
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8.8e+113)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.8e+113)
		tmp = (0.5 / a) * (pi / (b * (a + b)));
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 8.8e+113], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.80000000000000041e113

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. sub-neg 79.6%

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

      3. distribute-neg-frac 79.6%

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

      4. metadata-eval 79.6%

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

      5. *-commutative 79.6%

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

      6. associate-*r/ 79.6%

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

      7. associate-*l/ 79.7%

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

      8. *-lft-identity 79.7%

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

      9. difference-of-squares 86.9%

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

      10. associate-/r* 87.4%

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

   3. Simplified 87.4%

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

      1. *-commutative 87.4%

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

      2. associate-/l/ 87.4%

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

      3. frac-add 87.4%

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

      4. frac-times 99.1%

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

      5. +-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. associate-/l* 99.1%

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

      2. +-commutative 99.1%

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

      3. fma-define 99.1%

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

      4. associate-*l* 99.1%

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

   8. Applied egg-rr 99.1%

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

   9. Taylor expanded in a around 0 99.6%

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

       1. frac-times 99.2%

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

   11. Applied egg-rr 99.2%

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

       1. times-frac 99.6%

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

       2. *-commutative 99.6%

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

       3. associate-/l/ 99.6%

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

       4. *-commutative 99.6%

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

       5. times-frac 95.1%

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

       6. associate-/l* 95.0%

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

   13. Simplified 95.0%

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

   if 8.80000000000000041e113 < b

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. sub-neg 75.7%

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

      3. distribute-neg-frac 75.7%

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

      4. metadata-eval 75.7%

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

      5. *-commutative 75.7%

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

      6. associate-*r/ 75.7%

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

      7. associate-*l/ 75.7%

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

      8. *-lft-identity 75.7%

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

      9. difference-of-squares 88.2%

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

      10. associate-/r* 88.1%

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

   3. Simplified 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-/l/ 88.1%

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

      3. frac-add 88.1%

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

      4. frac-times 99.8%

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

      5. +-commutative 99.8%

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

      6. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-define 99.8%

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

      4. associate-*l* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in a around 0 99.8%

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

   10. Taylor expanded in a around 0 99.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= a -2.7e+14) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (a <= -2.7e+14) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = t_0 * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (a <= -2.7e+14) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if a <= -2.7e+14:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = t_0 * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (a <= -2.7e+14)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(t_0 * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (a <= -2.7e+14)
		tmp = t_0 * (pi / a);
	else
		tmp = t_0 * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.7e+14], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e14

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. sub-neg 83.3%

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

      3. distribute-neg-frac 83.3%

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

      4. metadata-eval 83.3%

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

      5. *-commutative 83.3%

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

      6. associate-*r/ 83.3%

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

      7. associate-*l/ 83.4%

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

      8. *-lft-identity 83.4%

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

      9. difference-of-squares 86.7%

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

      10. associate-/r* 88.5%

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

   3. Simplified 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-/l/ 88.6%

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

      3. frac-add 88.5%

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

      4. frac-times 98.9%

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

      5. +-commutative 98.9%

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

      6. *-un-lft-identity 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-/l* 98.9%

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

      2. +-commutative 98.9%

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

      3. fma-define 98.9%

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

      4. associate-*l* 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in a around 0 99.7%

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

   10. Taylor expanded in a around inf 88.4%

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

   if -2.7e14 < a

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. sub-neg 77.8%

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

      3. distribute-neg-frac 77.8%

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

      4. metadata-eval 77.8%

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

      5. *-commutative 77.8%

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

      6. associate-*r/ 77.8%

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

      7. associate-*l/ 77.9%

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

      8. *-lft-identity 77.9%

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

      9. difference-of-squares 87.1%

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

      10. associate-/r* 87.2%

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

   3. Simplified 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-/l/ 87.2%

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

      3. frac-add 87.1%

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

      4. frac-times 99.3%

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

      5. +-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-/l* 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-define 99.3%

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

      4. associate-*l* 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in a around 0 99.6%

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

   10. Taylor expanded in a around 0 66.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{a + b} \cdot \frac{0.5}{a \cdot b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ PI (+ a b)) (/ 0.5 (* a b))))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(pi / Float64(a + b)) * Float64(0.5 / Float64(a * b)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (pi / (a + b)) * (0.5 / (a * b));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. sub-neg 79.1%

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

   3. distribute-neg-frac 79.1%

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

   4. metadata-eval 79.1%

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

   5. *-commutative 79.1%

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

   6. associate-*r/ 79.1%

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

   7. associate-*l/ 79.2%

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

   8. *-lft-identity 79.2%

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

   9. difference-of-squares 87.0%

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

   10. associate-/r* 87.5%

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

3. Simplified 87.5%

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

   1. *-commutative 87.5%

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

   2. associate-/l/ 87.5%

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

   3. frac-add 87.5%

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

   4. frac-times 99.2%

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

   5. +-commutative 99.2%

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

   6. *-un-lft-identity 99.2%

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

6. Applied egg-rr 99.2%

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

   1. associate-/l* 99.2%

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

   2. +-commutative 99.2%

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

   3. fma-define 99.2%

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

   4. associate-*l* 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in a around 0 99.6%

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

10. Final simplification 99.6%

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

Alternative 4: 63.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{0.5}{a \cdot b} \cdot \frac{\pi}{a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ 0.5 (* a b)) (/ PI a)))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (0.5 / (a * b)) * (math.pi / a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / (a * b)) * (pi / a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. sub-neg 79.1%

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

   3. distribute-neg-frac 79.1%

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

   4. metadata-eval 79.1%

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

   5. *-commutative 79.1%

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

   6. associate-*r/ 79.1%

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

   7. associate-*l/ 79.2%

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

   8. *-lft-identity 79.2%

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

   9. difference-of-squares 87.0%

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

   10. associate-/r* 87.5%

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

3. Simplified 87.5%

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

   1. *-commutative 87.5%

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

   2. associate-/l/ 87.5%

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

   3. frac-add 87.5%

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

   4. frac-times 99.2%

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

   5. +-commutative 99.2%

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

   6. *-un-lft-identity 99.2%

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

6. Applied egg-rr 99.2%

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

   1. associate-/l* 99.2%

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

   2. +-commutative 99.2%

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

   3. fma-define 99.2%

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

   4. associate-*l* 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in a around 0 99.6%

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

10. Taylor expanded in a around inf 64.4%

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

11. Final simplification 64.4%

    \[\leadsto \frac{0.5}{a \cdot b} \cdot \frac{\pi}{a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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))))