NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 10.6s

Alternatives: 5

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: 78.1% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. un-div-inv 75.8%

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

   2. difference-of-squares 89.9%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

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

4. Applied egg-rr 90.2%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   3. *-commutative 99.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \pi}}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   4. +-commutative 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{a + b}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b - a} \]

   7. metadata-eval 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b - a} \]

8. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a}} \]

9. Taylor expanded in a around 0 99.7%

   \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{b \cdot a}} \]
12. Step-by-step derivation

    1. un-div-inv 99.8%

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

    2. associate-/l* 99.8%

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

    3. +-commutative 99.8%

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

13. Applied egg-rr 99.8%

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

Alternative 2: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-73}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.6e-73)
   (* (* 0.5 (/ PI a)) (/ 1.0 (* b a)))
   (/ (* 0.5 PI) (* (* b a) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.6e-73) {
		tmp = (0.5 * (((double) M_PI) / a)) * (1.0 / (b * a));
	} else {
		tmp = (0.5 * ((double) M_PI)) / ((b * a) * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8.6e-73) {
		tmp = (0.5 * (Math.PI / a)) * (1.0 / (b * a));
	} else {
		tmp = (0.5 * Math.PI) / ((b * a) * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8.6e-73:
		tmp = (0.5 * (math.pi / a)) * (1.0 / (b * a))
	else:
		tmp = (0.5 * math.pi) / ((b * a) * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.6e-73)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) * Float64(1.0 / Float64(b * a)));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(b * a) * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.6e-73)
		tmp = (0.5 * (pi / a)) * (1.0 / (b * a));
	else
		tmp = (0.5 * pi) / ((b * a) * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.6e-73], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5999999999999998e-73

   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. Add Preprocessing
   3. Step-by-step derivation

      1. un-div-inv 79.6%

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

      2. difference-of-squares 92.1%

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

      3. associate-/r* 92.1%

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

      4. div-inv 92.1%

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

      5. metadata-eval 92.1%

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

   4. Applied egg-rr 92.1%

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

      1. associate-*l/ 99.7%

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

      2. associate-/l* 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \pi}}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{a + b}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

      5. sub-neg 99.7%

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

      6. distribute-neg-frac 99.7%

         \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b - a} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b - a} \]

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a}} \]

   9. Taylor expanded in a around 0 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
   10. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

   11. Simplified 99.7%

       \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{b \cdot a}} \]

   12. Taylor expanded in a around inf 72.8%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{a}\right)} \cdot \frac{1}{b \cdot a} \]

   if 8.5999999999999998e-73 < b

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-*r* 69.5%

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

      3. associate-*r/ 69.5%

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

      4. associate-*r* 69.5%

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

      5. *-rgt-identity 69.5%

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

      6. sub-neg 69.5%

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

      7. distribute-neg-frac 69.5%

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

      8. metadata-eval 69.5%

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

   3. Simplified 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. difference-of-squares 86.2%

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

      3. times-frac 99.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 93.0%

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

      6. frac-times 93.0%

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

      7. metadata-eval 93.0%

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

      8. metadata-eval 93.0%

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

      9. frac-times 93.0%

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

      10. sqrt-unprod 93.0%

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

      11. add-sqr-sqrt 93.0%

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

      12. div-inv 93.0%

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

      13. metadata-eval 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. associate-*l/ 93.1%

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

      2. *-lft-identity 93.1%

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

      3. associate-/l* 93.1%

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

      4. associate-*l/ 93.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{1}{b}}{b + a} \cdot \frac{\pi \cdot 0.5}{b - a}} \]

      5. +-commutative 93.1%

         \[\leadsto \frac{\color{blue}{\frac{1}{b} + \frac{1}{a}}}{b + a} \cdot \frac{\pi \cdot 0.5}{b - a} \]

      6. +-commutative 93.1%

         \[\leadsto \frac{\frac{1}{b} + \frac{1}{a}}{\color{blue}{a + b}} \cdot \frac{\pi \cdot 0.5}{b - a} \]

      7. *-commutative 93.1%

         \[\leadsto \frac{\frac{1}{b} + \frac{1}{a}}{a + b} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b - a} \]

   8. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{b} + \frac{1}{a}}{a + b} \cdot \frac{0.5 \cdot \pi}{b - a}} \]

   9. Taylor expanded in b around 0 93.1%

      \[\leadsto \color{blue}{\frac{1}{a \cdot b}} \cdot \frac{0.5 \cdot \pi}{b - a} \]
   10. Step-by-step derivation

       1. frac-times 93.1%

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

       2. *-un-lft-identity 93.1%

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

   11. Applied egg-rr 93.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-73}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. un-div-inv 75.8%

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

   2. difference-of-squares 89.9%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

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

4. Applied egg-rr 90.2%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   3. *-commutative 99.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \pi}}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   4. +-commutative 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{a + b}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b - a} \]

   7. metadata-eval 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b - a} \]

8. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a}} \]

9. Taylor expanded in a around 0 99.7%

   \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{b \cdot a}} \]
12. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \color{blue}{\frac{1}{b \cdot a} \cdot \frac{0.5 \cdot \pi}{a + b}} \]

    2. +-commutative 99.7%

       \[\leadsto \frac{1}{b \cdot a} \cdot \frac{0.5 \cdot \pi}{\color{blue}{b + a}} \]

    3. frac-times 99.1%

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

    4. *-un-lft-identity 99.1%

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

13. Applied egg-rr 99.1%

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

14. Final simplification 99.1%

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

Alternative 4: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{b \cdot a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (* 0.5 (/ PI a)) (/ 1.0 (* b a))))

C

double code(double a, double b) {
	return (0.5 * (((double) M_PI) / a)) * (1.0 / (b * a));
}

Java

public static double code(double a, double b) {
	return (0.5 * (Math.PI / a)) * (1.0 / (b * a));
}

Python

def code(a, b):
	return (0.5 * (math.pi / a)) * (1.0 / (b * a))

Julia

function code(a, b)
	return Float64(Float64(0.5 * Float64(pi / a)) * Float64(1.0 / Float64(b * a)))
end

MATLAB

function tmp = code(a, b)
	tmp = (0.5 * (pi / a)) * (1.0 / (b * a));
end

Wolfram

code[a_, b_] := N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. un-div-inv 75.8%

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

   2. difference-of-squares 89.9%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

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

4. Applied egg-rr 90.2%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   3. *-commutative 99.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \pi}}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   4. +-commutative 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{a + b}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \]

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b - a} \]

   7. metadata-eval 99.7%

      \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b - a} \]

8. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a}} \]

9. Taylor expanded in a around 0 99.7%

   \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \color{blue}{\frac{1}{b \cdot a}} \]

12. Taylor expanded in a around inf 65.3%

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{a}\right)} \cdot \frac{1}{b \cdot a} \]
13. Add Preprocessing

Alternative 5: 30.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. associate-*r* 75.8%

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

   3. associate-*r/ 75.8%

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

   4. associate-*r* 75.8%

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

   5. *-rgt-identity 75.8%

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

   6. sub-neg 75.8%

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

   7. distribute-neg-frac 75.8%

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

   8. metadata-eval 75.8%

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

3. Simplified 75.8%

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

   1. *-un-lft-identity 75.8%

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

   2. difference-of-squares 89.9%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 46.6%

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

   5. sqrt-unprod 76.8%

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

   6. frac-times 76.8%

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

   7. metadata-eval 76.8%

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

   8. metadata-eval 76.8%

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

   9. frac-times 76.8%

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

   10. sqrt-unprod 38.6%

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

   11. add-sqr-sqrt 69.5%

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

   12. div-inv 69.5%

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

   13. metadata-eval 69.5%

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

6. Applied egg-rr 69.5%

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

   1. associate-*l/ 69.6%

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

   2. *-lft-identity 69.6%

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

   3. associate-/l* 69.6%

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

   4. associate-*l/ 69.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{1}{b}}{b + a} \cdot \frac{\pi \cdot 0.5}{b - a}} \]

   5. +-commutative 69.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{b} + \frac{1}{a}}}{b + a} \cdot \frac{\pi \cdot 0.5}{b - a} \]

   6. +-commutative 69.6%

      \[\leadsto \frac{\frac{1}{b} + \frac{1}{a}}{\color{blue}{a + b}} \cdot \frac{\pi \cdot 0.5}{b - a} \]

   7. *-commutative 69.6%

      \[\leadsto \frac{\frac{1}{b} + \frac{1}{a}}{a + b} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b - a} \]

8. Simplified 69.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{b} + \frac{1}{a}}{a + b} \cdot \frac{0.5 \cdot \pi}{b - a}} \]

9. Taylor expanded in b around 0 69.6%

   \[\leadsto \color{blue}{\frac{1}{a \cdot b}} \cdot \frac{0.5 \cdot \pi}{b - a} \]

10. Taylor expanded in b around 0 36.1%

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

    1. associate-*l/ 36.1%

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

    2. *-un-lft-identity 36.1%

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

    3. *-commutative 36.1%

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

    4. *-commutative 36.1%

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

12. Applied egg-rr 36.1%

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

Reproduce

herbie shell --seed 2024096 
(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))))