NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 11.6s

Alternatives: 8

Speedup: 1.8×

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: 78.5% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. *-commutative 79.2%

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

   2. associate-*r/ 79.2%

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

   3. *-rgt-identity 79.2%

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

   4. difference-of-squares 86.3%

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

   5. associate-/r* 86.6%

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

   6. associate-*r/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-rgt-identity 99.6%

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

   11. associate-/l* 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\pi \cdot 0.5}{a + b}}{b - a} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. *-commutative 79.2%

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

   2. associate-*r/ 79.2%

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

   3. *-rgt-identity 79.2%

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

   4. difference-of-squares 86.3%

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

   5. associate-/r* 86.6%

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

   6. associate-*r/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-rgt-identity 99.6%

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

   11. associate-/l* 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

   2. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot \frac{0.5}{a + b}\right)}{b - a} \]
10. Add Preprocessing

Alternative 3: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e+88)
   (* (/ -0.5 a) (/ (- PI) (* a b)))
   (* (/ -0.5 a) (/ (/ PI b) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e+88) {
		tmp = (-0.5 / a) * (-((double) M_PI) / (a * b));
	} else {
		tmp = (-0.5 / a) * ((((double) M_PI) / b) / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2e+88) {
		tmp = (-0.5 / a) * (-Math.PI / (a * b));
	} else {
		tmp = (-0.5 / a) * ((Math.PI / b) / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2e+88:
		tmp = (-0.5 / a) * (-math.pi / (a * b))
	else:
		tmp = (-0.5 / a) * ((math.pi / b) / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e+88)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(-pi) / Float64(a * b)));
	else
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(pi / b) / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e+88)
		tmp = (-0.5 / a) * (-pi / (a * b));
	else
		tmp = (-0.5 / a) * ((pi / b) / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e+88], N[(N[(-0.5 / a), $MachinePrecision] * N[((-Pi) / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.99999999999999992e88

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r/ 81.5%

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

      3. *-rgt-identity 81.5%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.7%

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

      6. associate-*r/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-rgt-identity 99.6%

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

      11. associate-/l* 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.6%

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

      2. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 64.4%

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

       1. associate-*r/ 64.4%

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

       2. times-frac 64.3%

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

   11. Simplified 64.3%

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

       1. associate-/l* 64.4%

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

   13. Applied egg-rr 64.4%

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

   14. Taylor expanded in b around 0 67.5%

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

       1. associate-*r/ 67.5%

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

       2. neg-mul-1 67.5%

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

   16. Simplified 67.5%

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

   if 1.99999999999999992e88 < b

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*r/ 66.6%

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

      3. *-rgt-identity 66.6%

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

      4. difference-of-squares 85.0%

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

      5. associate-/r* 85.8%

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

      6. associate-*r/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. associate-/l* 99.7%

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

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/l* 99.7%

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

      2. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around inf 64.6%

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

       1. associate-*r/ 64.6%

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

       2. times-frac 64.6%

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

   11. Simplified 64.6%

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

       1. associate-/l* 64.8%

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

   13. Applied egg-rr 64.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a \cdot \frac{b - a}{\frac{\pi}{b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e+88)
   (* (/ -0.5 a) (/ (- PI) (* a b)))
   (/ -0.5 (* a (/ (- b a) (/ PI b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.8e+88) {
		tmp = (-0.5 / a) * (-((double) M_PI) / (a * b));
	} else {
		tmp = -0.5 / (a * ((b - a) / (((double) M_PI) / b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e+88) {
		tmp = (-0.5 / a) * (-Math.PI / (a * b));
	} else {
		tmp = -0.5 / (a * ((b - a) / (Math.PI / b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.8e+88:
		tmp = (-0.5 / a) * (-math.pi / (a * b))
	else:
		tmp = -0.5 / (a * ((b - a) / (math.pi / b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.8e+88)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(-pi) / Float64(a * b)));
	else
		tmp = Float64(-0.5 / Float64(a * Float64(Float64(b - a) / Float64(pi / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e+88)
		tmp = (-0.5 / a) * (-pi / (a * b));
	else
		tmp = -0.5 / (a * ((b - a) / (pi / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.8e+88], N[(N[(-0.5 / a), $MachinePrecision] * N[((-Pi) / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(a * N[(N[(b - a), $MachinePrecision] / N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8000000000000001e88

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r/ 81.5%

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

      3. *-rgt-identity 81.5%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.7%

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

      6. associate-*r/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-rgt-identity 99.6%

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

      11. associate-/l* 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.6%

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

      2. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 64.4%

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

       1. associate-*r/ 64.4%

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

       2. times-frac 64.3%

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

   11. Simplified 64.3%

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

       1. associate-/l* 64.4%

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

   13. Applied egg-rr 64.4%

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

   14. Taylor expanded in b around 0 67.5%

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

       1. associate-*r/ 67.5%

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

       2. neg-mul-1 67.5%

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

   16. Simplified 67.5%

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

   if 1.8000000000000001e88 < b

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*r/ 66.6%

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

      3. *-rgt-identity 66.6%

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

      4. difference-of-squares 85.0%

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

      5. associate-/r* 85.8%

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

      6. associate-*r/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. associate-/l* 99.7%

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

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/l* 99.7%

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

      2. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around inf 64.6%

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

       1. associate-*r/ 64.6%

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

       2. times-frac 64.6%

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

   11. Simplified 64.6%

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

       1. associate-/l* 64.8%

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

   13. Applied egg-rr 64.8%

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

       1. *-commutative 64.8%

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

       2. clear-num 64.8%

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

       3. frac-times 64.8%

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

       4. metadata-eval 64.8%

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

   15. Applied egg-rr 64.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a \cdot \frac{b - a}{\frac{\pi}{b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(b - a\right) \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.4e-91)
   (* (/ -0.5 a) (/ (- PI) (* a b)))
   (/ (* PI 0.5) (* (- b a) (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.4e-91) {
		tmp = (-0.5 / a) * (-((double) M_PI) / (a * b));
	} else {
		tmp = (((double) M_PI) * 0.5) / ((b - a) * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.4e-91) {
		tmp = (-0.5 / a) * (-Math.PI / (a * b));
	} else {
		tmp = (Math.PI * 0.5) / ((b - a) * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.4e-91:
		tmp = (-0.5 / a) * (-math.pi / (a * b))
	else:
		tmp = (math.pi * 0.5) / ((b - a) * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.4e-91)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(-pi) / Float64(a * b)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(b - a) * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.4e-91)
		tmp = (-0.5 / a) * (-pi / (a * b));
	else
		tmp = (pi * 0.5) / ((b - a) * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.4e-91], N[(N[(-0.5 / a), $MachinePrecision] * N[((-Pi) / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.40000000000000027e-91

   1. Initial program 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*r/ 79.3%

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

      3. *-rgt-identity 79.3%

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

      4. difference-of-squares 85.2%

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

      5. associate-/r* 85.4%

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

      6. associate-*r/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-rgt-identity 99.6%

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

      11. associate-/l* 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.6%

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

      2. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 70.0%

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

       1. associate-*r/ 70.0%

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

       2. times-frac 69.9%

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

   11. Simplified 69.9%

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

       1. associate-/l* 70.0%

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

   13. Applied egg-rr 70.0%

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

   14. Taylor expanded in b around 0 71.7%

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

       1. associate-*r/ 71.7%

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

       2. neg-mul-1 71.7%

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

   16. Simplified 71.7%

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

   if 3.40000000000000027e-91 < b

   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. associate-*r/ 79.0%

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

      3. *-rgt-identity 79.0%

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

      4. difference-of-squares 89.0%

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

      5. associate-/r* 89.8%

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

      6. associate-*r/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. associate-/l* 99.7%

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

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/l* 89.8%

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

   10. Applied egg-rr 89.8%

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

       1. *-lft-identity 89.8%

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

       2. associate-/l* 89.8%

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

       3. +-commutative 89.8%

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

   12. Simplified 89.8%

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

       1. frac-add 89.7%

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

       2. associate-*r/ 89.7%

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

       3. frac-times 99.5%

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

       4. *-un-lft-identity 99.5%

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

       5. +-commutative 99.5%

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

   14. Applied egg-rr 99.5%

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

   15. Taylor expanded in b around inf 92.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(b - a\right) \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.6e-91)
   (* (/ -0.5 a) (/ (- PI) (* a b)))
   (/ (/ (* 0.5 (/ PI a)) b) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-91) {
		tmp = (-0.5 / a) * (-((double) M_PI) / (a * b));
	} else {
		tmp = ((0.5 * (((double) M_PI) / a)) / b) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-91) {
		tmp = (-0.5 / a) * (-Math.PI / (a * b));
	} else {
		tmp = ((0.5 * (Math.PI / a)) / b) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e-91:
		tmp = (-0.5 / a) * (-math.pi / (a * b))
	else:
		tmp = ((0.5 * (math.pi / a)) / b) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e-91)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(-pi) / Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(pi / a)) / b) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.6e-91)
		tmp = (-0.5 / a) * (-pi / (a * b));
	else
		tmp = ((0.5 * (pi / a)) / b) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e-91], N[(N[(-0.5 / a), $MachinePrecision] * N[((-Pi) / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.60000000000000023e-91

   1. Initial program 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*r/ 79.3%

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

      3. *-rgt-identity 79.3%

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

      4. difference-of-squares 85.2%

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

      5. associate-/r* 85.4%

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

      6. associate-*r/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-rgt-identity 99.6%

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

      11. associate-/l* 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.6%

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

      2. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 70.0%

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

       1. associate-*r/ 70.0%

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

       2. times-frac 69.9%

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

   11. Simplified 69.9%

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

       1. associate-/l* 70.0%

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

   13. Applied egg-rr 70.0%

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

   14. Taylor expanded in b around 0 71.7%

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

       1. associate-*r/ 71.7%

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

       2. neg-mul-1 71.7%

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

   16. Simplified 71.7%

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

   if 6.60000000000000023e-91 < b

   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. associate-*r/ 79.0%

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

      3. *-rgt-identity 79.0%

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

      4. difference-of-squares 89.0%

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

      5. associate-/r* 89.8%

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

      6. associate-*r/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-rgt-identity 99.7%

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

      11. associate-/l* 99.7%

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

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 92.4%

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

       1. associate-/r* 92.6%

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

       2. associate-*r/ 92.6%

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

   11. Simplified 92.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{-\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. *-commutative 79.2%

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

   2. associate-*r/ 79.2%

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

   3. *-rgt-identity 79.2%

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

   4. difference-of-squares 86.3%

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

   5. associate-/r* 86.6%

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

   6. associate-*r/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-rgt-identity 99.6%

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

   11. associate-/l* 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around inf 67.4%

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

   1. associate-*l/ 67.4%

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

   2. +-commutative 67.4%

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

   3. associate-*r/ 67.3%

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

7. Applied egg-rr 67.3%

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

   1. mul-1-neg 67.3%

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

   2. associate-*r/ 67.4%

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

   3. distribute-neg-frac 67.4%

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

   4. distribute-rgt-neg-in 67.4%

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

   5. metadata-eval 67.4%

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

   6. +-commutative 67.4%

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

9. Simplified 67.4%

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

    1. *-un-lft-identity 67.4%

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

    2. associate-/l/ 67.4%

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

    3. associate-/l* 67.4%

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

    4. +-commutative 67.4%

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

11. Applied egg-rr 67.4%

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

    1. *-lft-identity 67.4%

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

    2. associate-/l* 67.4%

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

    3. *-commutative 67.4%

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

13. Simplified 67.4%

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

14. Taylor expanded in b around 0 99.6%

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

    1. mul-1-neg 99.6%

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

    2. *-commutative 99.6%

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

    3. distribute-rgt-neg-in 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 8: 63.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. *-commutative 79.2%

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

   2. associate-*r/ 79.2%

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

   3. *-rgt-identity 79.2%

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

   4. difference-of-squares 86.3%

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

   5. associate-/r* 86.6%

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

   6. associate-*r/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-rgt-identity 99.6%

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

   11. associate-/l* 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

   2. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around inf 64.4%

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

    1. associate-*r/ 64.4%

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

    2. times-frac 64.4%

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

11. Simplified 64.4%

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

    1. associate-/l* 64.4%

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

13. Applied egg-rr 64.4%

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

14. Taylor expanded in b around 0 62.5%

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

    1. associate-*r/ 62.5%

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

    2. neg-mul-1 62.5%

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

16. Simplified 62.5%

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

17. Final simplification 62.5%

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

Reproduce

herbie shell --seed 2024036 
(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))))