NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.0% → 99.6%

Time: 12.1s

Alternatives: 4

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: 79.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. un-div-inv 81.7%

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

   2. difference-of-squares 89.5%

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

   3. associate-/r* 89.7%

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

   4. div-inv 89.7%

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

   5. metadata-eval 89.7%

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

4. Applied egg-rr 89.7%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{-0.5}{b}}{a} \cdot \frac{\pi}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.1e-124)
   (* (/ (/ -0.5 b) a) (/ PI (- b a)))
   (/ (* 0.5 (/ PI b)) (* a (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.1e-124) {
		tmp = ((-0.5 / b) / a) * (((double) M_PI) / (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 <= 5.1e-124) {
		tmp = ((-0.5 / b) / a) * (Math.PI / (b - a));
	} else {
		tmp = (0.5 * (Math.PI / b)) / (a * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.1e-124:
		tmp = ((-0.5 / b) / a) * (math.pi / (b - a))
	else:
		tmp = (0.5 * (math.pi / b)) / (a * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.1e-124)
		tmp = Float64(Float64(Float64(-0.5 / b) / a) * Float64(pi / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.1e-124)
		tmp = ((-0.5 / b) / a) * (pi / (b - a));
	else
		tmp = (0.5 * (pi / b)) / (a * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.1e-124], N[(N[(N[(-0.5 / b), $MachinePrecision] / a), $MachinePrecision] * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1000000000000001e-124

   1. Initial program 79.0%

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

      1. un-div-inv 79.0%

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

      2. difference-of-squares 87.2%

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

      3. associate-/r* 87.4%

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

      4. div-inv 87.4%

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

      5. metadata-eval 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. *-commutative 73.9%

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

      3. *-commutative 73.9%

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

      4. times-frac 73.9%

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

   9. Simplified 73.9%

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

       1. *-un-lft-identity 73.9%

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

       2. associate-*r/ 73.9%

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

       3. *-commutative 73.9%

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

       4. clear-num 73.9%

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

       5. un-div-inv 73.9%

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

   11. Applied egg-rr 73.9%

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

       1. *-lft-identity 73.9%

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

       2. associate-/l/ 65.2%

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

       3. associate-/r/ 65.2%

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

       4. *-commutative 65.2%

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

       5. times-frac 74.0%

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

   13. Simplified 74.0%

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

   if 5.1000000000000001e-124 < b

   1. Initial program 87.1%

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

      1. un-div-inv 87.2%

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

      2. difference-of-squares 94.3%

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

      3. associate-/r* 94.3%

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

      4. div-inv 94.3%

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

      5. metadata-eval 94.3%

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in a around 0 76.3%

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

      1. frac-times 80.7%

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

      2. associate-/l* 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. *-rgt-identity 80.7%

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

      2. associate-*r/ 80.7%

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

      3. +-commutative 80.7%

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

      4. *-commutative 80.7%

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

   9. Simplified 80.7%

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

   10. Taylor expanded in a around 0 80.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{-0.5}{b}}{a} \cdot \frac{\pi}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. associate-*r* 81.7%

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

   3. associate-*r/ 81.7%

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

   4. associate-*r* 81.7%

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

   5. *-rgt-identity 81.7%

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

   6. sub-neg 81.7%

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

   7. distribute-neg-frac 81.7%

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

   8. metadata-eval 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in a around inf 58.9%

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

   1. difference-of-squares 66.7%

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

7. Applied egg-rr 66.7%

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

   1. times-frac 72.9%

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

9. Applied egg-rr 72.9%

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

    1. +-commutative 72.9%

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

    2. associate-/l/ 72.9%

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

11. Simplified 72.9%

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

12. Taylor expanded in b around 0 99.7%

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

    1. associate-*r/ 99.7%

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

    2. mul-1-neg 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 4: 65.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. associate-*r* 81.7%

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

   3. associate-*r/ 81.7%

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

   4. associate-*r* 81.7%

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

   5. *-rgt-identity 81.7%

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

   6. sub-neg 81.7%

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

   7. distribute-neg-frac 81.7%

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

   8. metadata-eval 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in a around inf 58.9%

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

   1. difference-of-squares 66.7%

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

7. Applied egg-rr 66.7%

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

   1. times-frac 72.9%

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

9. Applied egg-rr 72.9%

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

    1. +-commutative 72.9%

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

    2. associate-/l/ 72.9%

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

11. Simplified 72.9%

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

12. Taylor expanded in a around inf 70.1%

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

13. Final simplification 70.1%

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

Reproduce

herbie shell --seed 2024081 
(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))))