NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 11.5s

Alternatives: 7

Speedup: 1.1×

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: 78.3% 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

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

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(Float64(Float64(0.5 / b) / a) * pi) / Float64(b + a))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. div-inv 80.1%

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

   2. clear-num 80.0%

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

   3. frac-sub 79.9%

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

   4. frac-times 75.0%

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

   5. *-un-lft-identity 75.0%

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

   6. *-un-lft-identity 75.0%

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

   7. div-inv 75.0%

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

   8. metadata-eval 75.0%

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

3. Applied egg-rr 75.0%

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

   1. *-rgt-identity 75.0%

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

   2. *-commutative 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-/l* 75.0%

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

   5. *-commutative 75.0%

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

   6. associate-*l* 75.0%

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

   7. *-commutative 75.0%

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

5. Simplified 75.0%

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

   1. *-un-lft-identity 75.0%

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

   2. times-frac 80.0%

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

   3. *-commutative 80.0%

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

7. Applied egg-rr 80.0%

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

   1. *-lft-identity 80.0%

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

   2. associate-*l/ 80.0%

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

   3. difference-of-squares 86.6%

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

   4. +-commutative 86.6%

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

   5. times-frac 99.6%

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

   6. *-commutative 99.6%

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

   7. times-frac 88.3%

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

9. Simplified 88.3%

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

    1. expm1-log1p-u 73.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{a + b} \cdot \frac{\frac{0.5}{a} \cdot \frac{b - a}{b}}{b - a}\right)\right)} \]

    2. expm1-udef 49.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{a + b} \cdot \frac{\frac{0.5}{a} \cdot \frac{b - a}{b}}{b - a}\right)} - 1} \]

    3. *-commutative 49.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{0.5}{a} \cdot \frac{b - a}{b}}{b - a} \cdot \frac{\pi}{a + b}}\right)} - 1 \]

    4. associate-/l* 49.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{0.5}{a}}{\frac{b - a}{\frac{b - a}{b}}}} \cdot \frac{\pi}{a + b}\right)} - 1 \]

11. Applied egg-rr 49.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.5}{a}}{\frac{b - a}{\frac{b - a}{b}}} \cdot \frac{\pi}{a + b}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 73.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.5}{a}}{\frac{b - a}{\frac{b - a}{b}}} \cdot \frac{\pi}{a + b}\right)\right)} \]

    2. expm1-log1p 88.3%

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

    3. associate-/l/ 88.3%

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

    4. associate-/r* 88.3%

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

    5. associate-/r/ 99.6%

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

    6. *-inverses 99.6%

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

13. Simplified 99.6%

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

    1. associate-*r/ 99.6%

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

    2. *-un-lft-identity 99.6%

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

15. Applied egg-rr 99.6%

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

16. Final simplification 99.6%

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

Alternative 2: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot b}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.7e-133)
   (* (/ (/ 0.5 a) b) (/ PI a))
   (* 0.5 (/ (/ PI (* b b)) a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.7e-133) {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / a);
	} else {
		tmp = 0.5 * ((((double) M_PI) / (b * b)) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.7e-133) {
		tmp = ((0.5 / a) / b) * (Math.PI / a);
	} else {
		tmp = 0.5 * ((Math.PI / (b * b)) / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.7e-133:
		tmp = ((0.5 / a) / b) * (math.pi / a)
	else:
		tmp = 0.5 * ((math.pi / (b * b)) / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.7e-133)
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / a));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / Float64(b * b)) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.7e-133)
		tmp = ((0.5 / a) / b) * (pi / a);
	else
		tmp = 0.5 * ((pi / (b * b)) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.7e-133], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.70000000000000003e-133

   1. Initial program 80.1%

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

      1. div-inv 80.2%

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

      2. clear-num 80.1%

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

      3. frac-sub 80.0%

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

      4. frac-times 74.2%

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

      5. *-un-lft-identity 74.2%

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

      6. *-un-lft-identity 74.2%

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

      7. div-inv 74.2%

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

      8. metadata-eval 74.2%

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

   3. Applied egg-rr 74.2%

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

      1. *-rgt-identity 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r/ 74.3%

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

      4. associate-/l* 74.3%

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

      5. *-commutative 74.3%

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

      6. associate-*l* 74.3%

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

      7. *-commutative 74.3%

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

   5. Simplified 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. times-frac 80.1%

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

      3. *-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. *-lft-identity 80.1%

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

      2. associate-*l/ 80.0%

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

      3. difference-of-squares 87.8%

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

      4. +-commutative 87.8%

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

      5. times-frac 99.6%

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

      6. *-commutative 99.6%

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

      7. times-frac 81.4%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in a around 0 99.6%

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

       1. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around inf 68.6%

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

   if 4.70000000000000003e-133 < b

   1. Initial program 80.0%

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

      1. inv-pow 80.0%

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

      2. difference-of-squares 84.9%

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

      3. unpow-prod-down 84.7%

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

      4. inv-pow 84.7%

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

      5. inv-pow 84.7%

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

   3. Applied egg-rr 84.7%

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

      1. associate-*r/ 84.8%

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

      2. *-rgt-identity 84.8%

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

      3. +-commutative 84.8%

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

   5. Simplified 84.8%

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

   6. Taylor expanded in a around 0 66.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 66.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. times-frac 66.1%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{\pi}{{b}^{2}}} \]

      3. unpow2 66.1%

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

      4. associate-*l/ 66.1%

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

      5. *-lft-identity 66.1%

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

      6. times-frac 66.1%

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

      7. metadata-eval 66.1%

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

   8. Simplified 66.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot b}}{a}\\ \end{array} \]

Alternative 3: 71.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.5}{a}}{b}\\ \mathbf{if}\;b \leq 4.7 \cdot 10^{-133}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ 0.5 a) b)))
   (if (<= b 4.7e-133) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

double code(double a, double b) {
	double t_0 = (0.5 / a) / b;
	double tmp;
	if (b <= 4.7e-133) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = t_0 * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (0.5 / a) / b;
	double tmp;
	if (b <= 4.7e-133) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (0.5 / a) / b
	tmp = 0
	if b <= 4.7e-133:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = t_0 * (math.pi / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(0.5 / a) / b)
	tmp = 0.0
	if (b <= 4.7e-133)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(t_0 * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (0.5 / a) / b;
	tmp = 0.0;
	if (b <= 4.7e-133)
		tmp = t_0 * (pi / a);
	else
		tmp = t_0 * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 4.7e-133], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.70000000000000003e-133

   1. Initial program 80.1%

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

      1. div-inv 80.2%

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

      2. clear-num 80.1%

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

      3. frac-sub 80.0%

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

      4. frac-times 74.2%

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

      5. *-un-lft-identity 74.2%

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

      6. *-un-lft-identity 74.2%

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

      7. div-inv 74.2%

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

      8. metadata-eval 74.2%

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

   3. Applied egg-rr 74.2%

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

      1. *-rgt-identity 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r/ 74.3%

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

      4. associate-/l* 74.3%

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

      5. *-commutative 74.3%

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

      6. associate-*l* 74.3%

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

      7. *-commutative 74.3%

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

   5. Simplified 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. times-frac 80.1%

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

      3. *-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. *-lft-identity 80.1%

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

      2. associate-*l/ 80.0%

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

      3. difference-of-squares 87.8%

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

      4. +-commutative 87.8%

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

      5. times-frac 99.6%

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

      6. *-commutative 99.6%

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

      7. times-frac 81.4%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in a around 0 99.6%

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

       1. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around inf 68.6%

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

   if 4.70000000000000003e-133 < b

   1. Initial program 80.0%

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

      1. div-inv 79.9%

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

      2. clear-num 79.9%

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

      3. frac-sub 79.8%

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

      4. frac-times 76.1%

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

      5. *-un-lft-identity 76.1%

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

      6. *-un-lft-identity 76.1%

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

      7. div-inv 76.1%

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

      8. metadata-eval 76.1%

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

   3. Applied egg-rr 76.1%

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

      1. *-rgt-identity 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*r/ 76.1%

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

      4. associate-/l* 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*l* 76.1%

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

      7. *-commutative 76.1%

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

   5. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. times-frac 79.9%

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

      3. *-commutative 79.9%

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

   7. Applied egg-rr 79.9%

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

      1. *-lft-identity 79.9%

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

      2. associate-*l/ 79.9%

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

      3. difference-of-squares 84.8%

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

      4. +-commutative 84.8%

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

      5. times-frac 99.6%

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

      6. *-commutative 99.6%

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

      7. times-frac 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in a around 0 99.7%

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

       1. associate-/r* 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in a around 0 80.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. div-inv 80.1%

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

   2. clear-num 80.0%

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

   3. frac-sub 79.9%

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

   4. frac-times 75.0%

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

   5. *-un-lft-identity 75.0%

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

   6. *-un-lft-identity 75.0%

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

   7. div-inv 75.0%

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

   8. metadata-eval 75.0%

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

3. Applied egg-rr 75.0%

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

   1. *-rgt-identity 75.0%

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

   2. *-commutative 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-/l* 75.0%

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

   5. *-commutative 75.0%

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

   6. associate-*l* 75.0%

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

   7. *-commutative 75.0%

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

5. Simplified 75.0%

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

   1. *-un-lft-identity 75.0%

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

   2. times-frac 80.0%

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

   3. *-commutative 80.0%

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

7. Applied egg-rr 80.0%

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

   1. *-lft-identity 80.0%

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

   2. associate-*l/ 80.0%

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

   3. difference-of-squares 86.6%

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

   4. +-commutative 86.6%

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

   5. times-frac 99.6%

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

   6. *-commutative 99.6%

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

   7. times-frac 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in a around 0 99.6%

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

11. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. div-inv 80.1%

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

   2. clear-num 80.0%

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

   3. frac-sub 79.9%

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

   4. frac-times 75.0%

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

   5. *-un-lft-identity 75.0%

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

   6. *-un-lft-identity 75.0%

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

   7. div-inv 75.0%

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

   8. metadata-eval 75.0%

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

3. Applied egg-rr 75.0%

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

   1. *-rgt-identity 75.0%

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

   2. *-commutative 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-/l* 75.0%

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

   5. *-commutative 75.0%

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

   6. associate-*l* 75.0%

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

   7. *-commutative 75.0%

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

5. Simplified 75.0%

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

   1. *-un-lft-identity 75.0%

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

   2. times-frac 80.0%

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

   3. *-commutative 80.0%

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

7. Applied egg-rr 80.0%

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

   1. *-lft-identity 80.0%

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

   2. associate-*l/ 80.0%

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

   3. difference-of-squares 86.6%

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

   4. +-commutative 86.6%

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

   5. times-frac 99.6%

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

   6. *-commutative 99.6%

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

   7. times-frac 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in a around 0 99.6%

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

    1. associate-/r* 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 6: 56.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. inv-pow 80.0%

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

   2. difference-of-squares 86.7%

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

   3. unpow-prod-down 86.8%

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

   4. inv-pow 86.8%

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

   5. inv-pow 86.8%

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

3. Applied egg-rr 86.8%

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

   1. associate-*r/ 86.9%

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

   2. *-rgt-identity 86.9%

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

   3. +-commutative 86.9%

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

5. Simplified 86.9%

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

6. Taylor expanded in a around 0 57.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
7. Step-by-step derivation

   1. associate-/r* 57.2%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{\pi}{a}}{{b}^{2}}} \]

   2. unpow2 57.2%

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

8. Simplified 57.2%

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

9. Final simplification 57.2%

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

Alternative 7: 56.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. inv-pow 80.0%

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

   2. difference-of-squares 86.7%

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

   3. unpow-prod-down 86.8%

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

   4. inv-pow 86.8%

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

   5. inv-pow 86.8%

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

3. Applied egg-rr 86.8%

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

   1. associate-*r/ 86.9%

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

   2. *-rgt-identity 86.9%

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

   3. +-commutative 86.9%

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

5. Simplified 86.9%

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

6. Taylor expanded in a around 0 57.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 57.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

   2. times-frac 57.2%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{\pi}{{b}^{2}}} \]

   3. unpow2 57.2%

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

   4. associate-*l/ 57.2%

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

   5. *-lft-identity 57.2%

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

   6. times-frac 57.2%

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

   7. metadata-eval 57.2%

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

8. Simplified 57.2%

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

9. Final simplification 57.2%

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

Reproduce

herbie shell --seed 2023287 
(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))))