NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.5% → 99.7%

Time: 9.5s

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: 77.5% 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: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $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. associate-*r/ 79.2%

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

   2. *-rgt-identity 79.2%

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

   3. associate-*l/ 79.2%

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

   4. difference-of-squares 90.9%

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

   5. *-commutative 90.9%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in b around 0 99.7%

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

   1. expm1-log1p-u 68.6%

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

   2. expm1-udef 58.9%

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

   3. associate-*r/ 58.9%

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

   4. times-frac 58.9%

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

8. Applied egg-rr 58.9%

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

   1. expm1-def 68.6%

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

   2. expm1-log1p 99.6%

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

   3. *-commutative 99.6%

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

10. Simplified 99.6%

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

    1. associate-*r/ 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 35.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+115} \lor \neg \left(a \leq -5.2 \cdot 10^{-204}\right) \land a \leq -1.58 \cdot 10^{-291}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.7e+115) (and (not (<= a -5.2e-204)) (<= a -1.58e-291)))
   (* (/ PI a) (/ -0.5 (* b a)))
   (* (/ PI (* b a)) (/ -0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.7e+115) || (!(a <= -5.2e-204) && (a <= -1.58e-291))) {
		tmp = (((double) M_PI) / a) * (-0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.7e+115) || (!(a <= -5.2e-204) && (a <= -1.58e-291))) {
		tmp = (Math.PI / a) * (-0.5 / (b * a));
	} else {
		tmp = (Math.PI / (b * a)) * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.7e+115) or (not (a <= -5.2e-204) and (a <= -1.58e-291)):
		tmp = (math.pi / a) * (-0.5 / (b * a))
	else:
		tmp = (math.pi / (b * a)) * (-0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.7e+115) || (!(a <= -5.2e-204) && (a <= -1.58e-291)))
		tmp = Float64(Float64(pi / a) * Float64(-0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.7e+115) || (~((a <= -5.2e-204)) && (a <= -1.58e-291)))
		tmp = (pi / a) * (-0.5 / (b * a));
	else
		tmp = (pi / (b * a)) * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.7e+115], And[N[Not[LessEqual[a, -5.2e-204]], $MachinePrecision], LessEqual[a, -1.58e-291]]], N[(N[(Pi / a), $MachinePrecision] * N[(-0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7e115 or -5.19999999999999965e-204 < a < -1.58000000000000004e-291

   1. Initial program 65.5%

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

      1. associate-*r/ 65.4%

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

      2. *-rgt-identity 65.4%

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

      3. associate-*l/ 65.4%

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

      4. difference-of-squares 89.3%

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

      5. *-commutative 89.3%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 68.5%

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

   5. Taylor expanded in b around 0 73.8%

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

      1. associate-*r/ 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. frac-2neg 73.8%

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

      3. metadata-eval 73.8%

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

      4. frac-times 73.3%

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

      5. *-un-lft-identity 73.3%

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

      6. *-commutative 73.3%

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

   9. Applied egg-rr 73.3%

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

       1. *-commutative 73.3%

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

       2. times-frac 73.8%

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

       3. add-sqr-sqrt 35.2%

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

       4. sqrt-unprod 60.4%

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

       5. sqr-neg 60.4%

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

       6. sqrt-unprod 25.2%

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

       7. add-sqr-sqrt 59.5%

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

       8. *-commutative 59.5%

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

   11. Applied egg-rr 59.5%

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

   if -1.7e115 < a < -5.19999999999999965e-204 or -1.58000000000000004e-291 < a

   1. Initial program 84.5%

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

      1. associate-*r/ 84.5%

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

      2. *-rgt-identity 84.5%

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

      3. associate-*l/ 84.5%

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

      4. difference-of-squares 91.5%

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

      5. *-commutative 91.5%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 60.4%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{-1}{a \cdot b}} \]
   5. Step-by-step derivation

      1. *-commutative 60.4%

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

      2. clear-num 60.4%

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

      3. frac-times 60.4%

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

      4. metadata-eval 60.4%

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

      5. *-commutative 60.4%

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

      6. div-inv 60.4%

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

      7. metadata-eval 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. associate-*r/ 60.5%

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

      2. associate-/l* 60.4%

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

      3. neg-mul-1 60.4%

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

      4. distribute-rgt-neg-in 60.4%

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

      5. metadata-eval 60.4%

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

      6. *-commutative 60.4%

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

      7. *-commutative 60.4%

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

   8. Simplified 60.4%

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

      1. associate-*r* 58.4%

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

      2. times-frac 58.3%

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

   10. Applied egg-rr 58.3%

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

   11. Taylor expanded in b around inf 26.0%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+115} \lor \neg \left(a \leq -5.2 \cdot 10^{-204}\right) \land a \leq -1.58 \cdot 10^{-291}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 3: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e+102)
   (/ 0.5 (* a (* a (/ b PI))))
   (* (/ PI (* b a)) (/ -0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+102) {
		tmp = 0.5 / (a * (a * (b / ((double) M_PI))));
	} else {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+102) {
		tmp = 0.5 / (a * (a * (b / Math.PI)));
	} else {
		tmp = (Math.PI / (b * a)) * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e+102:
		tmp = 0.5 / (a * (a * (b / math.pi)))
	else:
		tmp = (math.pi / (b * a)) * (-0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e+102)
		tmp = Float64(0.5 / Float64(a * Float64(a * Float64(b / pi))));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.8e+102)
		tmp = 0.5 / (a * (a * (b / pi)));
	else
		tmp = (pi / (b * a)) * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e+102], N[(0.5 / N[(a * N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000005e102

   1. Initial program 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-rgt-identity 83.9%

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

      3. associate-*l/ 83.9%

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

      4. difference-of-squares 92.1%

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

      5. *-commutative 92.1%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 62.8%

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

   5. Taylor expanded in b around 0 62.6%

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

      1. associate-*r/ 62.6%

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

   7. Simplified 62.6%

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

      1. associate-/l* 62.5%

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

      2. frac-times 62.3%

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

      3. metadata-eval 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. expm1-log1p-u 39.3%

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

       2. expm1-udef 23.5%

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

       3. *-commutative 23.5%

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

   11. Applied egg-rr 23.5%

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

       1. expm1-def 39.3%

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

       2. expm1-log1p 62.3%

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

       3. associate-*r* 62.3%

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

       4. associate-*l/ 62.4%

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

       5. associate-*r/ 62.3%

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

   13. Simplified 62.3%

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

   if 5.8000000000000005e102 < b

   1. Initial program 58.1%

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

      1. associate-*r/ 58.2%

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

      2. *-rgt-identity 58.2%

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

      3. associate-*l/ 58.1%

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

      4. difference-of-squares 85.8%

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

      5. *-commutative 85.8%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 62.3%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{-1}{a \cdot b}} \]
   5. Step-by-step derivation

      1. *-commutative 62.3%

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

      2. clear-num 62.3%

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

      3. frac-times 62.4%

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

      4. metadata-eval 62.4%

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

      5. *-commutative 62.4%

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

      6. div-inv 62.4%

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

      7. metadata-eval 62.4%

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

   6. Applied egg-rr 62.4%

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

      1. associate-*r/ 62.4%

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

      2. associate-/l* 62.4%

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

      3. neg-mul-1 62.4%

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

      4. distribute-rgt-neg-in 62.4%

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

      5. metadata-eval 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

   8. Simplified 62.4%

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

      1. associate-*r* 62.4%

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

      2. times-frac 62.3%

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

   10. Applied egg-rr 62.3%

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

   11. Taylor expanded in b around inf 62.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 4: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot \left(b \cdot a\right)}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e+101)
   (/ 0.5 (/ (* a (* b a)) PI))
   (* (/ PI (* b a)) (/ -0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+101) {
		tmp = 0.5 / ((a * (b * a)) / ((double) M_PI));
	} else {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+101) {
		tmp = 0.5 / ((a * (b * a)) / Math.PI);
	} else {
		tmp = (Math.PI / (b * a)) * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.4e+101:
		tmp = 0.5 / ((a * (b * a)) / math.pi)
	else:
		tmp = (math.pi / (b * a)) * (-0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e+101)
		tmp = Float64(0.5 / Float64(Float64(a * Float64(b * a)) / pi));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.4e+101)
		tmp = 0.5 / ((a * (b * a)) / pi);
	else
		tmp = (pi / (b * a)) * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e+101], N[(0.5 / N[(N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.39999999999999988e101

   1. Initial program 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-rgt-identity 83.9%

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

      3. associate-*l/ 83.9%

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

      4. difference-of-squares 92.1%

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

      5. *-commutative 92.1%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 62.8%

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

   5. Taylor expanded in b around 0 62.6%

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

      1. associate-*r/ 62.6%

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

   7. Simplified 62.6%

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

      1. associate-/l* 62.5%

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

      2. frac-times 62.3%

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

      3. metadata-eval 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. associate-*l/ 62.4%

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

       2. *-commutative 62.4%

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

   11. Applied egg-rr 62.4%

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

   if 2.39999999999999988e101 < b

   1. Initial program 58.1%

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

      1. associate-*r/ 58.2%

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

      2. *-rgt-identity 58.2%

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

      3. associate-*l/ 58.1%

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

      4. difference-of-squares 85.8%

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

      5. *-commutative 85.8%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 62.3%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{-1}{a \cdot b}} \]
   5. Step-by-step derivation

      1. *-commutative 62.3%

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

      2. clear-num 62.3%

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

      3. frac-times 62.4%

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

      4. metadata-eval 62.4%

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

      5. *-commutative 62.4%

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

      6. div-inv 62.4%

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

      7. metadata-eval 62.4%

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

   6. Applied egg-rr 62.4%

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

      1. associate-*r/ 62.4%

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

      2. associate-/l* 62.4%

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

      3. neg-mul-1 62.4%

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

      4. distribute-rgt-neg-in 62.4%

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

      5. metadata-eval 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

   8. Simplified 62.4%

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

      1. associate-*r* 62.4%

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

      2. times-frac 62.3%

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

   10. Applied egg-rr 62.3%

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

   11. Taylor expanded in b around inf 62.3%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot \left(b \cdot a\right)}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 5: 93.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * N[(b + a), $MachinePrecision]), $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. associate-*r/ 79.2%

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

   2. *-rgt-identity 79.2%

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

   3. associate-*l/ 79.2%

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

   4. difference-of-squares 90.9%

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

   5. *-commutative 90.9%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in b around 0 99.7%

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

   1. expm1-log1p-u 80.6%

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

   2. expm1-udef 56.9%

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

   3. associate-*r/ 56.9%

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

   4. times-frac 56.9%

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

   5. +-commutative 56.9%

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

8. Applied egg-rr 56.9%

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

   1. expm1-def 80.6%

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

   2. expm1-log1p 99.6%

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

   3. times-frac 99.7%

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

   4. associate-*r/ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-/r* 99.6%

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

   7. associate-/l/ 93.6%

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

10. Simplified 93.6%

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

11. Final simplification 93.6%

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

Alternative 6: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. associate-*r/ 79.2%

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

   2. *-rgt-identity 79.2%

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

   3. associate-*l/ 79.2%

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

   4. difference-of-squares 90.9%

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

   5. *-commutative 90.9%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in b around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 30.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(Pi / a), $MachinePrecision] * N[(-0.5 / 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. associate-*r/ 79.2%

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

   2. *-rgt-identity 79.2%

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

   3. associate-*l/ 79.2%

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

   4. difference-of-squares 90.9%

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

   5. *-commutative 90.9%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 62.7%

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

5. Taylor expanded in b around 0 60.9%

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

   1. associate-*r/ 60.9%

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

7. Simplified 60.9%

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

   1. *-commutative 60.9%

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

   2. frac-2neg 60.9%

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

   3. metadata-eval 60.9%

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

   4. frac-times 60.8%

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

   5. *-un-lft-identity 60.8%

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

   6. *-commutative 60.8%

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

9. Applied egg-rr 60.8%

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

    1. *-commutative 60.8%

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

    2. times-frac 60.9%

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

    3. add-sqr-sqrt 28.9%

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

    4. sqrt-unprod 41.4%

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

    5. sqr-neg 41.4%

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

    6. sqrt-unprod 16.2%

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

    7. add-sqr-sqrt 33.3%

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

    8. *-commutative 33.3%

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

11. Applied egg-rr 33.3%

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

12. Final simplification 33.3%

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

Reproduce

herbie shell --seed 2023332 
(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))))