NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.3% → 99.6%

Time: 10.1s

Alternatives: 10

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: 79.3% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (/ (/ PI 2.0) (- b a)) (/ (+ (/ 1.0 a) (/ -1.0 b)) (+ b a))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) / (b - a)) * (((1.0 / a) + (-1.0 / b)) / (b + a));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) / (b - a)) * (((1.0 / a) + (-1.0 / b)) / (b + a));
}

Python

def code(a, b):
	return ((math.pi / 2.0) / (b - a)) * (((1.0 / a) + (-1.0 / b)) / (b + a))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) / Float64(b - a)) * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(b + a)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) / (b - a)) * (((1.0 / a) + (-1.0 / b)) / (b + a));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. associate-*r/ 80.3%

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

   2. *-rgt-identity 80.3%

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

   3. associate-*l/ 80.3%

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

   4. difference-of-squares 90.4%

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

   5. *-commutative 90.4%

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

   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. Add Preprocessing

5. Final simplification 99.7%

   \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a} \]
6. Add Preprocessing

Alternative 2: 96.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\frac{-\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+98)
   (* (/ (- PI) a) (/ -0.5 (* b a)))
   (* 0.5 (/ PI (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+98) {
		tmp = (-((double) M_PI) / a) * (-0.5 / (b * a));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+98) {
		tmp = (-Math.PI / a) * (-0.5 / (b * a));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+98:
		tmp = (-math.pi / a) * (-0.5 / (b * a))
	else:
		tmp = 0.5 * (math.pi / (b * (a * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+98)
		tmp = Float64(Float64(Float64(-pi) / a) * Float64(-0.5 / Float64(b * a)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+98)
		tmp = (-pi / a) * (-0.5 / (b * a));
	else
		tmp = 0.5 * (pi / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+98], N[(N[((-Pi) / a), $MachinePrecision] * N[(-0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.9999999999999998e98

   1. Initial program 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-rgt-identity 70.6%

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

      3. associate-*l/ 70.6%

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

      4. difference-of-squares 88.6%

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

      5. *-commutative 88.6%

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

      6. times-frac 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 99.9%

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

      1. expm1-log1p-u 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 73.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

      3. *-commutative 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

      4. *-commutative 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

      5. div-inv 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

      6. metadata-eval 73.0%

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

      7. associate-/l* 73.0%

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

   7. Applied egg-rr 73.0%

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

      1. expm1-def 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

      2. expm1-log1p 99.9%

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

      3. associate-/l* 99.9%

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

      4. times-frac 99.8%

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

      5. *-commutative 99.8%

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

      6. neg-mul-1 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. times-frac 99.9%

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

      10. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in b around 0 99.9%

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

       1. associate-*r/ 99.9%

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

       2. mul-1-neg 99.9%

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

   12. Simplified 99.9%

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

   if -4.9999999999999998e98 < a

   1. Initial program 82.6%

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

      1. un-div-inv 82.6%

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

      2. difference-of-squares 90.9%

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

      3. associate-/l/ 91.1%

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

      4. frac-sub 91.1%

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

      5. frac-times 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

      8. *-un-lft-identity 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in b around 0 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. expm1-log1p-u 76.5%

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

      2. expm1-udef 54.1%

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

      3. *-commutative 54.1%

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

      4. *-un-lft-identity 54.1%

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

      5. times-frac 54.1%

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

      6. metadata-eval 54.1%

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

      7. *-commutative 54.1%

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

      8. *-commutative 54.1%

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

   9. Applied egg-rr 54.1%

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

       1. expm1-def 76.5%

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

       2. expm1-log1p 98.8%

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

       3. associate-*l* 94.7%

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

       4. +-commutative 94.7%

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

   11. Simplified 94.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\frac{-\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+133}:\\ \;\;\;\;\frac{-\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(b + a\right)} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+133)
   (* (/ (- PI) a) (/ -0.5 (* b a)))
   (* (/ PI (* a (+ b a))) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+133) {
		tmp = (-((double) M_PI) / a) * (-0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) / (a * (b + a))) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+133) {
		tmp = (-Math.PI / a) * (-0.5 / (b * a));
	} else {
		tmp = (Math.PI / (a * (b + a))) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+133:
		tmp = (-math.pi / a) * (-0.5 / (b * a))
	else:
		tmp = (math.pi / (a * (b + a))) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+133)
		tmp = Float64(Float64(Float64(-pi) / a) * Float64(-0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi / Float64(a * Float64(b + a))) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+133)
		tmp = (-pi / a) * (-0.5 / (b * a));
	else
		tmp = (pi / (a * (b + a))) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+133], N[(N[((-Pi) / a), $MachinePrecision] * N[(-0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999961e133

   1. Initial program 60.3%

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

      1. associate-*r/ 60.4%

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

      2. *-rgt-identity 60.4%

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

      3. associate-*l/ 60.3%

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

      4. difference-of-squares 84.7%

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

      5. *-commutative 84.7%

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

      6. times-frac 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 82.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

      3. *-commutative 82.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

      4. *-commutative 82.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

      5. div-inv 82.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

      6. metadata-eval 82.2%

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

      7. associate-/l* 82.2%

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

   7. Applied egg-rr 82.2%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/l* 100.0%

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

      4. times-frac 99.8%

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

      5. *-commutative 99.8%

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

      6. neg-mul-1 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. times-frac 100.0%

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

      10. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. associate-*r/ 100.0%

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

       2. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -4.99999999999999961e133 < a

   1. Initial program 83.6%

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

      1. un-div-inv 83.6%

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

      2. difference-of-squares 91.4%

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

      3. associate-/l/ 91.7%

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

      4. frac-sub 91.6%

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

      5. frac-times 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

      8. *-un-lft-identity 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in b around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

      1. associate-*r* 95.0%

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

      2. times-frac 95.8%

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

   9. Applied egg-rr 95.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+133}:\\ \;\;\;\;\frac{-\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(b + a\right)} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.36e-117)
   (* (/ (- PI) a) (/ -0.5 (* b a)))
   (/ (* PI (/ 0.5 b)) (* b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.36e-117) {
		tmp = (-((double) M_PI) / a) * (-0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) * (0.5 / b)) / (b * a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.36e-117) {
		tmp = (-Math.PI / a) * (-0.5 / (b * a));
	} else {
		tmp = (Math.PI * (0.5 / b)) / (b * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.36e-117:
		tmp = (-math.pi / a) * (-0.5 / (b * a))
	else:
		tmp = (math.pi * (0.5 / b)) / (b * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.36e-117)
		tmp = Float64(Float64(Float64(-pi) / a) * Float64(-0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi * Float64(0.5 / b)) / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.36e-117)
		tmp = (-pi / a) * (-0.5 / (b * a));
	else
		tmp = (pi * (0.5 / b)) / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.36e-117], N[(N[((-Pi) / a), $MachinePrecision] * N[(-0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35999999999999996e-117

   1. Initial program 80.6%

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

      1. associate-*r/ 80.7%

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

      2. *-rgt-identity 80.7%

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

      3. associate-*l/ 80.7%

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

      4. difference-of-squares 90.6%

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

      5. *-commutative 90.6%

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

      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. Add Preprocessing

   5. Taylor expanded in a around inf 90.4%

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

      1. expm1-log1p-u 82.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 67.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

      3. *-commutative 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

      4. *-commutative 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

      5. div-inv 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

      6. metadata-eval 67.3%

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

      7. associate-/l* 67.3%

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

   7. Applied egg-rr 67.3%

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

      1. expm1-def 82.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

      2. expm1-log1p 90.4%

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

      3. associate-/l* 90.4%

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

      4. times-frac 90.3%

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

      5. *-commutative 90.3%

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

      6. neg-mul-1 90.3%

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

      7. distribute-rgt-neg-in 90.3%

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

      8. metadata-eval 90.3%

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

      9. times-frac 90.4%

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

      10. *-commutative 90.4%

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

   9. Simplified 90.4%

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

   10. Taylor expanded in b around 0 79.7%

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

       1. associate-*r/ 79.7%

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

       2. mul-1-neg 79.7%

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

   12. Simplified 79.7%

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

   if -1.35999999999999996e-117 < a

   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. associate-*r/ 80.0%

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

      2. *-rgt-identity 80.0%

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

      3. associate-*l/ 80.0%

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

      4. difference-of-squares 90.3%

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

      5. *-commutative 90.3%

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

      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. Add Preprocessing

   5. Taylor expanded in a around inf 59.9%

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

      1. expm1-log1p-u 48.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 37.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

      3. *-commutative 37.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

      4. *-commutative 37.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

      5. div-inv 37.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

      6. metadata-eval 37.6%

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

      7. associate-/l* 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. expm1-def 48.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

      2. expm1-log1p 59.9%

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

      3. associate-/l* 59.9%

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

      4. times-frac 59.5%

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

      5. *-commutative 59.5%

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

      6. neg-mul-1 59.5%

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

      7. distribute-rgt-neg-in 59.5%

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

      8. metadata-eval 59.5%

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

      9. times-frac 59.9%

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

      10. *-commutative 59.9%

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

   9. Simplified 59.9%

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

   10. Taylor expanded in b around inf 26.1%

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

       1. *-commutative 26.1%

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

       2. associate-/l/ 26.1%

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

       3. frac-times 26.1%

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

       4. add-sqr-sqrt 14.3%

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

       5. sqrt-unprod 49.2%

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

       6. frac-times 49.2%

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

       7. metadata-eval 49.2%

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{b \cdot b}} \cdot \pi}{a \cdot b} \]

       8. metadata-eval 49.2%

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

       9. frac-times 49.2%

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

       10. sqrt-unprod 38.1%

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

       11. add-sqr-sqrt 71.0%

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

       12. *-commutative 71.0%

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

   12. Applied egg-rr 71.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\pi}{a} \cdot \frac{-0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(b \cdot a\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \frac{b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7e+126)
   (/ (/ PI a) (* (* b a) -2.0))
   (/ (/ 0.5 b) (* a (/ b PI)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e+126) {
		tmp = (((double) M_PI) / a) / ((b * a) * -2.0);
	} else {
		tmp = (0.5 / b) / (a * (b / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7e+126) {
		tmp = (Math.PI / a) / ((b * a) * -2.0);
	} else {
		tmp = (0.5 / b) / (a * (b / Math.PI));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7e+126:
		tmp = (math.pi / a) / ((b * a) * -2.0)
	else:
		tmp = (0.5 / b) / (a * (b / math.pi))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e+126)
		tmp = Float64(Float64(pi / a) / Float64(Float64(b * a) * -2.0));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(a * Float64(b / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e+126)
		tmp = (pi / a) / ((b * a) * -2.0);
	else
		tmp = (0.5 / b) / (a * (b / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7e+126], N[(N[(Pi / a), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.0000000000000005e126

   1. Initial program 61.4%

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

      1. associate-*r/ 61.4%

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

      2. *-rgt-identity 61.4%

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

      3. associate-*l/ 61.4%

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

      4. difference-of-squares 85.1%

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

      5. *-commutative 85.1%

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

      6. times-frac 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. clear-num 99.9%

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

      3. frac-times 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. add-sqr-sqrt 60.4%

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

      6. sqrt-unprod 79.5%

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

      7. frac-times 79.5%

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

      8. metadata-eval 79.5%

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

      9. metadata-eval 79.5%

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

      10. frac-times 79.5%

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

      11. sqrt-unprod 32.0%

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

      12. add-sqr-sqrt 79.8%

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

   6. Applied egg-rr 79.8%

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

   7. Taylor expanded in a around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. *-commutative 79.8%

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

   9. Simplified 79.8%

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

       1. expm1-log1p-u 79.8%

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

       2. expm1-udef 80.1%

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

       3. times-frac 80.1%

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

   11. Applied egg-rr 80.1%

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

       1. expm1-def 79.8%

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

       2. expm1-log1p 79.8%

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

       3. associate-/l* 79.8%

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

   13. Simplified 79.8%

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

   14. Taylor expanded in b around 0 79.8%

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

   if -7.0000000000000005e126 < a

   1. Initial program 83.5%

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

      1. associate-*r/ 83.6%

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

      2. *-rgt-identity 83.6%

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

      3. associate-*l/ 83.6%

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

      4. difference-of-squares 91.4%

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

      5. *-commutative 91.4%

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

      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. Add Preprocessing

   5. Taylor expanded in a around inf 65.7%

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

      1. expm1-log1p-u 53.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 42.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

      3. *-commutative 42.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

      4. *-commutative 42.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

      5. div-inv 42.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

      6. metadata-eval 42.6%

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

      7. associate-/l* 42.6%

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

   7. Applied egg-rr 42.6%

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

      1. expm1-def 53.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

      2. expm1-log1p 65.7%

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

      3. associate-/l* 65.7%

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

      4. times-frac 65.3%

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

      5. *-commutative 65.3%

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

      6. neg-mul-1 65.3%

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

      7. distribute-rgt-neg-in 65.3%

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

      8. metadata-eval 65.3%

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

      9. times-frac 65.7%

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

      10. *-commutative 65.7%

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

   9. Simplified 65.7%

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

   10. Taylor expanded in b around inf 29.8%

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

       1. clear-num 29.8%

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

       2. associate-/l/ 29.8%

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

       3. frac-times 29.8%

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

       4. *-un-lft-identity 29.8%

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

       5. add-sqr-sqrt 15.3%

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

       6. sqrt-unprod 46.4%

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

       7. frac-times 46.4%

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

       8. metadata-eval 46.4%

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{b \cdot b}}}{\frac{b}{\pi} \cdot a} \]

       9. metadata-eval 46.4%

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

       10. frac-times 46.4%

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

       11. sqrt-unprod 33.8%

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

       12. add-sqr-sqrt 65.4%

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

   12. Applied egg-rr 65.4%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(b \cdot a\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \frac{b}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. un-div-inv 80.3%

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

   2. difference-of-squares 90.4%

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

   3. associate-/l/ 91.2%

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

   4. frac-sub 91.2%

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

   5. frac-times 99.0%

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

   6. div-inv 99.0%

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

   7. metadata-eval 99.0%

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

   8. *-un-lft-identity 99.0%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in b around 0 99.0%

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

   1. *-commutative 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 7: 30.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*r/ 80.3%

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

   2. *-rgt-identity 80.3%

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

   3. associate-*l/ 80.3%

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

   4. difference-of-squares 90.4%

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

   5. *-commutative 90.4%

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

   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. Add Preprocessing

5. Taylor expanded in a around inf 70.7%

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

   1. expm1-log1p-u 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 48.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

   3. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

   4. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

   5. div-inv 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

   6. metadata-eval 48.2%

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

   7. associate-/l* 48.2%

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

7. Applied egg-rr 48.2%

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

   1. expm1-def 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

   2. expm1-log1p 70.7%

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

   3. associate-/l* 70.7%

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

   4. times-frac 70.5%

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

   5. *-commutative 70.5%

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

   6. neg-mul-1 70.5%

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

   7. distribute-rgt-neg-in 70.5%

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

   8. metadata-eval 70.5%

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

   9. times-frac 70.7%

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

   10. *-commutative 70.7%

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

9. Simplified 70.7%

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

10. Taylor expanded in b around inf 32.3%

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

11. Final simplification 32.3%

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

Alternative 8: 62.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*r/ 80.3%

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

   2. *-rgt-identity 80.3%

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

   3. associate-*l/ 80.3%

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

   4. difference-of-squares 90.4%

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

   5. *-commutative 90.4%

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

   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. Add Preprocessing

5. Taylor expanded in a around inf 70.7%

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

   1. expm1-log1p-u 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 48.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

   3. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

   4. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

   5. div-inv 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

   6. metadata-eval 48.2%

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

   7. associate-/l* 48.2%

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

7. Applied egg-rr 48.2%

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

   1. expm1-def 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

   2. expm1-log1p 70.7%

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

   3. associate-/l* 70.7%

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

   4. times-frac 70.5%

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

   5. *-commutative 70.5%

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

   6. neg-mul-1 70.5%

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

   7. distribute-rgt-neg-in 70.5%

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

   8. metadata-eval 70.5%

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

   9. times-frac 70.7%

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

   10. *-commutative 70.7%

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

9. Simplified 70.7%

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

10. Taylor expanded in b around inf 32.3%

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

    1. *-commutative 32.3%

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

    2. associate-/l/ 32.3%

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

    3. clear-num 32.3%

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

    4. frac-times 32.3%

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

    5. *-un-lft-identity 32.3%

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

    6. add-sqr-sqrt 16.4%

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

    7. sqrt-unprod 46.4%

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

    8. frac-times 46.4%

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

    9. metadata-eval 46.4%

       \[\leadsto \frac{\pi}{\frac{a}{\sqrt{\frac{\color{blue}{0.25}}{b \cdot b}}} \cdot b} \]

    10. metadata-eval 46.4%

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

    11. frac-times 46.4%

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

    12. sqrt-unprod 32.2%

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

    13. add-sqr-sqrt 62.3%

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

12. Applied egg-rr 62.3%

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

13. Final simplification 62.3%

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

Alternative 9: 62.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*r/ 80.3%

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

   2. *-rgt-identity 80.3%

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

   3. associate-*l/ 80.3%

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

   4. difference-of-squares 90.4%

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

   5. *-commutative 90.4%

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

   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. Add Preprocessing

5. Taylor expanded in a around inf 70.7%

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

   1. expm1-log1p-u 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 48.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

   3. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

   4. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

   5. div-inv 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

   6. metadata-eval 48.2%

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

   7. associate-/l* 48.2%

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

7. Applied egg-rr 48.2%

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

   1. expm1-def 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

   2. expm1-log1p 70.7%

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

   3. associate-/l* 70.7%

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

   4. times-frac 70.5%

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

   5. *-commutative 70.5%

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

   6. neg-mul-1 70.5%

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

   7. distribute-rgt-neg-in 70.5%

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

   8. metadata-eval 70.5%

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

   9. times-frac 70.7%

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

   10. *-commutative 70.7%

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

9. Simplified 70.7%

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

10. Taylor expanded in b around inf 32.3%

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

    1. *-commutative 32.3%

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

    2. associate-/l/ 32.3%

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

    3. frac-times 32.3%

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

    4. add-sqr-sqrt 16.4%

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

    5. sqrt-unprod 46.4%

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

    6. frac-times 46.4%

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

    7. metadata-eval 46.4%

       \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{b \cdot b}} \cdot \pi}{a \cdot b} \]

    8. metadata-eval 46.4%

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

    9. frac-times 46.4%

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

    10. sqrt-unprod 32.4%

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

    11. add-sqr-sqrt 62.6%

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

    12. *-commutative 62.6%

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

12. Applied egg-rr 62.6%

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

13. Final simplification 62.6%

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

Alternative 10: 62.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*r/ 80.3%

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

   2. *-rgt-identity 80.3%

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

   3. associate-*l/ 80.3%

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

   4. difference-of-squares 90.4%

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

   5. *-commutative 90.4%

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

   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. Add Preprocessing

5. Taylor expanded in a around inf 70.7%

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

   1. expm1-log1p-u 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 48.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

   3. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{a \cdot b} \cdot \frac{\frac{\pi}{2}}{b - a}}\right)} - 1 \]

   4. *-commutative 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{\color{blue}{b \cdot a}} \cdot \frac{\frac{\pi}{2}}{b - a}\right)} - 1 \]

   5. div-inv 48.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a}\right)} - 1 \]

   6. metadata-eval 48.2%

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

   7. associate-/l* 48.2%

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

7. Applied egg-rr 48.2%

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

   1. expm1-def 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b \cdot a} \cdot \frac{\pi}{\frac{b - a}{0.5}}\right)\right)} \]

   2. expm1-log1p 70.7%

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

   3. associate-/l* 70.7%

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

   4. times-frac 70.5%

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

   5. *-commutative 70.5%

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

   6. neg-mul-1 70.5%

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

   7. distribute-rgt-neg-in 70.5%

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

   8. metadata-eval 70.5%

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

   9. times-frac 70.7%

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

   10. *-commutative 70.7%

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

9. Simplified 70.7%

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

10. Taylor expanded in b around inf 32.3%

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

    1. clear-num 32.3%

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

    2. associate-/l/ 32.3%

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

    3. frac-times 32.3%

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

    4. *-un-lft-identity 32.3%

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

    5. add-sqr-sqrt 16.4%

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

    6. sqrt-unprod 46.4%

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

    7. frac-times 46.4%

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

    8. metadata-eval 46.4%

       \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{b \cdot b}}}{\frac{b}{\pi} \cdot a} \]

    9. metadata-eval 46.4%

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

    10. frac-times 46.4%

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

    11. sqrt-unprod 32.4%

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

    12. add-sqr-sqrt 62.7%

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

12. Applied egg-rr 62.7%

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

13. Final simplification 62.7%

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

Reproduce

herbie shell --seed 2024019 
(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))))