NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.1%

Time: 15.2s

Alternatives: 10

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.7% 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: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 8.5e+82)
   (* (/ 0.5 (* a (* b (+ a b)))) PI)
   (/ (/ (* PI 0.5) (* a b)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8.5e+82) {
		tmp = (0.5 / (a * (b * (a + b)))) * ((double) M_PI);
	} else {
		tmp = ((((double) M_PI) * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8.5e+82) {
		tmp = (0.5 / (a * (b * (a + b)))) * Math.PI;
	} else {
		tmp = ((Math.PI * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8.5e+82:
		tmp = (0.5 / (a * (b * (a + b)))) * math.pi
	else:
		tmp = ((math.pi * 0.5) / (a * b)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8.5e+82)
		tmp = Float64(Float64(0.5 / Float64(a * Float64(b * Float64(a + b)))) * pi);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.5e+82)
		tmp = (0.5 / (a * (b * (a + b)))) * pi;
	else
		tmp = ((pi * 0.5) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 8.5e+82], N[(N[(0.5 / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.4999999999999995e82

   1. Initial program 81.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 8.4999999999999995e82 < b

   1. Initial program 49.7%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{b + a}}{b - a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (/ (/ (/ (- (/ PI a) (/ PI b)) 2.0) (+ b a)) (- b a)))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return ((((math.pi / a) - (math.pi / b)) / 2.0) / (b + a)) / (b - a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(Float64(Float64(pi / a) - Float64(pi / b)) / 2.0) / Float64(b + a)) / Float64(b - a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((((pi / a) - (pi / b)) / 2.0) / (b + a)) / (b - a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(N[(N[(Pi / a), $MachinePrecision] - N[(Pi / b), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a} \cdot \frac{0.5}{b - a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (* (/ (- (/ PI a) (/ PI b)) (+ b a)) (/ 0.5 (- b a))))

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (((pi / a) - (pi / b)) / (b + a)) * (0.5 / (b - a));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(N[(Pi / a), $MachinePrecision] - N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot b}}{\frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-22) (/ (/ 0.5 (* a b)) (/ a PI)) (/ (/ (* PI 0.5) b) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-22) {
		tmp = (0.5 / (a * b)) / (a / ((double) M_PI));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (a * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-22) {
		tmp = (0.5 / (a * b)) / (a / Math.PI);
	} else {
		tmp = ((Math.PI * 0.5) / b) / (a * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.1e-22:
		tmp = (0.5 / (a * b)) / (a / math.pi)
	else:
		tmp = ((math.pi * 0.5) / b) / (a * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.1e-22)
		tmp = Float64(Float64(0.5 / Float64(a * b)) / Float64(a / pi));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(a * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e-22)
		tmp = (0.5 / (a * b)) / (a / pi);
	else
		tmp = ((pi * 0.5) / b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.1e-22], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000008e-22

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.10000000000000008e-22 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot b}}{\frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\pi}}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2e-22) (/ (/ 0.5 (* a b)) (/ a PI)) (/ (/ 0.5 (/ b PI)) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2e-22) {
		tmp = (0.5 / (a * b)) / (a / ((double) M_PI));
	} else {
		tmp = (0.5 / (b / ((double) M_PI))) / (a * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2e-22) {
		tmp = (0.5 / (a * b)) / (a / Math.PI);
	} else {
		tmp = (0.5 / (b / Math.PI)) / (a * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2e-22:
		tmp = (0.5 / (a * b)) / (a / math.pi)
	else:
		tmp = (0.5 / (b / math.pi)) / (a * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2e-22)
		tmp = Float64(Float64(0.5 / Float64(a * b)) / Float64(a / pi));
	else
		tmp = Float64(Float64(0.5 / Float64(b / pi)) / Float64(a * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e-22)
		tmp = (0.5 / (a * b)) / (a / pi);
	else
		tmp = (0.5 / (b / pi)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2e-22], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b / Pi), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.0000000000000001e-22

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.0000000000000001e-22 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot b}}{\frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e-22) (/ (/ 0.5 (* a b)) (/ a PI)) (* (/ (/ 0.5 a) b) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.5e-22) {
		tmp = (0.5 / (a * b)) / (a / ((double) M_PI));
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.5e-22) {
		tmp = (0.5 / (a * b)) / (a / Math.PI);
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.5e-22:
		tmp = (0.5 / (a * b)) / (a / math.pi)
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.5e-22)
		tmp = Float64(Float64(0.5 / Float64(a * b)) / Float64(a / pi));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.5e-22)
		tmp = (0.5 / (a * b)) / (a / pi);
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.5e-22], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999977e-22

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.49999999999999977e-22 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e-22) (* (/ PI (* a b)) (/ 0.5 a)) (* (/ (/ 0.5 a) b) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-22) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-22) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.9e-22:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.9e-22)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e-22)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.9e-22], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9000000000000002e-22

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.9000000000000002e-22 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-22) (* (/ PI (* a b)) (/ 0.5 a)) (* PI (/ 0.5 (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-22) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-22) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.3e-22:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.3e-22)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.3e-22)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = pi * (0.5 / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.3e-22], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999998e-22

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.2999999999999998e-22 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{0.5}{\frac{a + b}{\frac{\pi}{a \cdot b}}} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ 0.5 (/ (+ a b) (/ PI (* a b)))))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(0.5 / Float64(Float64(a + b) / Float64(pi / Float64(a * b))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = 0.5 / ((a + b) / (pi / (a * b)));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(0.5 / N[(N[(a + b), $MachinePrecision] / N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in b around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 10: 62.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* PI (/ 0.5 (* b (* a b)))))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.pi * (0.5 / (b * (a * b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(pi * Float64(0.5 / Float64(b * Float64(a * b))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = pi * (0.5 / (b * (a * b)));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

3. Taylor expanded in b around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

7. Applied egg-rr 0

   \[\leadsto expr\]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))