NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 11.7s

Alternatives: 5

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: 78.3% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 86.6%

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

   3. associate-/r* 87.4%

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

   4. div-inv 87.4%

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

   5. metadata-eval 87.4%

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

4. Applied egg-rr 87.4%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.7%

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

    1. un-div-inv 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e-193)
   (/ (* -0.5 (/ PI (* a b))) (- b a))
   (* (/ PI a) (/ 0.5 (* b (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-193) {
		tmp = (-0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-193) {
		tmp = (-0.5 * (Math.PI / (a * b))) / (b - a);
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * (b - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e-193:
		tmp = (-0.5 * (math.pi / (a * b))) / (b - a)
	else:
		tmp = (math.pi / a) * (0.5 / (b * (b - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e-193)
		tmp = Float64(Float64(-0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e-193)
		tmp = (-0.5 * (pi / (a * b))) / (b - a);
	else
		tmp = (pi / a) * (0.5 / (b * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e-193], N[(N[(-0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.2e-193

   1. Initial program 78.6%

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

      1. un-div-inv 78.7%

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

      2. difference-of-squares 86.4%

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

      3. associate-/r* 88.5%

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

      4. div-inv 88.5%

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

      5. metadata-eval 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 76.0%

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

   if -1.2e-193 < a

   1. Initial program 77.6%

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

      1. associate-*l* 77.7%

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

      2. *-rgt-identity 77.7%

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

      3. associate-/l* 77.7%

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

      4. metadata-eval 77.7%

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

      5. associate-*l/ 77.7%

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

      6. *-lft-identity 77.7%

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

      7. sub-neg 77.7%

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

      8. distribute-neg-frac 77.7%

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

      9. metadata-eval 77.7%

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

   3. Simplified 77.7%

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

      1. metadata-eval 77.7%

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

      2. div-inv 77.7%

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

      3. associate-*r/ 77.8%

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

      4. *-commutative 77.8%

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

      5. difference-of-squares 86.9%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in a around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. *-commutative 75.4%

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

      3. times-frac 75.4%

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

   9. Simplified 75.4%

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

       1. *-un-lft-identity 75.4%

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

       2. associate-/l* 65.2%

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

   11. Applied egg-rr 65.2%

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

       1. *-lft-identity 65.2%

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

       2. associate-/l/ 65.3%

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

   13. Simplified 65.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. associate-*l* 78.0%

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

   2. *-rgt-identity 78.0%

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

   3. associate-/l* 78.0%

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

   4. metadata-eval 78.0%

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

   5. associate-*l/ 78.1%

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

   6. *-lft-identity 78.1%

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

   7. sub-neg 78.1%

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

   8. distribute-neg-frac 78.1%

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

   9. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. metadata-eval 78.1%

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

   2. div-inv 78.1%

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

   3. associate-*r/ 78.1%

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

   4. *-commutative 78.1%

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

   5. difference-of-squares 86.7%

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

   6. associate-/r* 99.7%

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

6. Applied egg-rr 70.4%

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

7. Taylor expanded in a around 0 70.4%

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

   1. associate-*r/ 70.4%

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

   2. *-commutative 70.4%

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

   3. times-frac 70.3%

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

9. Simplified 70.3%

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

    1. *-un-lft-identity 70.3%

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

    2. associate-/l* 62.2%

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

11. Applied egg-rr 62.2%

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

    1. *-lft-identity 62.2%

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

    2. associate-/l/ 62.0%

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

13. Simplified 62.0%

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

14. Final simplification 62.0%

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

Alternative 4: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 86.6%

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

   3. associate-/r* 87.4%

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

   4. div-inv 87.4%

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

   5. metadata-eval 87.4%

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

4. Applied egg-rr 87.4%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-*r/ 99.7%

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

    3. frac-times 99.6%

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

    4. *-un-lft-identity 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 31.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. associate-*l* 78.0%

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

   2. *-rgt-identity 78.0%

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

   3. associate-/l* 78.0%

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

   4. metadata-eval 78.0%

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

   5. associate-*l/ 78.1%

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

   6. *-lft-identity 78.1%

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

   7. sub-neg 78.1%

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

   8. distribute-neg-frac 78.1%

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

   9. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. metadata-eval 78.1%

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

   2. div-inv 78.1%

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

   3. associate-*r/ 78.1%

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

   4. *-commutative 78.1%

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

   5. difference-of-squares 86.7%

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

   6. associate-/r* 99.7%

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

6. Applied egg-rr 70.4%

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

7. Taylor expanded in a around 0 70.4%

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

   1. associate-*r/ 70.4%

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

   2. *-commutative 70.4%

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

   3. times-frac 70.3%

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

9. Simplified 70.3%

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

    1. *-un-lft-identity 70.3%

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

    2. associate-/l* 62.2%

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

11. Applied egg-rr 62.2%

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

    1. *-lft-identity 62.2%

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

    2. associate-/l/ 62.0%

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

13. Simplified 62.0%

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

14. Taylor expanded in b around 0 34.0%

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

    1. *-commutative 34.0%

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

16. Simplified 34.0%

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

17. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2024054 
(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))))