NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.7%

Time: 10.0s

Alternatives: 7

Speedup: 2.4×

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: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a}}{2 \cdot \left(b \cdot a\right)}} \]
7. Add Preprocessing

Alternative 2: 95.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+122}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.5e+122)
   (* PI (/ 0.5 (* a (* b a))))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.5e+122) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7.5e+122) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.5e+122:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.5e+122)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.5e+122)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.5e+122], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5000000000000002e122

   1. Initial program 60.4%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 80.4%

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

   9. Applied rewrites 98.7%

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

   if -7.5000000000000002e122 < a

   1. Initial program 81.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{b + a}}{2 \cdot \left(b \cdot a\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a}}}{2 \cdot \left(b \cdot a\right)} \]

      3. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(b + a\right)\right)}}}{2 \cdot \left(b \cdot a\right)} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(2 \cdot \left(b \cdot a\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(2 \cdot \left(b \cdot a\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{\left(2 \cdot \left(b \cdot a\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(2 \cdot \left(b \cdot a\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(2 \cdot \left(b \cdot a\right)\right)} \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\left(b \cdot a\right) \cdot 2\right)} \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(\color{blue}{\left(b \cdot a\right)} \cdot 2\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(\color{blue}{\left(a \cdot b\right)} \cdot 2\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(a \cdot \left(b \cdot 2\right)\right)} \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(a \cdot \left(b \cdot \color{blue}{\frac{1}{\frac{1}{2}}}\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      14. div-inv N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(a \cdot \color{blue}{\frac{b}{\frac{1}{2}}}\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(a \cdot \frac{b}{\frac{1}{2}}\right)} \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(a \cdot \color{blue}{\left(b \cdot \frac{1}{\frac{1}{2}}\right)}\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(a \cdot \left(b \cdot \color{blue}{2}\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(a \cdot \color{blue}{\left(b \cdot 2\right)}\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)} \]

      19. lower-neg.f64 98.9

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

   8. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\left(a \cdot \left(b \cdot 2\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(a \cdot \left(b \cdot 2\right)\right) \cdot \left(\mathsf{neg}\left(\left(b + a\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{a \cdot \left(b \cdot 2\right)}}{\mathsf{neg}\left(\left(b + a\right)\right)}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{a \cdot \left(b \cdot 2\right)}}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{a \cdot \left(b \cdot 2\right)}\right)}}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot \left(b \cdot 2\right)}}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{a \cdot \color{blue}{\left(b \cdot 2\right)}}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(a \cdot b\right) \cdot 2}}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot a\right)} \cdot 2}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot a\right)} \cdot 2}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      11. associate-/l/ N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot a}}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      12. div-inv N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}}{b \cdot a}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}}{b \cdot a}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}}{b \cdot a}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{b \cdot a}}\right)}{\mathsf{neg}\left(\left(b + a\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{b \cdot a}\right)}{\color{blue}{\mathsf{neg}\left(\left(b + a\right)\right)}} \]

      17. frac-2neg N/A

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

   10. Applied rewrites 95.3%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+122}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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))))