NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.6%

Time: 4.7s

Alternatives: 4

Speedup: 2.5×

Specification

Math

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

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: 78.1% accurate, 1.0× speedup

Math

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

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, 2.0× speedup

Math

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

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 * Float64(0.5 / 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 * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 99.6%

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 2: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.9%

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 98.9%

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.9%

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

7. Applied rewrites 98.9%

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

Alternative 3: 81.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(a, b\right) \leq -1.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1.5707963267948966}{\mathsf{max}\left(a, b\right) \cdot \left(\left(\mathsf{min}\left(a, b\right) + \mathsf{max}\left(a, b\right)\right) \cdot \left(\mathsf{max}\left(a, b\right) - \mathsf{min}\left(a, b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\mathsf{max}\left(a, b\right) \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)} \cdot \pi\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= (fmin a b) -1.3e-127)
  (/
   -1.5707963267948966
   (*
    (fmax a b)
    (* (+ (fmin a b) (fmax a b)) (- (fmax a b) (fmin a b)))))
  (* (/ 0.5 (* (* (fmax a b) (fmax a b)) (fmin a b))) PI)))

C

double code(double a, double b) {
	double tmp;
	if (fmin(a, b) <= -1.3e-127) {
		tmp = -1.5707963267948966 / (fmax(a, b) * ((fmin(a, b) + fmax(a, b)) * (fmax(a, b) - fmin(a, b))));
	} else {
		tmp = (0.5 / ((fmax(a, b) * fmax(a, b)) * fmin(a, b))) * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (fmin(a, b) <= -1.3e-127) {
		tmp = -1.5707963267948966 / (fmax(a, b) * ((fmin(a, b) + fmax(a, b)) * (fmax(a, b) - fmin(a, b))));
	} else {
		tmp = (0.5 / ((fmax(a, b) * fmax(a, b)) * fmin(a, b))) * Math.PI;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if fmin(a, b) <= -1.3e-127:
		tmp = -1.5707963267948966 / (fmax(a, b) * ((fmin(a, b) + fmax(a, b)) * (fmax(a, b) - fmin(a, b))))
	else:
		tmp = (0.5 / ((fmax(a, b) * fmax(a, b)) * fmin(a, b))) * math.pi
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (fmin(a, b) <= -1.3e-127)
		tmp = Float64(-1.5707963267948966 / Float64(fmax(a, b) * Float64(Float64(fmin(a, b) + fmax(a, b)) * Float64(fmax(a, b) - fmin(a, b)))));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(fmax(a, b) * fmax(a, b)) * fmin(a, b))) * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (min(a, b) <= -1.3e-127)
		tmp = -1.5707963267948966 / (max(a, b) * ((min(a, b) + max(a, b)) * (max(a, b) - min(a, b))));
	else
		tmp = (0.5 / ((max(a, b) * max(a, b)) * min(a, b))) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Min[a, b], $MachinePrecision], -1.3e-127], N[(-1.5707963267948966 / N[(N[Max[a, b], $MachinePrecision] * N[(N[(N[Min[a, b], $MachinePrecision] + N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[(N[Max[a, b], $MachinePrecision] - N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(N[Max[a, b], $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e-127

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sub-to-fraction N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 64.0%

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

   6. Applied rewrites 64.0%

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

   7. Evaluated real constant 64.0%

      \[\leadsto \frac{-1.5707963267948966}{b \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)} \]

   if -1.3e-127 < a

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 57.2%

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

Alternative 4: 57.2% accurate, 0.2× speedup

Math

\[\frac{0.5}{\left(\mathsf{max}\left(a, b\right) \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)} \cdot \pi \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-*.f64 N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l/ N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-/l/ N/A

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

   17. lift-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-*.f64 N/A

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

7. Applied rewrites 93.3%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 57.2%

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

Reproduce

herbie shell --seed 2025258 
(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))))