NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.2%

Time: 4.9s

Alternatives: 8

Speedup: N/A×

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: 78.4% 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: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a} - \frac{1}{b}\\ t_1 := \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot t\_0\\ t_2 := \frac{\frac{\pi}{2}}{\left(b + a\right) \cdot \left(b - a\right)} \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-269}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\mathsf{fma}\left(2, {a}^{-1}, 2 \cdot {b}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 a) (/ 1.0 b)))
        (t_1 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) t_0))
        (t_2 (* (/ (/ PI 2.0) (* (+ b a) (- b a))) t_0)))
   (if (<= t_1 -5e-269)
     t_2
     (if (<= t_1 0.0)
       (/ (/ PI (pow (* a b) 2.0)) (fma 2.0 (pow a -1.0) (* 2.0 (pow b -1.0))))
       t_2))))

C

double code(double a, double b) {
	double t_0 = (1.0 / a) - (1.0 / b);
	double t_1 = ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * t_0;
	double t_2 = ((((double) M_PI) / 2.0) / ((b + a) * (b - a))) * t_0;
	double tmp;
	if (t_1 <= -5e-269) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (((double) M_PI) / pow((a * b), 2.0)) / fma(2.0, pow(a, -1.0), (2.0 * pow(b, -1.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(1.0 / a) - Float64(1.0 / b))
	t_1 = Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * t_0)
	t_2 = Float64(Float64(Float64(pi / 2.0) / Float64(Float64(b + a) * Float64(b - a))) * t_0)
	tmp = 0.0
	if (t_1 <= -5e-269)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(pi / (Float64(a * b) ^ 2.0)) / fma(2.0, (a ^ -1.0), Float64(2.0 * (b ^ -1.0))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(Pi / 2.0), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-269], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(Pi / N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[a, -1.0], $MachinePrecision] + N[(2.0 * N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b))) < -4.99999999999999979e-269 or 0.0 < (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b)))

   1. Initial program 86.2%

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

      1. 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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if -4.99999999999999979e-269 < (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b))) < 0.0

   1. Initial program 66.7%

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-*.f64 85.4

         \[\leadsto \frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\frac{\mathsf{fma}\left(-1, b, \left(-a\right) \cdot 1\right)}{\left(-a\right) \cdot b} \cdot 2} \]

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in a around inf

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

      1. lower-fma.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\mathsf{fma}\left(2, {a}^{-1}, 2 \cdot \frac{1}{b}\right)} \]

      5. inv-pow N/A

         \[\leadsto \frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\mathsf{fma}\left(2, {a}^{-1}, 2 \cdot {b}^{-1}\right)} \]

      6. lift-pow.f64 99.7

         \[\leadsto \frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\mathsf{fma}\left(2, {a}^{-1}, 2 \cdot {b}^{-1}\right)} \]

   10. Applied rewrites 99.7%

       \[\leadsto \frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\color{blue}{\mathsf{fma}\left(2, {a}^{-1}, 2 \cdot {b}^{-1}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \leq -5 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{\pi}{2}}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{elif}\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \leq 0:\\ \;\;\;\;\frac{\frac{\pi}{{\left(a \cdot b\right)}^{2}}}{\mathsf{fma}\left(2, {a}^{-1}, 2 \cdot {b}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{2}}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{2}}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ t_1 := \left(a \cdot b\right) \cdot 2\\ \mathbf{if}\;a \leq -7 \cdot 10^{-209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t\_1}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+127}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI 2.0) (* (+ b a) (- b a))) (- (/ 1.0 a) (/ 1.0 b))))
        (t_1 (* (* a b) 2.0)))
   (if (<= a -7e-209)
     t_0
     (if (<= a 1.3e-171)
       (/ (/ PI b) t_1)
       (if (<= a 2e+127) t_0 (/ (/ PI a) t_1))))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / 2.0) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b));
	double t_1 = (a * b) * 2.0;
	double tmp;
	if (a <= -7e-209) {
		tmp = t_0;
	} else if (a <= 1.3e-171) {
		tmp = (((double) M_PI) / b) / t_1;
	} else if (a <= 2e+127) {
		tmp = t_0;
	} else {
		tmp = (((double) M_PI) / a) / t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / 2.0) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b));
	double t_1 = (a * b) * 2.0;
	double tmp;
	if (a <= -7e-209) {
		tmp = t_0;
	} else if (a <= 1.3e-171) {
		tmp = (Math.PI / b) / t_1;
	} else if (a <= 2e+127) {
		tmp = t_0;
	} else {
		tmp = (Math.PI / a) / t_1;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / 2.0) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b))
	t_1 = (a * b) * 2.0
	tmp = 0
	if a <= -7e-209:
		tmp = t_0
	elif a <= 1.3e-171:
		tmp = (math.pi / b) / t_1
	elif a <= 2e+127:
		tmp = t_0
	else:
		tmp = (math.pi / a) / t_1
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / 2.0) / Float64(Float64(b + a) * Float64(b - a))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
	t_1 = Float64(Float64(a * b) * 2.0)
	tmp = 0.0
	if (a <= -7e-209)
		tmp = t_0;
	elseif (a <= 1.3e-171)
		tmp = Float64(Float64(pi / b) / t_1);
	elseif (a <= 2e+127)
		tmp = t_0;
	else
		tmp = Float64(Float64(pi / a) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / 2.0) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b));
	t_1 = (a * b) * 2.0;
	tmp = 0.0;
	if (a <= -7e-209)
		tmp = t_0;
	elseif (a <= 1.3e-171)
		tmp = (pi / b) / t_1;
	elseif (a <= 2e+127)
		tmp = t_0;
	else
		tmp = (pi / a) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[a, -7e-209], t$95$0, If[LessEqual[a, 1.3e-171], N[(N[(Pi / b), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[a, 2e+127], t$95$0, N[(N[(Pi / a), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.00000000000000004e-209 or 1.30000000000000002e-171 < a < 1.99999999999999991e127

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if -7.00000000000000004e-209 < a < 1.30000000000000002e-171

   1. Initial program 64.0%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if 1.99999999999999991e127 < a

   1. Initial program 43.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{\pi}{2}}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\pi}{2}}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{+107}:\\ \;\;\;\;\left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1e+107)
   (* (* (/ (/ PI 2.0) (+ b a)) (/ 1.0 (- b a))) (- (/ 1.0 a) (/ 1.0 b)))
   (/ (/ PI a) (* (* a b) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1e+107) {
		tmp = (((((double) M_PI) / 2.0) / (b + a)) * (1.0 / (b - a))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1e+107) {
		tmp = (((Math.PI / 2.0) / (b + a)) * (1.0 / (b - a))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1e+107:
		tmp = (((math.pi / 2.0) / (b + a)) * (1.0 / (b - a))) * ((1.0 / a) - (1.0 / b))
	else:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1e+107)
		tmp = Float64(Float64(Float64(Float64(pi / 2.0) / Float64(b + a)) * Float64(1.0 / Float64(b - a))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)));
	else
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1e+107)
		tmp = (((pi / 2.0) / (b + a)) * (1.0 / (b - a))) * ((1.0 / a) - (1.0 / b));
	else
		tmp = (pi / a) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1e+107], N[(N[(N[(N[(Pi / 2.0), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.9999999999999997e106

   1. Initial program 86.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 91.3

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

   4. Applied rewrites 91.3%

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

   if 9.9999999999999997e106 < a

   1. Initial program 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 4: 88.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. 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. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. difference-of-squares N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 88.0

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

4. Applied rewrites 88.0%

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

Alternative 5: 87.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. difference-of-squares N/A

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

   8. lower-*.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower--.f64 87.8

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

4. Applied rewrites 87.8%

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

Alternative 6: 84.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (*
          (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a))))
          (- (/ 1.0 a) (/ 1.0 b)))))
   (if (<= t_0 INFINITY) t_0 (* (/ PI (* (* b b) a)) 0.5))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	return tmp

Julia

function code(a, b)
	t_0 = 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)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (pi / ((b * b) * a)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b))) < +inf.0

   1. Initial program 84.5%

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

   if +inf.0 < (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b)))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

         \[\leadsto \frac{\pi}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      7. pow2 N/A

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

      8. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 77.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\\ \mathbf{if}\;t\_0 \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{fma}\left(\frac{a}{b}, -1, 1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a))))))
   (if (<= (* t_0 (- (/ 1.0 a) (/ 1.0 b))) INFINITY)
     (* t_0 (/ (fma (/ a b) -1.0 1.0) a))
     (* (/ PI (* (* b b) a)) 0.5))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)));
	double tmp;
	if ((t_0 * ((1.0 / a) - (1.0 / b))) <= ((double) INFINITY)) {
		tmp = t_0 * (fma((a / b), -1.0, 1.0) / a);
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a))))
	tmp = 0.0
	if (Float64(t_0 * Float64(Float64(1.0 / a) - Float64(1.0 / b))) <= Inf)
		tmp = Float64(t_0 * Float64(fma(Float64(a / b), -1.0, 1.0) / a));
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(N[(a / b), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b))) < +inf.0

   1. Initial program 84.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if +inf.0 < (*.f64 (*.f64 (/.f64 (PI.f64) #s(literal 2 binary64)) (/.f64 #s(literal 1 binary64) (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 #s(literal 1 binary64) a) (/.f64 #s(literal 1 binary64) b)))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

         \[\leadsto \frac{\pi}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      7. pow2 N/A

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

      8. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 8: 57.3% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

   7. pow2 N/A

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

   8. lift-*.f64 57.8

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

5. Applied rewrites 57.8%

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

Reproduce

herbie shell --seed 2025065 
(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))))