NMSE Section 6.1 mentioned, B

Average Error: 14.8 → 10.0

Time: 8.4s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\left(-0.5 \cdot \frac{\pi}{{a}^{2}}\right) \cdot \frac{-1}{b}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} \cdot \frac{1}{{a}^{2} \cdot b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+142)
   (* (* -0.5 (/ PI (pow a 2.0))) (/ -1.0 b))
   (if (<= a 3.5e+101)
     (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b)))
     (* (/ PI 2.0) (/ 1.0 (* (pow a 2.0) 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));
}

↓

double code(double a, double b) {
	double tmp;
	if (a <= -5e+142) {
		tmp = (-0.5 * (((double) M_PI) / pow(a, 2.0))) * (-1.0 / b);
	} else if (a <= 3.5e+101) {
		tmp = ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (((double) M_PI) / 2.0) * (1.0 / (pow(a, 2.0) * b));
	}
	return tmp;
}

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));
}

↓

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+142) {
		tmp = (-0.5 * (Math.PI / Math.pow(a, 2.0))) * (-1.0 / b);
	} else if (a <= 3.5e+101) {
		tmp = ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (Math.PI / 2.0) * (1.0 / (Math.pow(a, 2.0) * b));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if a <= -5e+142:
		tmp = (-0.5 * (math.pi / math.pow(a, 2.0))) * (-1.0 / b)
	elif a <= 3.5e+101:
		tmp = ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))
	else:
		tmp = (math.pi / 2.0) * (1.0 / (math.pow(a, 2.0) * b))
	return tmp

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

↓

function code(a, b)
	tmp = 0.0
	if (a <= -5e+142)
		tmp = Float64(Float64(-0.5 * Float64(pi / (a ^ 2.0))) * Float64(-1.0 / b));
	elseif (a <= 3.5e+101)
		tmp = 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)));
	else
		tmp = Float64(Float64(pi / 2.0) * Float64(1.0 / Float64((a ^ 2.0) * b)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+142)
		tmp = (-0.5 * (pi / (a ^ 2.0))) * (-1.0 / b);
	elseif (a <= 3.5e+101)
		tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	else
		tmp = (pi / 2.0) * (1.0 / ((a ^ 2.0) * b));
	end
	tmp_2 = tmp;
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]

↓

code[a_, b_] := If[LessEqual[a, -5e+142], N[(N[(-0.5 * N[(Pi / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.5e+101], 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], N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[Power[a, 2.0], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.0000000000000001e142

   1. Initial program 30.5

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

   2. Taylor expanded in b around 0 15.0

      \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{\pi}{{a}^{2}}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. Taylor expanded in a around inf 15.0

      \[\leadsto \left(-0.5 \cdot \frac{\pi}{{a}^{2}}\right) \cdot \color{blue}{\frac{-1}{b}} \]

   if -5.0000000000000001e142 < a < 3.50000000000000023e101

   1. Initial program 7.9

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

   if 3.50000000000000023e101 < a

   1. Initial program 24.5

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

   2. Applied egg-rr 24.5

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

   3. Simplified 24.5

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

      [Start] 24.5	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) + \frac{1}{b} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(-\frac{\pi}{2}\right)\right) \]
      rational_best-simplify-113 [=>] 24.5	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) + \color{blue}{\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{b} \cdot \left(-\frac{\pi}{2}\right)\right)} \]
      rational_best-simplify-53 [=>] 24.5	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) + \frac{1}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(-\frac{\pi}{2} \cdot \frac{1}{b}\right)} \]
      rational_best-simplify-52 [<=] 24.5	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) + \color{blue}{\left(-\left(\frac{\pi}{2} \cdot \frac{1}{b}\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]
      rational_best-simplify-3 [<=] 24.5	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) + \left(-\color{blue}{\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b}\right)}\right) \]
      rational_best-simplify-61 [<=] 24.5	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) - \frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b}\right)} \]
      rational_best-simplify-113 [=>] 24.5	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a}\right) - \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{b}\right)} \]
      rational_best-simplify-110 [=>] 24.5	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{a} - \frac{1}{b \cdot b - a \cdot a} \cdot \frac{1}{b}\right)} \]

   4. Taylor expanded in b around 0 12.4

      \[\leadsto \frac{\pi}{2} \cdot \color{blue}{\frac{1}{{a}^{2} \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\left(-0.5 \cdot \frac{\pi}{{a}^{2}}\right) \cdot \frac{-1}{b}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} \cdot \frac{1}{{a}^{2} \cdot b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	20996

\[\begin{array}{l} t_0 := \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{if}\;t_0 \leq 10^{+286}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	13512

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\left(-0.5 \cdot \frac{\pi}{{a}^{2}}\right) \cdot \frac{-1}{b}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	13384

\[\begin{array}{l} t_0 := -0.5 \cdot \frac{\pi}{{b}^{3}}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.8
Cost	7680

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

Alternative 6
Error	19.3
Cost	7560

\[\begin{array}{l} t_0 := \frac{1}{b \cdot b - a \cdot a}\\ t_1 := t_0 \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{-45}:\\ \;\;\;\;t_0 \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	28.9
Cost	7296

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

Reproduce

herbie shell --seed 2023104 
(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))))