NMSE Section 6.1 mentioned, B

Average Error: 14.5 → 9.8

Time: 15.4s

Precision: binary64

Cost: 7944

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\left(b - a\right) \cdot \left(b + a\right)}\\ \mathbf{if}\;a \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\frac{\pi}{2} \cdot \left(\frac{-1}{b} \cdot t_0\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (- b a) (+ b a)))))
   (if (<= a -2e+95)
     (* (/ PI 2.0) (* (/ -1.0 b) t_0))
     (if (<= a 5e+140)
       (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b)))
       (* t_0 (* -0.5 (/ PI 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 t_0 = 1.0 / ((b - a) * (b + a));
	double tmp;
	if (a <= -2e+95) {
		tmp = (((double) M_PI) / 2.0) * ((-1.0 / b) * t_0);
	} else if (a <= 5e+140) {
		tmp = ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = t_0 * (-0.5 * (((double) M_PI) / 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 t_0 = 1.0 / ((b - a) * (b + a));
	double tmp;
	if (a <= -2e+95) {
		tmp = (Math.PI / 2.0) * ((-1.0 / b) * t_0);
	} else if (a <= 5e+140) {
		tmp = ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = t_0 * (-0.5 * (Math.PI / 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):
	t_0 = 1.0 / ((b - a) * (b + a))
	tmp = 0
	if a <= -2e+95:
		tmp = (math.pi / 2.0) * ((-1.0 / b) * t_0)
	elif a <= 5e+140:
		tmp = ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))
	else:
		tmp = t_0 * (-0.5 * (math.pi / 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)
	t_0 = Float64(1.0 / Float64(Float64(b - a) * Float64(b + a)))
	tmp = 0.0
	if (a <= -2e+95)
		tmp = Float64(Float64(pi / 2.0) * Float64(Float64(-1.0 / b) * t_0));
	elseif (a <= 5e+140)
		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(t_0 * Float64(-0.5 * Float64(pi / 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)
	t_0 = 1.0 / ((b - a) * (b + a));
	tmp = 0.0;
	if (a <= -2e+95)
		tmp = (pi / 2.0) * ((-1.0 / b) * t_0);
	elseif (a <= 5e+140)
		tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
	else
		tmp = t_0 * (-0.5 * (pi / 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_] := Block[{t$95$0 = N[(1.0 / N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e+95], N[(N[(Pi / 2.0), $MachinePrecision] * N[(N[(-1.0 / b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e+140], 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[(t$95$0 * N[(-0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000004e95

   1. Initial program 22.9

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

   2. Simplified 22.9

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

      [Start] 22.9	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 22.9	\[ \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 22.9	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]

   3. Applied egg-rr 11.5

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

   4. Taylor expanded in a around inf 11.6

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

   if -2.00000000000000004e95 < a < 5.00000000000000008e140

   1. Initial program 8.0

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

   if 5.00000000000000008e140 < a

   1. Initial program 29.4

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

   2. Simplified 29.4

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

      [Start] 29.4	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 29.4	\[ \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 29.4	\[ \left(\frac{1}{a} - \frac{1}{b}\right) \cdot \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 29.4	\[ \color{blue}{\frac{1}{b \cdot b - a \cdot a} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\pi}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 29.4	\[ \frac{1}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{\pi}{2} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   3. Taylor expanded in a around inf 29.4

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

   4. Applied egg-rr 14.6

      \[\leadsto \frac{1}{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}} \cdot \left(-0.5 \cdot \frac{\pi}{b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\frac{\pi}{2} \cdot \left(\frac{-1}{b} \cdot \frac{1}{\left(b - a\right) \cdot \left(b + a\right)}\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b - a\right) \cdot \left(b + a\right)} \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	14.5
Cost	7952

\[\begin{array}{l} t_0 := \frac{1}{b \cdot b - a \cdot a}\\ t_1 := \frac{1}{\left(b - a\right) \cdot \left(b + a\right)} \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \mathbf{if}\;b \leq -3.25 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{a} \cdot \left(t_0 \cdot \frac{\pi}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.5
Cost	7824

\[\begin{array}{l} t_0 := \frac{1}{b \cdot b - a \cdot a} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ t_1 := \frac{1}{\left(b - a\right) \cdot \left(b + a\right)} \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \mathbf{if}\;b \leq -3.25 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7688

\[\begin{array}{l} t_0 := \frac{1}{\left(b - a\right) \cdot \left(b + a\right)}\\ t_1 := \frac{\pi}{2} \cdot \left(\frac{1}{a} \cdot t_0\right)\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-35}:\\ \;\;\;\;t_0 \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	9.7
Cost	7680

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

Alternative 5
Error	24.0
Cost	7296

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

Reproduce

herbie shell --seed 2023090 
(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))))