NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.3

Time: 11.0s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \mathbf{if}\;a \leq -2.2134972907283006 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 4.8211285527296116 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot \left(b + a\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (/ (/ (* 0.5 (/ PI a)) b) a)))
   (if (<= a -2.2134972907283006e+131)
     t_0
     (if (<= a 4.8211285527296116e+85)
       (/ (/ (* PI 0.5) (* a (+ b a))) b)
       t_0))))

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 = ((0.5 * (((double) M_PI) / a)) / b) / a;
	double tmp;
	if (a <= -2.2134972907283006e+131) {
		tmp = t_0;
	} else if (a <= 4.8211285527296116e+85) {
		tmp = ((((double) M_PI) * 0.5) / (a * (b + a))) / b;
	} else {
		tmp = t_0;
	}
	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 = ((0.5 * (Math.PI / a)) / b) / a;
	double tmp;
	if (a <= -2.2134972907283006e+131) {
		tmp = t_0;
	} else if (a <= 4.8211285527296116e+85) {
		tmp = ((Math.PI * 0.5) / (a * (b + a))) / b;
	} else {
		tmp = t_0;
	}
	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 = ((0.5 * (math.pi / a)) / b) / a
	tmp = 0
	if a <= -2.2134972907283006e+131:
		tmp = t_0
	elif a <= 4.8211285527296116e+85:
		tmp = ((math.pi * 0.5) / (a * (b + a))) / b
	else:
		tmp = t_0
	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(Float64(Float64(0.5 * Float64(pi / a)) / b) / a)
	tmp = 0.0
	if (a <= -2.2134972907283006e+131)
		tmp = t_0;
	elseif (a <= 4.8211285527296116e+85)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * Float64(b + a))) / b);
	else
		tmp = t_0;
	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 = ((0.5 * (pi / a)) / b) / a;
	tmp = 0.0;
	if (a <= -2.2134972907283006e+131)
		tmp = t_0;
	elseif (a <= 4.8211285527296116e+85)
		tmp = ((pi * 0.5) / (a * (b + a))) / b;
	else
		tmp = t_0;
	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[(N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -2.2134972907283006e+131], t$95$0, If[LessEqual[a, 4.8211285527296116e+85], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2134972907283006e131 or 4.8211285527296116e85 < a

   1. Initial program 24.6

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

   2. Applied egg-rr 29.5

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\right)}^{3}}} \]

   3. Taylor expanded in a around inf 12.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]

   4. Simplified 12.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}} \]
      Proof
      (*.f64 1/2 (/.f64 (/.f64 (PI.f64) b) (*.f64 a a))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (/.f64 (/.f64 (PI.f64) b) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite=> associate-/l/_binary64 (/.f64 (PI.f64) (*.f64 (pow.f64 a 2) b)))): 23 points increase in error, 26 points decrease in error

   5. Taylor expanded in b around 0 12.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]

   6. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}} \]
      Proof
      (*.f64 (/.f64 (PI.f64) a) (/.f64 1/2 (*.f64 a b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 a (*.f64 a b)))): 33 points increase in error, 33 points decrease in error
      (/.f64 (*.f64 (PI.f64) 1/2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a a) b))): 50 points increase in error, 23 points decrease in error
      (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) b)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (PI.f64) (*.f64 (pow.f64 a 2) b)) 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (PI.f64) (*.f64 (pow.f64 a 2) b)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 0.3

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

   if -2.2134972907283006e131 < a < 4.8211285527296116e85

   1. Initial program 7.7

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

   2. Applied egg-rr 31.3

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\right)}^{3}}} \]

   3. Applied egg-rr 0.8

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

   4. Applied egg-rr 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2134972907283006 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \mathbf{elif}\;a \leq 4.8211285527296116 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot \left(b + a\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	7552

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

Alternative 2
Error	4.8
Cost	7304

\[\begin{array}{l} t_0 := \frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b - a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.3664376830880614 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.3
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7814011105811632 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b}\\ \mathbf{elif}\;b \leq 4.023039642702119 \cdot 10^{+25}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot \left(b \cdot \left(b + a\right)\right)}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b - a}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	7176

\[\begin{array}{l} t_0 := \frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \mathbf{if}\;a \leq -4.86301502830551 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.0845449820359813 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -4.86301502830551 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \mathbf{elif}\;a \leq 1.0845449820359813 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{a}\\ \end{array} \]

Alternative 6
Error	7.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -4.86301502830551 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{a}}{b}}{a}\\ \mathbf{elif}\;a \leq 1.0845449820359813 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{a}\\ \end{array} \]

Alternative 7
Error	0.7
Cost	7040

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

Alternative 8
Error	0.2
Cost	7040

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

Alternative 9
Error	25.7
Cost	6912

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

Reproduce

herbie shell --seed 2022300 
(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))))