NMSE Section 6.1 mentioned, B

Average Error: 14.0 → 0.3

Time: 11.7s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2625160415156985 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \mathbf{elif}\;b \leq 7.930436204185564 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot \left(a + b\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]

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 (<= b -5.2625160415156985e+97)
   (/ (/ (* PI 0.5) (* a b)) b)
   (if (<= b 7.930436204185564e+128)
     (/ (/ (/ PI a) (* b (+ a b))) 2.0)
     (* (/ 0.5 b) (/ (/ PI a) 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 (b <= -5.2625160415156985e+97) {
		tmp = ((((double) M_PI) * 0.5) / (a * b)) / b;
	} else if (b <= 7.930436204185564e+128) {
		tmp = ((((double) M_PI) / a) / (b * (a + b))) / 2.0;
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / 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 (b <= -5.2625160415156985e+97) {
		tmp = ((Math.PI * 0.5) / (a * b)) / b;
	} else if (b <= 7.930436204185564e+128) {
		tmp = ((Math.PI / a) / (b * (a + b))) / 2.0;
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / 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 b <= -5.2625160415156985e+97:
		tmp = ((math.pi * 0.5) / (a * b)) / b
	elif b <= 7.930436204185564e+128:
		tmp = ((math.pi / a) / (b * (a + b))) / 2.0
	else:
		tmp = (0.5 / b) * ((math.pi / a) / 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 (b <= -5.2625160415156985e+97)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / b);
	elseif (b <= 7.930436204185564e+128)
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * Float64(a + b))) / 2.0);
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / 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 (b <= -5.2625160415156985e+97)
		tmp = ((pi * 0.5) / (a * b)) / b;
	elseif (b <= 7.930436204185564e+128)
		tmp = ((pi / a) / (b * (a + b))) / 2.0;
	else
		tmp = (0.5 / b) * ((pi / a) / 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[b, -5.2625160415156985e+97], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 7.930436204185564e+128], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2625160415156985e97

   1. Initial program 23.2

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

   2. Simplified 11.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2 \cdot \mathsf{fma}\left(b, b, a \cdot \left(-a\right)\right)}} \]
      Proof
      (/.f64 (fma.f64 (PI.f64) (/.f64 -1 b) (/.f64 (PI.f64) a)) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (PI.f64) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) b) (/.f64 (PI.f64) a)) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (PI.f64) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 b))) (/.f64 (PI.f64) a)) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (PI.f64) (neg.f64 (/.f64 1 b)) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (PI.f64) 1)) a)) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (PI.f64) (neg.f64 (/.f64 1 b)) (Rewrite<= associate-*r/_binary64 (*.f64 (PI.f64) (/.f64 1 a)))) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 9 points increase in error, 7 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (PI.f64) (neg.f64 (/.f64 1 b))) (*.f64 (PI.f64) (/.f64 1 a)))) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 (PI.f64) (+.f64 (neg.f64 (/.f64 1 b)) (/.f64 1 a)))) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (PI.f64) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 a) (neg.f64 (/.f64 1 b))))) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (PI.f64) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 a) (/.f64 1 b)))) (*.f64 2 (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (PI.f64) (-.f64 (/.f64 1 a) (/.f64 1 b))) (*.f64 2 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a a)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (PI.f64) (-.f64 (/.f64 1 a) (/.f64 1 b))) (*.f64 2 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 a a))))): 20 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (PI.f64) 2) (/.f64 (-.f64 (/.f64 1 a) (/.f64 1 b)) (-.f64 (*.f64 b b) (*.f64 a a))))): 17 points increase in error, 23 points decrease in error
      (*.f64 (/.f64 (PI.f64) 2) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (/.f64 1 a) (/.f64 1 b)))) (-.f64 (*.f64 b b) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (PI.f64) 2) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a))) (-.f64 (/.f64 1 a) (/.f64 1 b))))): 14 points increase in error, 9 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (PI.f64) 2) (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 1 a) (/.f64 1 b)))): 23 points increase in error, 15 points decrease in error

   3. Taylor expanded in b around inf 11.4

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

   4. Simplified 11.4

      \[\leadsto \color{blue}{\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}} \]
      Proof
      (*.f64 (PI.f64) (/.f64 1/2 (*.f64 a (*.f64 b b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (/.f64 1/2 (*.f64 a (Rewrite<= unpow2_binary64 (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1/2 (*.f64 a (pow.f64 b 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1/2 (PI.f64)) (*.f64 a (pow.f64 b 2)))): 18 points increase in error, 10 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (PI.f64) (*.f64 a (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.3

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

   if -5.2625160415156985e97 < b < 7.93043620418556419e128

   1. Initial program 7.5

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

   2. Applied egg-rr 7.5

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

   3. Applied egg-rr 0.8

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

   4. Applied egg-rr 0.4

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

   5. Applied egg-rr 0.3

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

   if 7.93043620418556419e128 < b

   1. Initial program 27.1

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

   2. Applied egg-rr 62.9

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\frac{{b}^{4} - \left(a \cdot \left(-a\right)\right) \cdot \left(a \cdot \left(-a\right)\right)}{b \cdot b - a \cdot \left(-a\right)}}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. Taylor expanded in b around inf 13.0

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

   4. Simplified 0.2

      \[\leadsto \color{blue}{\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}} \]
      Proof
      (*.f64 (/.f64 1/2 b) (/.f64 (/.f64 (PI.f64) a) b)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (/.f64 (PI.f64) a)) (*.f64 b b))): 54 points increase in error, 26 points decrease in error
      (/.f64 (*.f64 1/2 (/.f64 (PI.f64) a)) (Rewrite<= unpow2_binary64 (pow.f64 b 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (/.f64 (PI.f64) a) (pow.f64 b 2)))): 1 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= associate-/r*_binary64 (/.f64 (PI.f64) (*.f64 a (pow.f64 b 2))))): 22 points increase in error, 22 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2625160415156985 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \mathbf{elif}\;b \leq 7.930436204185564 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot \left(a + b\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -4.502623231139432 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot \left(a \cdot b\right)}}{2}\\ \mathbf{elif}\;a \leq 2.9827146369553694 \cdot 10^{-9}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{a}\\ \end{array} \]

Alternative 2
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -4.502623231139432 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot \left(a \cdot b\right)}}{2}\\ \mathbf{elif}\;a \leq 2.9827146369553694 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{a}\\ \end{array} \]

Alternative 3
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -4.502623231139432 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot \left(a \cdot b\right)}}{2}\\ \mathbf{elif}\;a \leq 2.9827146369553694 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{a}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7168

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

Alternative 5
Error	0.8
Cost	7040

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

Alternative 6
Error	25.0
Cost	6912

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

Alternative 7
Error	24.8
Cost	6912

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

Alternative 8
Error	24.8
Cost	6912

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

Reproduce

herbie shell --seed 2022317 
(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))))