NMSE Section 6.1 mentioned, B

Average Error: 14.4 → 0.3

Time: 10.9s

Precision: binary64

Cost: 7040

Math

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

↓

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

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 (/ (/ (/ 0.5 b) a) (/ (+ b a) PI)))

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) {
	return ((0.5 / b) / a) / ((b + a) / ((double) M_PI));
}

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) {
	return ((0.5 / b) / a) / ((b + a) / Math.PI);
}

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):
	return ((0.5 / b) / a) / ((b + a) / math.pi)

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)
	return Float64(Float64(Float64(0.5 / b) / a) / Float64(Float64(b + a) / pi))
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 = code(a, b)
	tmp = ((0.5 / b) / a) / ((b + a) / pi);
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_] := N[(N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.4

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

2. Simplified 9.4

   \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{\mathsf{fma}\left(b, b, a \cdot \left(-a\right)\right)}} \]
   Proof
   (*.f64 (/.f64 (PI.f64) 2) (/.f64 (+.f64 (/.f64 1 a) (/.f64 -1 b)) (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) 2) (/.f64 (+.f64 (/.f64 1 a) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) b)) (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) 2) (/.f64 (+.f64 (/.f64 1 a) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 b)))) (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) 2) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 a) (/.f64 1 b))) (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) 2) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (/.f64 1 a) (/.f64 1 b)))) (fma.f64 b b (*.f64 a (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) 2) (/.f64 (*.f64 1 (-.f64 (/.f64 1 a) (/.f64 1 b))) (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) 2) (/.f64 (*.f64 1 (-.f64 (/.f64 1 a) (/.f64 1 b))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 a a))))): 24 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))))): 12 points increase in error, 8 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)))): 21 points increase in error, 12 points decrease in error

3. Applied egg-rr 0.8

   \[\leadsto \color{blue}{0 + \frac{\pi \cdot 0.5}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]

4. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{a + b}}{a \cdot b}} \]
   Proof
   (/.f64 (/.f64 (*.f64 1/2 (PI.f64)) (+.f64 a b)) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (PI.f64) 1/2)) (+.f64 a b)) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 (+.f64 a b) (*.f64 a b)))): 20 points increase in error, 33 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 (+.f64 a b) (*.f64 a b))))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.2

   \[\leadsto \color{blue}{0 + \frac{0.5 \cdot \frac{\pi}{a + b}}{a \cdot b}} \]

6. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\pi}{b + a} \cdot \frac{0.5}{b \cdot a}} \]
   Proof
   (*.f64 (/.f64 (PI.f64) (+.f64 b a)) (/.f64 1/2 (*.f64 b a))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) (Rewrite<= +-commutative_binary64 (+.f64 a b))) (/.f64 1/2 (*.f64 b a))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) (+.f64 a b)) (/.f64 1/2 (Rewrite<= *-commutative_binary64 (*.f64 a b)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (/.f64 (PI.f64) (+.f64 a b)) 1/2) (*.f64 a b))): 16 points increase in error, 25 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (PI.f64) (+.f64 a b)))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 (*.f64 1/2 (/.f64 (PI.f64) (+.f64 a b))) (*.f64 a b)))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 0.3

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

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	16.3
Cost	7176

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b \cdot b}}{a}\\ \mathbf{if}\;b \leq -9 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	12.0
Cost	7176

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b \cdot b}}{a}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.1
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.0
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.3
Cost	7176

\[\begin{array}{l} t_0 := \frac{0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \mathbf{elif}\;b \leq 1.86 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 7
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \pi}{b \cdot a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 8
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{b}}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 9
Error	7.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(b \cdot a\right)}}{2}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 10
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(b \cdot a\right)}}{2}\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\frac{0.5}{b}}{a}}{\frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 11
Error	0.3
Cost	7040

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

Alternative 12
Error	0.3
Cost	7040

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

Alternative 13
Error	0.2
Cost	7040

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

Alternative 14
Error	30.2
Cost	6912

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

Reproduce

herbie shell --seed 2022332 
(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))))