NMSE Section 6.1 mentioned, B

Average Error: 14.0 → 0.3

Time: 12.0s

Precision: binary64

Cost: 7040

Math

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

↓

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

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

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

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 (math.pi / (a + b)) * (0.5 / (a * b))

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(pi / Float64(a + b)) * Float64(0.5 / Float64(a * b)))
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 = (pi / (a + b)) * (0.5 / (a * b));
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[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.0

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

2. Simplified 14.0

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   Proof
   (*.f64 (/.f64 (/.f64 (PI.f64) 2) (-.f64 (*.f64 b b) (*.f64 a a))) (+.f64 (/.f64 1 a) (/.f64 -1 b))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (PI.f64) 2) (-.f64 (*.f64 b b) (*.f64 a a))) (+.f64 (/.f64 1 a) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) b))): 6 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (PI.f64) 2) (-.f64 (*.f64 b b) (*.f64 a a))) (+.f64 (/.f64 1 a) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 b))))): 0 points increase in error, 6 points decrease in error
   (*.f64 (/.f64 (/.f64 (PI.f64) 2) (-.f64 (*.f64 b b) (*.f64 a a))) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 a) (/.f64 1 b)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (PI.f64) 2) 1)) (-.f64 (*.f64 b b) (*.f64 a a))) (-.f64 (/.f64 1 a) (/.f64 1 b))): 6 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (PI.f64) 2) (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a))))) (-.f64 (/.f64 1 a) (/.f64 1 b))): 0 points increase in error, 6 points decrease in error

3. Applied egg-rr 29.2

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]

4. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\pi}{a + b} \cdot \frac{0.5}{a \cdot b}} \]
   Proof
   (*.f64 (/.f64 (PI.f64) (+.f64 a b)) (/.f64 1/2 (*.f64 a b))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) (Rewrite<= +-commutative_binary64 (+.f64 b a))) (/.f64 1/2 (*.f64 a b))): 5 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (PI.f64) (+.f64 b a)) (/.f64 1/2 (Rewrite<= *-commutative_binary64 (*.f64 b a)))): 0 points increase in error, 5 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 (+.f64 b a) (*.f64 b a)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 (+.f64 b a) (*.f64 b a)))))): 3 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 (+.f64 b a) (*.f64 b a))))) 1)): 0 points increase in error, 3 points decrease in error

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	15.2
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -7 \lor \neg \left(a \leq 0.000155\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 2
Error	10.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -10.5 \lor \neg \left(a \leq 0.0042\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 3
Error	6.5
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \lor \neg \left(a \leq 0.0064\right):\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 4
Error	6.6
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -2.1 \lor \neg \left(a \leq 0.0042\right):\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 5
Error	29.6
Cost	6912

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

Reproduce

herbie shell --seed 2022343 
(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))))