Average Error: 14.2 → 0.3
Time: 9.8s
Precision: binary64
Cost: 7040
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
↓
\[\frac{\frac{\frac{\pi}{\frac{a}{0.5}}}{b}}{a + b} \]
(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 0.5)) b) (+ a b)))
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 / 0.5)) / b) / (a + b);
}

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 / 0.5)) / b) / (a + b);
}

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

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(pi / Float64(a / 0.5)) / b) / Float64(a + b))
end

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 / 0.5)) / b) / (a + b);
end

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[(Pi / N[(a / 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

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

  2. Simplified14.2

    \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]
    Proof
    (*.f64 (/.f64 (PI.f64) 2) (*.f64 (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a))) (-.f64 (/.f64 1 a) (/.f64 1 b)))): 0 points increase in error, 0 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)))): 20 points increase in error, 18 points decrease in error

  3. Applied egg-rr18.7

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

  4. Simplified0.3

    \[\leadsto \frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{a \cdot b}} \]
    Proof
    (/.f64 (/.f64 1 (+.f64 a b)) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 (neg.f64 (-.f64 b a)) (neg.f64 (-.f64 b a)))) (+.f64 a b)) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (/.f64 (neg.f64 (-.f64 b a)) (neg.f64 (-.f64 b a))) (Rewrite<= +-commutative_binary64 (+.f64 b a))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 (-.f64 b a)) (*.f64 (neg.f64 (-.f64 b a)) (+.f64 b a)))) (*.f64 a b)): 48 points increase in error, 10 points decrease in error
    (/.f64 (/.f64 (neg.f64 (-.f64 b a)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 b a) (neg.f64 (-.f64 b a))))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 b a))) (*.f64 (+.f64 b a) (neg.f64 (-.f64 b a)))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 b a) -1)) (*.f64 (+.f64 b a) (neg.f64 (-.f64 b a)))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (-.f64 b a) (+.f64 b a)) (/.f64 -1 (neg.f64 (-.f64 b a))))) (*.f64 a b)): 10 points increase in error, 48 points decrease in error
    (/.f64 (*.f64 (/.f64 (-.f64 b a) (+.f64 b a)) (/.f64 -1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 b a))))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (*.f64 (/.f64 (-.f64 b a) (+.f64 b a)) (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 -1 -1) (-.f64 b a)))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (*.f64 (/.f64 (-.f64 b a) (+.f64 b a)) (/.f64 (Rewrite=> metadata-eval 1) (-.f64 b a))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 b a) 1) (*.f64 (+.f64 b a) (-.f64 b a)))) (*.f64 a b)): 48 points increase in error, 10 points decrease in error
    (/.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (-.f64 b a)) (*.f64 (+.f64 b a) (-.f64 b a))) (*.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (-.f64 b a) (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 b b) (*.f64 a a)))) (*.f64 a b)): 22 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (-.f64 b a) (-.f64 (*.f64 b b) (*.f64 a a))) (Rewrite<= *-commutative_binary64 (*.f64 b a))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 b a) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (*.f64 b a)))): 42 points increase in error, 17 points decrease in error

  5. Applied egg-rr5.1

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

  6. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{\frac{a}{0.5}}}{b}}{a + b}} \]
    Proof
    (/.f64 (/.f64 (/.f64 (PI.f64) (/.f64 a 1/2)) b) (+.f64 a b)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (PI.f64) 1/2) a)) b) (+.f64 a b)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 (*.f64 (PI.f64) 1/2) a) (*.f64 b (+.f64 a b)))): 39 points increase in error, 21 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 a (*.f64 b (+.f64 a b))))): 20 points increase in error, 17 points decrease in error
    (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 (*.f64 (PI.f64) 1/2) (*.f64 a (*.f64 b (+.f64 a b)))))): 0 points increase in error, 0 points decrease in error

  7. Final simplification0.3

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

Alternatives

Alternative 1
Error11.6
Cost7176
\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \mathbf{if}\;b \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error11.6
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{b}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]
Alternative 3
Error11.5
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+29}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]
Alternative 4
Error7.1
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+30}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]
Alternative 5
Error7.0
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]
Alternative 6
Error6.9
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -1100:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]
Alternative 7
Error0.2
Cost7040
\[0.5 \cdot \frac{\frac{\pi}{a + b}}{a \cdot b} \]
Alternative 8
Error30.1
Cost6912
\[0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)} \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))