Average Error: 14.5 → 0.2
Time: 9.8s
Precision: binary64
Cost: 7040
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
↓
\[\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot a} \]
(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 0.5) (+ b a)) (* b a)))
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) * 0.5) / (b + a)) / (b * a);
}

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

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

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 * 0.5) / Float64(b + a)) / Float64(b * a))
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 * 0.5) / (b + a)) / (b * a);
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 * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $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{\pi \cdot 0.5}{b + a}}{b \cdot a}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.5

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

  2. Applied egg-rr8.9

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

  3. Applied egg-rr0.2

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

  4. Final simplification0.2

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

Alternatives

Alternative 1
Error7.2
Cost7176
\[\begin{array}{l} t_0 := \frac{0.5}{b \cdot \left(b \cdot \frac{a}{\pi}\right)}\\ \mathbf{if}\;b \leq -8.727399046818619 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.11857673086143837:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error7.0
Cost7176
\[\begin{array}{l} t_0 := \frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \mathbf{if}\;b \leq -8.727399046818619 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.11857673086143837:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error7.0
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -8.727399046818619 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \mathbf{elif}\;b \leq 0.11857673086143837:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b}\\ \end{array} \]
Alternative 4
Error7.0
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -8.727399046818619 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \mathbf{elif}\;b \leq 0.11857673086143837:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b}\\ \end{array} \]
Alternative 5
Error0.3
Cost7040
\[\frac{\pi}{b + a} \cdot \frac{\frac{0.5}{a}}{b} \]
Alternative 6
Error0.3
Cost7040
\[\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b + a} \]
Alternative 7
Error30.1
Cost6912
\[\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)} \]
Alternative 8
Error25.4
Cost6912
\[\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{a} \]

Error

Reproduce

herbie shell --seed 2022316 
(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))))