?

Average Error: 13.9 → 0.3
Time: 12.8s
Precision: binary64
Cost: 20160

?

\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a} \]
(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
 (/ (/ (/ (fma PI (/ -1.0 b) (/ PI a)) 2.0) (+ 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 ((fma(((double) M_PI), (-1.0 / b), (((double) M_PI) / a)) / 2.0) / (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(fma(pi, Float64(-1.0 / b), Float64(pi / a)) / 2.0) / Float64(b + a)) / Float64(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[(N[(Pi * N[(-1.0 / b), $MachinePrecision] + N[(Pi / a), $MachinePrecision]), $MachinePrecision] / 2.0), $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{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a}

Error?

Derivation?

  1. Initial program 13.9

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a}} \]
    Proof

    [Start]13.9

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

    associate-*r/ [=>]13.9

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

    difference-of-squares [=>]9.3

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

    associate-/r* [=>]8.7

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

    *-rgt-identity [=>]8.7

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

    associate-*l/ [=>]0.3

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

  3. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a} \]

Alternatives

Alternative 1
Error0.3
Cost7552
\[\frac{\frac{\frac{-1}{b} + \frac{1}{a}}{b + a}}{b - a} \cdot \frac{\pi}{2} \]
Alternative 2
Error5.5
Cost7305
\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+18} \lor \neg \left(a \leq 1.18 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\frac{-0.5}{\frac{b \cdot a}{\pi}}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{b}\\ \end{array} \]
Alternative 3
Error5.5
Cost7305
\[\begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{+17} \lor \neg \left(a \leq 6.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{b}\\ \end{array} \]
Alternative 4
Error16.0
Cost7177
\[\begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+18} \lor \neg \left(a \leq 2.8 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{0.5}{a \cdot a} \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \]
Alternative 5
Error11.8
Cost7177
\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+17} \lor \neg \left(a \leq 2.6 \cdot 10^{-13}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \]
Alternative 6
Error7.4
Cost7177
\[\begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+18} \lor \neg \left(a \leq 2.6 \cdot 10^{-13}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
Alternative 7
Error7.0
Cost7177
\[\begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+18} \lor \neg \left(a \leq 3.3 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\frac{\frac{0.5}{b}}{a}}{\frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{b}\\ \end{array} \]
Alternative 8
Error7.3
Cost7176
\[\begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]
Alternative 9
Error7.3
Cost7176
\[\begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]
Alternative 10
Error7.1
Cost7176
\[\begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]
Alternative 11
Error7.0
Cost7176
\[\begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \end{array} \]
Alternative 12
Error7.0
Cost7176
\[\begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \end{array} \]
Alternative 13
Error0.3
Cost7168
\[\pi \cdot \frac{\frac{\frac{1}{b + a}}{b \cdot a}}{2} \]
Alternative 14
Error0.2
Cost7040
\[\frac{\frac{\frac{\pi}{2}}{b + a}}{b \cdot a} \]
Alternative 15
Error30.3
Cost6912
\[0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b} \]

Error

Reproduce?

herbie shell --seed 2023031 
(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))))