Average Error: 16.5 → 0.1

Time: 3.2min

Precision: binary64

Cost: 30337

\[\alpha > -1 \land \beta > -1\]

\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999883319836423:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{4}{\alpha}\right)\right) + \left(\left(\frac{-2}{\alpha} - \frac{\beta}{\alpha}\right) - \left(\frac{8}{{\alpha}^{3}} + \left(12 \cdot \frac{\beta}{{\alpha}^{3}} + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + 6 \cdot \left(\beta \cdot \frac{\beta}{{\alpha}^{3}}\right)\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array}\]

\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}

↓

\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999883319836423:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{4}{\alpha}\right)\right) + \left(\left(\frac{-2}{\alpha} - \frac{\beta}{\alpha}\right) - \left(\frac{8}{{\alpha}^{3}} + \left(12 \cdot \frac{\beta}{{\alpha}^{3}} + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + 6 \cdot \left(\beta \cdot \frac{\beta}{{\alpha}^{3}}\right)\right)\right)\right)\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\

\end{array}

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999883319836423)
   (/
    (-
     (/ beta (+ (+ beta alpha) 2.0))
     (+
      (+
       (/ 4.0 (* alpha alpha))
       (* (/ beta alpha) (+ (/ beta alpha) (/ 4.0 alpha))))
      (-
       (- (/ -2.0 alpha) (/ beta alpha))
       (+
        (/ 8.0 (pow alpha 3.0))
        (+
         (* 12.0 (/ beta (pow alpha 3.0)))
         (+
          (pow (/ beta alpha) 3.0)
          (* 6.0 (* beta (/ beta (pow alpha 3.0))))))))))
    2.0)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ (+ beta alpha) 2.0)))) 2.0)))

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999883319836423) {
		tmp = ((beta / ((beta + alpha) + 2.0)) - (((4.0 / (alpha * alpha)) + ((beta / alpha) * ((beta / alpha) + (4.0 / alpha)))) + (((-2.0 / alpha) - (beta / alpha)) - ((8.0 / pow(alpha, 3.0)) + ((12.0 * (beta / pow(alpha, 3.0))) + (pow((beta / alpha), 3.0) + (6.0 * (beta * (beta / pow(alpha, 3.0)))))))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / ((beta + alpha) + 2.0)))) / 2.0;
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `alpha`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.1
Cost	15361

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998512818455:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + -2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\alpha}{\sqrt{\left(\beta + \alpha\right) + 2}} - 1\right)}{2}\\ \end{array}\]

Alternative 2
Error	0.1
Cost	14849

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998512818455:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + -2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}\right) - 1\right)}{2}\\ \end{array}\]

Alternative 3
Error	0.1
Cost	14849

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998512818455:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + -2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 + \left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)}\right)}{2}\\ \end{array}\]

Alternative 4
Error	0.1
Cost	14465

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998512818455:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + -2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}\right)}{2}\\ \end{array}\]

Alternative 5
Error	0.1
Cost	2817

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998512818455:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + -2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]

Alternative 6
Error	0.2
Cost	2177

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999999790002:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]

Alternative 7
Error	0.2
Cost	2049

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999999790002:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)}{2}\\ \end{array}\]

Alternative 8
Error	0.2
Cost	1793

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999999790002:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array}\]

Alternative 9
Error	0.2
Cost	1665

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999999790002:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array}\]

Alternative 10
Error	4.1
Cost	897

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4858852822362.639:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array}\]

Alternative 11
Error	17.8
Cost	576

\[\frac{1 + \frac{\beta}{\beta + 2}}{2}\]

Alternative 12
Error	40.9
Cost	64

\[1\]

Alternative 13
Error	61.6
Cost	64

\[0\]

Derivation

Split input into 2 regimes
if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999988331983642342
1. Initial program 59.3
  \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
2. Using strategy rm
3. Applied div-sub_binary64_76559.3
  \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]
4. Applied associate-+l-_binary64_69557.4
  \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
5. Simplified57.4
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2}\]
6. Using strategy rm
7. Applied add-log-exp_binary64_79957.4
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - \color{blue}{\log \left(e^{1}\right)}\right)}{2}\]
8. Applied add-log-exp_binary64_79957.5
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\color{blue}{\log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}\right)} - \log \left(e^{1}\right)\right)}{2}\]
9. Applied diff-log_binary64_85257.5
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(\frac{e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}}{e^{1}}\right)}}{2}\]
10. Simplified57.4
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \color{blue}{\left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}}{2}\]
11. Taylor expanded around inf 6.0
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(4 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right) - \left(\frac{\beta}{\alpha} + \left(2 \cdot \frac{1}{\alpha} + \left(8 \cdot \frac{1}{{\alpha}^{3}} + \left(12 \cdot \frac{\beta}{{\alpha}^{3}} + \left(\frac{{\beta}^{3}}{{\alpha}^{3}} + 6 \cdot \frac{{\beta}^{2}}{{\alpha}^{3}}\right)\right)\right)\right)\right)\right)}}{2}\]
12. Simplified0.0
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{4}{\alpha}\right)\right) + \left(\left(\frac{-2}{\alpha} - \frac{\beta}{\alpha}\right) - \left(\frac{8}{{\alpha}^{3}} + \left(12 \cdot \frac{\beta}{{\alpha}^{3}} + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + 6 \cdot \left(\frac{\beta}{{\alpha}^{3}} \cdot \beta\right)\right)\right)\right)\right)\right)}}{2}\]
13. Simplified0.0
  \[\leadsto \color{blue}{\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{4}{\alpha}\right)\right) + \left(\left(\frac{-2}{\alpha} - \frac{\beta}{\alpha}\right) - \left(\frac{8}{{\alpha}^{3}} + \left(12 \cdot \frac{\beta}{{\alpha}^{3}} + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + 6 \cdot \left(\frac{\beta}{{\alpha}^{3}} \cdot \beta\right)\right)\right)\right)\right)\right)}{2}}\]
if -0.999988331983642342 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))
1. Initial program 0.1
  \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
2. Using strategy rm
3. Applied flip--_binary64_73513.3
  \[\leadsto \frac{\frac{\color{blue}{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\beta + \alpha}}}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
4. Applied associate-/l/_binary64_70713.3
  \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\beta + \alpha\right)}} + 1}{2}\]
5. Simplified13.3
  \[\leadsto \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} + 1}{2}\]
6. Using strategy rm
7. Applied flip-+_binary64_73422.3
  \[\leadsto \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\beta - \alpha}} \cdot \left(\left(\beta + \alpha\right) + 2\right)} + 1}{2}\]
8. Applied associate-*l/_binary64_70325.6
  \[\leadsto \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\frac{\left(\beta \cdot \beta - \alpha \cdot \alpha\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}{\beta - \alpha}}} + 1}{2}\]
9. Applied associate-/r/_binary64_70625.6
  \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\left(\beta \cdot \beta - \alpha \cdot \alpha\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \cdot \left(\beta - \alpha\right)} + 1}{2}\]
10. Simplified0.1
  \[\leadsto \frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) + 2}} \cdot \left(\beta - \alpha\right) + 1}{2}\]
11. Simplified0.1
  \[\leadsto \color{blue}{\frac{\frac{1}{\left(\beta + \alpha\right) + 2} \cdot \left(\beta - \alpha\right) + 1}{2}}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999883319836423:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{4}{\alpha}\right)\right) + \left(\left(\frac{-2}{\alpha} - \frac{\beta}{\alpha}\right) - \left(\frac{8}{{\alpha}^{3}} + \left(12 \cdot \frac{\beta}{{\alpha}^{3}} + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + 6 \cdot \left(\beta \cdot \frac{\beta}{{\alpha}^{3}}\right)\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

Error

Try it out

Alternatives

Error

Time

Derivation

`if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999988331983642342`

`if -0.999988331983642342 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))`

Reproduce