Average Error: 3.7 → 0.1
Time: 4.3s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\frac{\left(\alpha + 1\right) \cdot \frac{\left(1 + \beta\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\frac{\left(\alpha + 1\right) \cdot \frac{\left(1 + \beta\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}
(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))
↓
(FPCore (alpha beta)
 :precision binary64
 (/
  (*
   (+ alpha 1.0)
   (/ (* (+ 1.0 beta) (/ 1.0 (+ (+ alpha beta) 2.0))) (+ (+ alpha beta) 2.0)))
  (+ alpha (+ beta 3.0))))
double code(double alpha, double beta) {
	return (((((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0)));
}
↓
double code(double alpha, double beta) {
	return (((double) (((double) (alpha + 1.0)) * (((double) (((double) (1.0 + beta)) * (1.0 / ((double) (((double) (alpha + beta)) + 2.0))))) / ((double) (((double) (alpha + beta)) + 2.0))))) / ((double) (alpha + ((double) (beta + 3.0)))));
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 3.7

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

  2. Simplified2.2

    \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{\alpha + \left(\beta + 3\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*_binary640.1

    \[\leadsto \frac{\left(\alpha + 1\right) \cdot \color{blue}{\frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\alpha + \left(\beta + 3\right)}\]

  5. Simplified0.1

    \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{\frac{\beta + 1}{\left(\beta + \alpha\right) + 2}}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\]
  6. Using strategy rm

  7. Applied div-inv_binary640.1

    \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{\left(\beta + 1\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\]

  8. Simplified0.1

    \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\left(\beta + 1\right) \cdot \color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\]

  9. Final simplification0.1

    \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\left(1 + \beta\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))