Average Error: 19.8 → 0.6
Time: 5.6s
Precision: binary64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.3674838296607385 \cdot 10^{+79} \lor \neg \left(z \leq 1.7846832667500659 \cdot 10^{-09}\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.3674838296607385 \cdot 10^{+79} \lor \neg \left(z \leq 1.7846832667500659 \cdot 10^{-09}\right):\\
\;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\\

\end{array}
(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3674838296607385e+79) (not (<= z 1.7846832667500659e-09)))
   (+
    x
    (+
     (* y 0.0692910599291889)
     (* (/ y z) (- 0.07512208616047561 (/ 0.4046220386999212 z)))))
   (+
    x
    (*
     (/ y (sqrt (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))
     (/
      (+
       (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
       0.279195317918525)
      (sqrt (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))))))
double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3674838296607385e+79) || !(z <= 1.7846832667500659e-09)) {
		tmp = x + ((y * 0.0692910599291889) + ((y / z) * (0.07512208616047561 - (0.4046220386999212 / z))));
	} else {
		tmp = x + ((y / sqrt((z * (z + 6.012459259764103)) + 3.350343815022304)) * (((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525) / sqrt((z * (z + 6.012459259764103)) + 3.350343815022304)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.3674838296607385e79 or 1.78468326675006587e-9 < z

    1. Initial program 43.6

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    2. Taylor expanded around inf 0.9

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291889 \cdot y\right) - 0.4046220386999212 \cdot \frac{y}{{z}^{2}}\right)}\]

    3. Simplified0.9

      \[\leadsto x + \color{blue}{\left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)}\]

    if -1.3674838296607385e79 < z < 1.78468326675006587e-9

    1. Initial program 1.0

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary64_108821.4

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\color{blue}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \cdot \sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}}\]

    4. Applied times-frac_binary64_108670.3

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}}\]

    5. Simplified0.3

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\sqrt{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\]

    6. Simplified0.3

      \[\leadsto x + \frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \color{blue}{\frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3674838296607385 \cdot 10^{+79} \lor \neg \left(z \leq 1.7846832667500659 \cdot 10^{-09}\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))