Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.4

Time: 1.4min

Precision: binary64

\[z \leq 0.5\]

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

↓

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{\left(\left(\left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + \left(\left(2 - z\right) \cdot \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(3 - z\right) \cdot \left(\left(2 - z\right) \cdot \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) \cdot -176.6150291621406}{\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(2 - z\right) \cdot \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (*
     (sqrt (* PI 2.0))
     (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5)))
    (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5))))
   (+
    (+
     (+
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0)))
          (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0)))
         (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0)))
        (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0)))
       (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0)))
      (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0)))
     (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0)))
    (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))

↓

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z))) (exp (+ z -7.5)))
   (+
    (+
     (+
      (+
       (/
        (+
         (*
          (+
           (*
            (+
             (*
              (+ 0.9999999999994298 (pow (/ 676.5203681218851 (- 1.0 z)) 3.0))
              (- 2.0 z))
             (*
              (+
               0.9999999999996199
               (-
                (*
                 (/ 676.5203681218851 (- 1.0 z))
                 (/ 676.5203681218851 (- 1.0 z)))
                (* 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))))
              -1259.1392167224028))
            (- 3.0 z))
           (*
            (*
             (- 2.0 z)
             (+
              0.9999999999996199
              (-
               (*
                (/ 676.5203681218851 (- 1.0 z))
                (/ 676.5203681218851 (- 1.0 z)))
               (* 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z))))))
            771.3234287776531))
          (- 4.0 z))
         (*
          (*
           (- 3.0 z)
           (*
            (- 2.0 z)
            (+
             0.9999999999996199
             (-
              (*
               (/ 676.5203681218851 (- 1.0 z))
               (/ 676.5203681218851 (- 1.0 z)))
              (* 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))))))
          -176.6150291621406))
        (*
         (- 4.0 z)
         (*
          (- 3.0 z)
          (*
           (- 2.0 z)
           (+
            0.9999999999996199
            (-
             (*
              (/ 676.5203681218851 (- 1.0 z))
              (/ 676.5203681218851 (- 1.0 z)))
             (* 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))))))))
       (/ 12.507343278686905 (- 5.0 z)))
      (/ -0.13857109526572012 (- 6.0 z)))
     (/ 9.984369578019572e-06 (- 7.0 z)))
    (/ 1.5056327351493116e-07 (- 8.0 z))))))

C

double code(double z) {
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_PI) * 2.0) * pow(((((1.0 - z) - 1.0) + 7.0) + 0.5), (((1.0 - z) - 1.0) + 0.5))) * exp(-((((1.0 - z) - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (((1.0 - z) - 1.0) + 1.0))) + (-1259.1392167224028 / (((1.0 - z) - 1.0) + 2.0))) + (771.3234287776531 / (((1.0 - z) - 1.0) + 3.0))) + (-176.6150291621406 / (((1.0 - z) - 1.0) + 4.0))) + (12.507343278686905 / (((1.0 - z) - 1.0) + 5.0))) + (-0.13857109526572012 / (((1.0 - z) - 1.0) + 6.0))) + (9.984369578019572e-06 / (((1.0 - z) - 1.0) + 7.0))) + (1.5056327351493116e-07 / (((1.0 - z) - 1.0) + 8.0))));
}

↓

double code(double z) {
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_PI) * 2.0) * pow((7.5 - z), (0.5 - z))) * exp(z + -7.5)) * ((((((((((((0.9999999999994298 + pow((676.5203681218851 / (1.0 - z)), 3.0)) * (2.0 - z)) + ((0.9999999999996199 + (((676.5203681218851 / (1.0 - z)) * (676.5203681218851 / (1.0 - z))) - (0.9999999999998099 * (676.5203681218851 / (1.0 - z))))) * -1259.1392167224028)) * (3.0 - z)) + (((2.0 - z) * (0.9999999999996199 + (((676.5203681218851 / (1.0 - z)) * (676.5203681218851 / (1.0 - z))) - (0.9999999999998099 * (676.5203681218851 / (1.0 - z)))))) * 771.3234287776531)) * (4.0 - z)) + (((3.0 - z) * ((2.0 - z) * (0.9999999999996199 + (((676.5203681218851 / (1.0 - z)) * (676.5203681218851 / (1.0 - z))) - (0.9999999999998099 * (676.5203681218851 / (1.0 - z))))))) * -176.6150291621406)) / ((4.0 - z) * ((3.0 - z) * ((2.0 - z) * (0.9999999999996199 + (((676.5203681218851 / (1.0 - z)) * (676.5203681218851 / (1.0 - z))) - (0.9999999999998099 * (676.5203681218851 / (1.0 - z))))))))) + (12.507343278686905 / (5.0 - z))) + (-0.13857109526572012 / (6.0 - z))) + (9.984369578019572e-06 / (7.0 - z))) + (1.5056327351493116e-07 / (8.0 - z))));
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

2. Simplified 1.8

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)}\]
3. Using strategy rm

4. Applied flip3-+_binary64_2468 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\frac{{0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}}{0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)}} + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

5. Applied frac-add_binary64_2473 1.2

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)}} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

6. Applied frac-add_binary64_2473 1.2

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\frac{\left(\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531}{\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)}} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

7. Applied frac-add_binary64_2473 0.4

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\color{blue}{\frac{\left(\left(\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406}{\left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)}} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

8. Final simplification 0.4

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{\left(\left(\left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + \left(\left(2 - z\right) \cdot \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(3 - z\right) \cdot \left(\left(2 - z\right) \cdot \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) \cdot -176.6150291621406}{\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(2 - z\right) \cdot \left(0.9999999999996199 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

Reproduce

herbie shell --seed 2021045 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))