Average Error: 2.0 → 2.3
Time: 9.2s
Precision: binary64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left(x \cdot {\left({\left(e^{\frac{1}{\sqrt{2}}}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}\right)}^{\left(\frac{1}{\sqrt{2}}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\left(x \cdot {\left({\left(e^{\frac{1}{\sqrt{2}}}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}\right)}^{\left(\frac{1}{\sqrt{2}}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}
(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))
↓
(FPCore (x y z t a b)
 :precision binary64
 (/
  (*
   (*
    x
    (pow
     (pow
      (exp (/ 1.0 (sqrt 2.0)))
      (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
     (/ 1.0 (sqrt 2.0))))
   (pow E (/ (- (+ (* y (log z)) (* (- t 1.0) (log a))) b) 2.0)))
  y))
double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp(((y * log(z)) + ((t - 1.0) * log(a))) - b)) / y;
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * pow(pow(exp(1.0 / sqrt(2.0)), (((y * log(z)) + ((t - 1.0) * log(a))) - b)), (1.0 / sqrt(2.0)))) * pow(((double) M_E), ((((y * log(z)) + ((t - 1.0) * log(a))) - b) / 2.0))) / y;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.0
Target10.9
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Initial program 2.0

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity_binary642.0

    \[\leadsto \frac{x \cdot e^{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

  4. Applied exp-prod_binary642.0

    \[\leadsto \frac{x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

  5. Simplified2.0

    \[\leadsto \frac{x \cdot {\color{blue}{e}}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{y}\]
  6. Using strategy rm

  7. Applied sqr-pow_binary642.0

    \[\leadsto \frac{x \cdot \color{blue}{\left({e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)} \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}\right)}}{y}\]

  8. Applied associate-*r*_binary642.0

    \[\leadsto \frac{\color{blue}{\left(x \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}}{y}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt_binary642.2

    \[\leadsto \frac{\left(x \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

  11. Applied *-un-lft-identity_binary642.2

    \[\leadsto \frac{\left(x \cdot {e}^{\left(\frac{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt{2} \cdot \sqrt{2}}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

  12. Applied times-frac_binary642.2

    \[\leadsto \frac{\left(x \cdot {e}^{\color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{\sqrt{2}}\right)}}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

  13. Applied pow-unpow_binary642.2

    \[\leadsto \frac{\left(x \cdot \color{blue}{{\left({e}^{\left(\frac{1}{\sqrt{2}}\right)}\right)}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{\sqrt{2}}\right)}}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]
  14. Using strategy rm

  15. Applied div-inv_binary642.2

    \[\leadsto \frac{\left(x \cdot {\left({e}^{\left(\frac{1}{\sqrt{2}}\right)}\right)}^{\color{blue}{\left(\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right) \cdot \frac{1}{\sqrt{2}}\right)}}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

  16. Applied pow-unpow_binary642.3

    \[\leadsto \frac{\left(x \cdot \color{blue}{{\left({\left({e}^{\left(\frac{1}{\sqrt{2}}\right)}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}\right)}^{\left(\frac{1}{\sqrt{2}}\right)}}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

  17. Simplified2.3

    \[\leadsto \frac{\left(x \cdot {\color{blue}{\left({\left(e^{\frac{1}{\sqrt{2}}}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}\right)}}^{\left(\frac{1}{\sqrt{2}}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

  18. Final simplification2.3

    \[\leadsto \frac{\left(x \cdot {\left({\left(e^{\frac{1}{\sqrt{2}}}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}\right)}^{\left(\frac{1}{\sqrt{2}}\right)}\right) \cdot {e}^{\left(\frac{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}{2}\right)}}{y}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))