Average Error: 29.8 → 0.1
Time: 57.8s
Precision: 64
Internal Precision: 1344
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 0.00361961541110823:\\ \;\;\;\;{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left(-2\right) \cdot \left(2 \cdot 2\right) + {\left(e^{x}\right)}^{3}} \cdot e^{x}\right) \cdot \left(\sqrt[3]{\left(-2\right) \cdot \left(2 \cdot 2\right) + {\left(e^{x}\right)}^{3}} \cdot \sqrt[3]{e^{x} - 2}\right) + \sqrt[3]{2 \cdot 2 + e^{x} \cdot \left(2 + e^{x}\right)} \cdot \sqrt[3]{2 \cdot 2 + e^{x} \cdot \left(2 + e^{x}\right)}}{\left(\sqrt[3]{2 \cdot 2 + \left(2 + e^{x}\right) \cdot e^{x}} \cdot \sqrt[3]{2 \cdot 2 + \left(2 + e^{x}\right) \cdot e^{x}}\right) \cdot e^{x}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.1
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (exp x) 2) (exp (- x))) < 0.00361961541110823

    1. Initial program 30.1

      \[\left(e^{x} - 2\right) + e^{-x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

    if 0.00361961541110823 < (+ (- (exp x) 2) (exp (- x)))

    1. Initial program 1.0

      \[\left(e^{x} - 2\right) + e^{-x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt1.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{e^{x} - 2}\right) \cdot \sqrt[3]{e^{x} - 2}} + e^{-x}\]
    4. Using strategy rm

    5. Applied exp-neg1.8

      \[\leadsto \left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{e^{x} - 2}\right) \cdot \sqrt[3]{e^{x} - 2} + \color{blue}{\frac{1}{e^{x}}}\]

    6. Applied flip3--4.7

      \[\leadsto \left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{e^{x} - 2}\right) \cdot \sqrt[3]{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {2}^{3}}{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}}} + \frac{1}{e^{x}}\]

    7. Applied cbrt-div4.7

      \[\leadsto \left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{e^{x} - 2}\right) \cdot \color{blue}{\frac{\sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}}} + \frac{1}{e^{x}}\]

    8. Applied flip3--4.7

      \[\leadsto \left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {2}^{3}}{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}}}\right) \cdot \frac{\sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}} + \frac{1}{e^{x}}\]

    9. Applied cbrt-div4.8

      \[\leadsto \left(\sqrt[3]{e^{x} - 2} \cdot \color{blue}{\frac{\sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}}}\right) \cdot \frac{\sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}} + \frac{1}{e^{x}}\]

    10. Applied associate-*r/4.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}}} \cdot \frac{\sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}} + \frac{1}{e^{x}}\]

    11. Applied frac-times4.7

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}\right) \cdot \sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)} \cdot \sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}}} + \frac{1}{e^{x}}\]

    12. Applied frac-add5.5

      \[\leadsto \color{blue}{\frac{\left(\left(\sqrt[3]{e^{x} - 2} \cdot \sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}\right) \cdot \sqrt[3]{{\left(e^{x}\right)}^{3} - {2}^{3}}\right) \cdot e^{x} + \left(\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)} \cdot \sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}\right) \cdot 1}{\left(\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)} \cdot \sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}\right) \cdot e^{x}}}\]

    13. Applied simplify5.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(-2\right) \cdot \left(2 \cdot 2\right) + {\left(e^{x}\right)}^{3}} \cdot e^{x}\right) \cdot \left(\sqrt[3]{\left(-2\right) \cdot \left(2 \cdot 2\right) + {\left(e^{x}\right)}^{3}} \cdot \sqrt[3]{e^{x} - 2}\right) + \sqrt[3]{2 \cdot 2 + e^{x} \cdot \left(2 + e^{x}\right)} \cdot \sqrt[3]{2 \cdot 2 + e^{x} \cdot \left(2 + e^{x}\right)}}}{\left(\sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)} \cdot \sqrt[3]{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}\right) \cdot e^{x}}\]

    14. Applied simplify5.5

      \[\leadsto \frac{\left(\sqrt[3]{\left(-2\right) \cdot \left(2 \cdot 2\right) + {\left(e^{x}\right)}^{3}} \cdot e^{x}\right) \cdot \left(\sqrt[3]{\left(-2\right) \cdot \left(2 \cdot 2\right) + {\left(e^{x}\right)}^{3}} \cdot \sqrt[3]{e^{x} - 2}\right) + \sqrt[3]{2 \cdot 2 + e^{x} \cdot \left(2 + e^{x}\right)} \cdot \sqrt[3]{2 \cdot 2 + e^{x} \cdot \left(2 + e^{x}\right)}}{\color{blue}{\left(\sqrt[3]{2 \cdot 2 + \left(2 + e^{x}\right) \cdot e^{x}} \cdot \sqrt[3]{2 \cdot 2 + \left(2 + e^{x}\right) \cdot e^{x}}\right) \cdot e^{x}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 57.8s)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))