Average Error: 12.8 → 4.4
Time: 52.7s
Precision: 64
Internal Precision: 576
\[x \cdot \sqrt{\cos t \cdot \cos t + \left(\left(h \cdot h\right) \cdot \sin t\right) \cdot \sin t}\]
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\sqrt{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t}} \cdot \sqrt[3]{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t} \le 9.918820324233946 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(\sqrt[3]{\sqrt{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t}} \cdot \sqrt[3]{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(h \cdot \sin t + \frac{1}{2} \cdot \frac{{\left(\cos t\right)}^{2}}{h \cdot \sin t}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus t

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if (* (cbrt (sqrt (+ (* (* (sin t) h) (* (sin t) h)) (* (cos t) (cos t))))) (cbrt (+ (* (* (sin t) h) (* (sin t) h)) (* (cos t) (cos t))))) < 9.918820324233946e+132

    1. Initial program 6.5

      \[x \cdot \sqrt{\cos t \cdot \cos t + \left(\left(h \cdot h\right) \cdot \sin t\right) \cdot \sin t}\]
    2. Using strategy rm
    3. Applied add-cbrt-cube8.4

      \[\leadsto x \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\cos t \cdot \cos t + \left(\left(h \cdot h\right) \cdot \sin t\right) \cdot \sin t} \cdot \sqrt{\cos t \cdot \cos t + \left(\left(h \cdot h\right) \cdot \sin t\right) \cdot \sin t}\right) \cdot \sqrt{\cos t \cdot \cos t + \left(\left(h \cdot h\right) \cdot \sin t\right) \cdot \sin t}}}\]
    4. Applied simplify2.8

      \[\leadsto x \cdot \sqrt[3]{\color{blue}{\sqrt{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t} \cdot \left(\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t\right)}}\]
    5. Using strategy rm
    6. Applied cbrt-prod0.4

      \[\leadsto x \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t}} \cdot \sqrt[3]{\left(\sin t \cdot h\right) \cdot \left(\sin t \cdot h\right) + \cos t \cdot \cos t}\right)}\]

    if 9.918820324233946e+132 < (* (cbrt (sqrt (+ (* (* (sin t) h) (* (sin t) h)) (* (cos t) (cos t))))) (cbrt (+ (* (* (sin t) h) (* (sin t) h)) (* (cos t) (cos t)))))

    1. Initial program 54.9

      \[x \cdot \sqrt{\cos t \cdot \cos t + \left(\left(h \cdot h\right) \cdot \sin t\right) \cdot \sin t}\]
    2. Taylor expanded around inf 30.6

      \[\leadsto x \cdot \color{blue}{\left(h \cdot \sin t + \frac{1}{2} \cdot \frac{{\left(\cos t\right)}^{2}}{h \cdot \sin t}\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 52.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x t h)
  :name "x*sqrt(cos(t)*cos(t)+h*h*sin(t)*sin(t))"
  (* x (sqrt (+ (* (cos t) (cos t)) (* (* (* h h) (sin t)) (sin t))))))