Average Error: 0.1 → 0.1
Time: 11.8s
Precision: binary64
\[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
↓
\[\left|\mathsf{fma}\left(\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), \left(-eh\right) \cdot \sin t, \left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right)}^{3}\right)\right| \]
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew))))
   (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))
↓
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   (sin (atan (* eh (/ (tan t) (- ew)))))
   (* (- eh) (sin t))
   (* (* ew (cos t)) (pow (cbrt (cos (atan (* eh (/ (tan t) ew))))) 3.0)))))
double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))));
}
↓
double code(double eh, double ew, double t) {
	return fabs(fma(sin(atan((eh * (tan(t) / -ew)))), (-eh * sin(t)), ((ew * cos(t)) * pow(cbrt(cos(atan((eh * (tan(t) / ew))))), 3.0))));
}

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))))))
end
↓
function code(eh, ew, t)
	return abs(fma(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))), Float64(Float64(-eh) * sin(t)), Float64(Float64(ew * cos(t)) * (cbrt(cos(atan(Float64(eh * Float64(tan(t) / ew))))) ^ 3.0))))
end

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|
↓
\left|\mathsf{fma}\left(\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), \left(-eh\right) \cdot \sin t, \left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right)}^{3}\right)\right|

Error

Bits error versus eh

Bits error versus ew

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left|\mathsf{fma}\left(\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), \left(-eh\right) \cdot \sin t, \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right)\right|} \]

  3. Applied egg-rr0.1

    \[\leadsto \left|\mathsf{fma}\left(\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), \left(-eh\right) \cdot \sin t, \left(ew \cdot \cos t\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right)}^{3}}\right)\right| \]

  4. Final simplification0.1

    \[\leadsto \left|\mathsf{fma}\left(\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), \left(-eh\right) \cdot \sin t, \left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right)}^{3}\right)\right| \]

Reproduce

herbie shell --seed 2022159 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))