Average Error: 0.1 → 0.1
Time: 11.0s
Precision: binary64
\[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \left|\mathsf{fma}\left(\sin t_1, eh \cdot \left(-\sin t\right), \left(ew \cdot \cos t\right) \cdot \cos t_1\right)\right| \end{array} \]
(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
 (let* ((t_1 (atan (* eh (/ (tan t) (- ew))))))
   (fabs (fma (sin t_1) (* eh (- (sin t))) (* (* ew (cos t)) (cos t_1))))))
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) {
	double t_1 = atan((eh * (tan(t) / -ew)));
	return fabs(fma(sin(t_1), (eh * -sin(t)), ((ew * cos(t)) * cos(t_1))));
}

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)
	t_1 = atan(Float64(eh * Float64(tan(t) / Float64(-ew))))
	return abs(fma(sin(t_1), Float64(eh * Float64(-sin(t))), Float64(Float64(ew * cos(t)) * cos(t_1))))
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_] := Block[{t$95$1 = N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * N[(eh * (-N[Sin[t], $MachinePrecision])), $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $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|
↓
\begin{array}{l}
t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\
\left|\mathsf{fma}\left(\sin t_1, eh \cdot \left(-\sin t\right), \left(ew \cdot \cos t\right) \cdot \cos t_1\right)\right|
\end{array}

Error

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-rr0.9

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

  4. Taylor expanded in t around inf 0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022192 
(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)))))))