Average Error: 0.1 → 0.1
Time: 14.9s
Precision: binary64
\[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
↓
\[\left|ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh \cdot \tan t}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right| \]
(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
  (-
   (* ew (/ (cos t) (hypot 1.0 (/ (* eh (tan t)) ew))))
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) (- eh)) ew)))))))
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(((ew * (cos(t) / hypot(1.0, ((eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))));
}

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) * Math.cos(Math.atan(((-eh * Math.tan(t)) / ew)))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((-eh * Math.tan(t)) / ew))))));
}
↓
public static double code(double eh, double ew, double t) {
	return Math.abs(((ew * (Math.cos(t) / Math.hypot(1.0, ((eh * Math.tan(t)) / ew)))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((Math.tan(t) * -eh) / ew))))));
}

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) * math.cos(math.atan(((-eh * math.tan(t)) / ew)))) - ((eh * math.sin(t)) * math.sin(math.atan(((-eh * math.tan(t)) / ew))))))
↓
def code(eh, ew, t):
	return math.fabs(((ew * (math.cos(t) / math.hypot(1.0, ((eh * math.tan(t)) / ew)))) - ((eh * math.sin(t)) * math.sin(math.atan(((math.tan(t) * -eh) / ew))))))

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(Float64(Float64(ew * Float64(cos(t) / hypot(1.0, Float64(Float64(eh * tan(t)) / ew)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * Float64(-eh)) / ew))))))
end

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))));
end
↓
function tmp = code(eh, ew, t)
	tmp = abs(((ew * (cos(t) / hypot(1.0, ((eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))));
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[(ew * N[(N[Cos[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $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|ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh \cdot \tan t}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right|

Error

Bits error versus eh

Bits error versus ew

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Applied egg-rr0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022133 
(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)))))))