Average Error: 0.1 → 0.1
Time: 11.3s
Precision: binary64
\[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\\ \left|\left(eh \cdot \sin t\right) \cdot \sin t_1 - \cos t \cdot \left(ew \cdot \cos t_1\right)\right| \end{array} \]
(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 (/ (* (tan t) (- eh)) ew))))
   (fabs (- (* (* eh (sin t)) (sin t_1)) (* (cos t) (* ew (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(((tan(t) * -eh) / ew));
	return fabs((((eh * sin(t)) * sin(t_1)) - (cos(t) * (ew * cos(t_1)))));
}

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((((ew * cos(t)) * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))))
end function
↓
real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((tan(t) * -eh) / ew))
    code = abs((((eh * sin(t)) * sin(t_1)) - (cos(t) * (ew * cos(t_1)))))
end function

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) {
	double t_1 = Math.atan(((Math.tan(t) * -eh) / ew));
	return Math.abs((((eh * Math.sin(t)) * Math.sin(t_1)) - (Math.cos(t) * (ew * Math.cos(t_1)))));
}

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

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. Taylor expanded in ew around 0 0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022153 
(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)))))))