Average Error: 0.1 → 0.1
Time: 12.7s
Precision: binary64
\[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \frac{eh \cdot \tan t}{ew}\\ \left|\frac{ew \cdot \cos t}{\sqrt{1 + t_1 \cdot t_1}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \end{array} \]
(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 (/ (* eh (tan t)) ew)))
   (fabs
    (-
     (/ (* ew (cos t)) (sqrt (+ 1.0 (* t_1 t_1))))
     (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) 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) {
	double t_1 = (eh * tan(t)) / ew;
	return fabs((((ew * cos(t)) / sqrt((1.0 + (t_1 * t_1)))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))));
}

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 = (eh * tan(t)) / ew
    code = abs((((ew * cos(t)) / sqrt((1.0d0 + (t_1 * t_1)))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))))
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 = (eh * Math.tan(t)) / ew;
	return Math.abs((((ew * Math.cos(t)) / Math.sqrt((1.0 + (t_1 * t_1)))) - ((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.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 = (eh * math.tan(t)) / ew
	return math.fabs((((ew * math.cos(t)) / math.sqrt((1.0 + (t_1 * t_1)))) - ((eh * math.sin(t)) * math.sin(math.atan(((-eh * math.tan(t)) / 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)
	t_1 = Float64(Float64(eh * tan(t)) / ew)
	return abs(Float64(Float64(Float64(ew * cos(t)) / sqrt(Float64(1.0 + Float64(t_1 * t_1)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * tan(t)) / 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)
	t_1 = (eh * tan(t)) / ew;
	tmp = abs((((ew * cos(t)) / sqrt((1.0 + (t_1 * t_1)))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / 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_] := Block[{t$95$1 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $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]]

\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 := \frac{eh \cdot \tan t}{ew}\\
\left|\frac{ew \cdot \cos t}{\sqrt{1 + t_1 \cdot t_1}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\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. Applied cos-atan_binary640.1

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

  3. Applied un-div-inv_binary640.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022137 
(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)))))))