Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 24.5s

Alternatives: 8

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t_1 - \left(eh \cdot \sin t\right) \cdot \sin t_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

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(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t_1 - \left(eh \cdot \sin t\right) \cdot \sin t_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

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(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (log1p (expm1 (cos (atan (* eh (/ (tan t) ew)))))))
   (* (* eh (sin t)) (sin (atan (/ (* eh (- (tan t))) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * log1p(expm1(cos(atan((eh * (tan(t) / ew))))))) - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) * Math.log1p(Math.expm1(Math.cos(Math.atan((eh * (Math.tan(t) / ew))))))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((eh * -Math.tan(t)) / ew))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) * math.log1p(math.expm1(math.cos(math.atan((eh * (math.tan(t) / ew))))))) - ((eh * math.sin(t)) * math.sin(math.atan(((eh * -math.tan(t)) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) * log1p(expm1(cos(atan(Float64(eh * Float64(tan(t) / ew))))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $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]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. log1p-expm1-u 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   2. associate-/l* 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \color{blue}{\left(\frac{-eh}{\frac{ew}{\tan t}}\right)}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   3. div-inv 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{1}{\frac{ew}{\tan t}}\right)}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. add-sqr-sqrt 47.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. sqrt-unprod 94.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. sqr-neg 94.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   7. sqrt-unprod 52.6%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   8. add-sqr-sqrt 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\color{blue}{eh} \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   9. associate-/l* 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(eh \cdot \color{blue}{\frac{1 \cdot \tan t}{ew}}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   10. *-un-lft-identity 99.8%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(eh \cdot \frac{\color{blue}{\tan t}}{ew}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

3. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

4. Final simplification 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right| \]

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (/ 1.0 (hypot 1.0 (/ (tan t) (/ ew eh)))))
   (* (* eh (sin t)) (sin (atan (/ (* eh (- (tan t))) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * (1.0 / hypot(1.0, (tan(t) / (ew / eh))))) - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
}

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * (1.0 / hypot(1.0, (tan(t) / (ew / eh))))) - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] / N[(ew / eh), $MachinePrecision]), $MachinePrecision] ^ 2], $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]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. div-inv 99.8%

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

   7. add-sqr-sqrt 47.2%

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

   8. sqrt-unprod 94.0%

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

   9. sqr-neg 94.0%

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

   10. sqrt-unprod 52.6%

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

   11. add-sqr-sqrt 99.8%

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

   12. associate-/l* 99.8%

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

   13. *-un-lft-identity 99.8%

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

3. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r/ 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (cos (atan (/ (* eh (- (tan t))) ew))))
   (* (* eh (sin t)) (sin (atan (/ (* t (- eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * cos(atan(((eh * -tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
}

Fortran

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(((t * -eh) / ew))))))
end function

Java

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(((t * -eh) / ew))))));
}

Python

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(((t * -eh) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(eh * Float64(-tan(t))) / ew)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * cos(atan(((eh * -tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
end

Wolfram

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[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 99.0%

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

   1. associate-*r/ 77.6%

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

   2. mul-1-neg 77.6%

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

   3. distribute-rgt-neg-in 77.6%

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

4. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* ew (cos t))
   (* (* eh (sin t)) (sin (atan (/ (* eh (- (tan t))) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
}

Fortran

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)) - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $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]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. div-inv 99.8%

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

   7. add-sqr-sqrt 47.2%

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

   8. sqrt-unprod 94.0%

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

   9. sqr-neg 94.0%

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

   10. sqrt-unprod 52.6%

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

   11. add-sqr-sqrt 99.8%

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

   12. associate-/l* 99.8%

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

   13. *-un-lft-identity 99.8%

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

3. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 99.0%

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

7. Final simplification 99.0%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (- (* ew (cos t)) (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / ew))))));
}

Fortran

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)) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / ew))))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $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]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. div-inv 99.8%

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

   7. add-sqr-sqrt 47.2%

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

   8. sqrt-unprod 94.0%

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

   9. sqr-neg 94.0%

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

   10. sqrt-unprod 52.6%

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

   11. add-sqr-sqrt 99.8%

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

   12. associate-/l* 99.8%

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

   13. *-un-lft-identity 99.8%

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

3. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. neg-sub0 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{0 - eh \cdot \tan t}}{ew}\right)\right| \]

   3. remove-double-neg 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 - \color{blue}{\left(-\left(-eh \cdot \tan t\right)\right)}}{ew}\right)\right| \]

   4. distribute-lft-neg-out 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 - \left(-\color{blue}{\left(-eh\right) \cdot \tan t}\right)}{ew}\right)\right| \]

   5. neg-sub0 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 - \color{blue}{\left(0 - \left(-eh\right) \cdot \tan t\right)}}{ew}\right)\right| \]

   6. associate--r- 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(0 - 0\right) + \left(-eh\right) \cdot \tan t}}{ew}\right)\right| \]

   7. metadata-eval 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{0} + \left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   8. add-sqr-sqrt 46.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 + \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \tan t}{ew}\right)\right| \]

   9. sqrt-unprod 97.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 + \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \tan t}{ew}\right)\right| \]

   10. sqr-neg 97.2%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 + \sqrt{\color{blue}{eh \cdot eh}} \cdot \tan t}{ew}\right)\right| \]

   11. sqrt-unprod 52.6%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 + \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \tan t}{ew}\right)\right| \]

   12. add-sqr-sqrt 99.0%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{0 + \color{blue}{eh} \cdot \tan t}{ew}\right)\right| \]

8. Applied egg-rr 99.0%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{0 + eh \cdot \tan t}}{ew}\right)\right| \]
9. Step-by-step derivation

   1. +-lft-identity 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 6: 98.2% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (- (* ew (cos t)) (* (* eh (sin t)) (sin (atan (/ (* t (- eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
}

Fortran

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)) - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. div-inv 99.8%

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

   7. add-sqr-sqrt 47.2%

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

   8. sqrt-unprod 94.0%

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

   9. sqr-neg 94.0%

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

   10. sqrt-unprod 52.6%

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

   11. add-sqr-sqrt 99.8%

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

   12. associate-/l* 99.8%

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

   13. *-un-lft-identity 99.8%

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

3. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 99.0%

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

7. Taylor expanded in t around 0 98.5%

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

   1. associate-*r/ 77.6%

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

   2. mul-1-neg 77.6%

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

   3. distribute-rgt-neg-in 77.6%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 7: 78.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (- (* (* eh (sin t)) (sin (atan (/ (* eh (- (tan t))) ew)))) ew)))

C

double code(double eh, double ew, double t) {
	return fabs((((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew)))) - ew));
}

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew)))) - ew))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew)))) - ew));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(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] - ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. div-inv 99.8%

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

   7. add-sqr-sqrt 47.2%

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

   8. sqrt-unprod 94.0%

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

   9. sqr-neg 94.0%

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

   10. sqrt-unprod 52.6%

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

   11. add-sqr-sqrt 99.8%

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

   12. associate-/l* 99.8%

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

   13. *-un-lft-identity 99.8%

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

3. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 99.0%

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

7. Taylor expanded in t around 0 77.6%

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

8. Final simplification 77.6%

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

Alternative 8: 78.7% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (- ew (* (* eh (sin t)) (sin (atan (/ (* t (- eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((ew - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
}

Fortran

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 - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))))
end function

Java

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

Python

def code(eh, ew, t):
	return math.fabs((ew - ((eh * math.sin(t)) * math.sin(math.atan(((t * -eh) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(ew - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. div-inv 99.8%

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

   7. add-sqr-sqrt 47.2%

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

   8. sqrt-unprod 94.0%

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

   9. sqr-neg 94.0%

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

   10. sqrt-unprod 52.6%

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

   11. add-sqr-sqrt 99.8%

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

   12. associate-/l* 99.8%

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

   13. *-un-lft-identity 99.8%

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

3. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 99.0%

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

7. Taylor expanded in t around 0 77.6%

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

8. Taylor expanded in t around 0 77.6%

   \[\leadsto \left|ew \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right| \]
9. Step-by-step derivation

   1. associate-*r/ 77.6%

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

   2. mul-1-neg 77.6%

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

   3. distribute-rgt-neg-in 77.6%

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

10. Simplified 77.6%

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

11. Final simplification 77.6%

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

Reproduce

herbie shell --seed 2023301 
(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)))))))