Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 30.0s

Alternatives: 17

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(eh, ew, t):
	return math.fabs((((1.0 / math.hypot(1.0, (math.tan(t) * (eh / ew)))) * (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(Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))) * 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((((1.0 / hypot(1.0, (tan(t) * (eh / ew)))) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
end

Wolfram

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

3. Taylor expanded in t around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. mul-1-neg 99.2%

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

5. Simplified 99.2%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot t}}{ew}\right)\right| \]
6. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

7. Applied egg-rr 99.2%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	return math.fabs(((ew * math.cos(t)) - ((eh * math.sin(t)) * math.sin(math.atan(((math.tan(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(tan(t) * Float64(-eh)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((tan(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[(N[Tan[t], $MachinePrecision] * (-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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - ((eh * sin(t)) * sin(atan(((tan(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[(N[Tan[t], $MachinePrecision] * 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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

   1. add-log-exp 91.4%

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

   2. *-un-lft-identity 91.4%

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

   3. log-prod 91.4%

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

   4. metadata-eval 91.4%

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

   5. add-log-exp 98.6%

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

   6. add-sqr-sqrt 54.5%

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

   7. sqrt-unprod 97.8%

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

   8. sqr-neg 97.8%

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

   9. sqrt-unprod 44.0%

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

   10. add-sqr-sqrt 98.5%

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

9. Applied egg-rr 98.5%

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

    1. +-lft-identity 98.5%

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

11. Simplified 98.5%

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

12. Final simplification 98.5%

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

Alternative 5: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot eh}{ew}\\ t_2 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq -1.75 \cdot 10^{+122}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.2 \cdot 10^{-69}:\\ \;\;\;\;\left|ew \cdot \cos t - t\_1 \cdot \frac{t\_2}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - t\_2 \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* t eh) ew)) (t_2 (* eh (sin t))))
   (if (<= eh -1.75e+122)
     (fabs (- ew (* eh (* (sin t) (sin (atan (* eh (/ (tan t) ew))))))))
     (if (<= eh 5.2e-69)
       (fabs (- (* ew (cos t)) (* t_1 (/ t_2 (hypot 1.0 t_1)))))
       (fabs (- ew (* t_2 (sin (atan (/ (* t (- eh)) ew))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (t * eh) / ew;
	double t_2 = eh * sin(t);
	double tmp;
	if (eh <= -1.75e+122) {
		tmp = fabs((ew - (eh * (sin(t) * sin(atan((eh * (tan(t) / ew))))))));
	} else if (eh <= 5.2e-69) {
		tmp = fabs(((ew * cos(t)) - (t_1 * (t_2 / hypot(1.0, t_1)))));
	} else {
		tmp = fabs((ew - (t_2 * sin(atan(((t * -eh) / ew))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (t * eh) / ew;
	double t_2 = eh * Math.sin(t);
	double tmp;
	if (eh <= -1.75e+122) {
		tmp = Math.abs((ew - (eh * (Math.sin(t) * Math.sin(Math.atan((eh * (Math.tan(t) / ew))))))));
	} else if (eh <= 5.2e-69) {
		tmp = Math.abs(((ew * Math.cos(t)) - (t_1 * (t_2 / Math.hypot(1.0, t_1)))));
	} else {
		tmp = Math.abs((ew - (t_2 * Math.sin(Math.atan(((t * -eh) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (t * eh) / ew
	t_2 = eh * math.sin(t)
	tmp = 0
	if eh <= -1.75e+122:
		tmp = math.fabs((ew - (eh * (math.sin(t) * math.sin(math.atan((eh * (math.tan(t) / ew))))))))
	elif eh <= 5.2e-69:
		tmp = math.fabs(((ew * math.cos(t)) - (t_1 * (t_2 / math.hypot(1.0, t_1)))))
	else:
		tmp = math.fabs((ew - (t_2 * math.sin(math.atan(((t * -eh) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(t * eh) / ew)
	t_2 = Float64(eh * sin(t))
	tmp = 0.0
	if (eh <= -1.75e+122)
		tmp = abs(Float64(ew - Float64(eh * Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / ew))))))));
	elseif (eh <= 5.2e-69)
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(t_1 * Float64(t_2 / hypot(1.0, t_1)))));
	else
		tmp = abs(Float64(ew - Float64(t_2 * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (t * eh) / ew;
	t_2 = eh * sin(t);
	tmp = 0.0;
	if (eh <= -1.75e+122)
		tmp = abs((ew - (eh * (sin(t) * sin(atan((eh * (tan(t) / ew))))))));
	elseif (eh <= 5.2e-69)
		tmp = abs(((ew * cos(t)) - (t_1 * (t_2 / hypot(1.0, t_1)))));
	else
		tmp = abs((ew - (t_2 * sin(atan(((t * -eh) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -1.75e+122], N[Abs[N[(ew - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 5.2e-69], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(t$95$2 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.75000000000000007e122

   1. Initial program 99.6%

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

      1. cos-atan 99.6%

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

      2. hypot-1-def 99.6%

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

      3. associate-/l* 99.6%

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

      4. add-sqr-sqrt 99.6%

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

      5. sqrt-unprod 85.2%

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

      6. sqr-neg 85.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 97.7%

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

   8. Taylor expanded in t around 0 93.0%

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

      1. pow1 93.0%

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

      2. associate-*l* 93.0%

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

      3. associate-/l* 93.0%

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

      4. add-sqr-sqrt 93.0%

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

      5. sqrt-unprod 81.5%

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

      6. sqr-neg 81.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 93.0%

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

   10. Applied egg-rr 93.0%

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

       1. unpow1 93.0%

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

   12. Simplified 93.0%

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

   if -1.75000000000000007e122 < eh < 5.2000000000000004e-69

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 70.6%

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

      5. sqrt-unprod 99.6%

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

      6. sqr-neg 99.6%

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

      7. sqrt-unprod 29.3%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 99.0%

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

      1. add-log-exp 92.8%

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

      2. *-un-lft-identity 92.8%

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

      3. log-prod 92.8%

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

      4. metadata-eval 92.8%

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

      5. add-log-exp 99.0%

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

      6. add-sqr-sqrt 70.3%

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

      7. sqrt-unprod 97.8%

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

      8. sqr-neg 97.8%

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

      9. sqrt-unprod 28.7%

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

      10. add-sqr-sqrt 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. +-lft-identity 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in t around 0 98.6%

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

       1. sin-atan 88.2%

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

       2. associate-*r/ 88.1%

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

       3. associate-/l* 83.6%

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

       4. hypot-1-def 83.7%

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

       5. associate-/l* 83.6%

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

   14. Applied egg-rr 83.6%

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

       1. *-commutative 83.6%

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

       2. associate-/l* 84.9%

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

       3. associate-*r/ 89.5%

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

       4. associate-*r/ 89.7%

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

   16. Simplified 89.7%

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

   if 5.2000000000000004e-69 < eh

   1. Initial program 99.9%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/l* 99.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 93.2%

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

      6. sqr-neg 93.2%

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

      7. sqrt-unprod 99.9%

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

      8. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t around 0 98.0%

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

   8. Taylor expanded in t around 0 89.4%

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

   9. Taylor expanded in t around 0 89.4%

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

       1. associate-*r* 99.9%

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

       2. mul-1-neg 99.9%

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

   11. Simplified 89.4%

       \[\leadsto \left|ew \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot t}}{ew}\right)\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.75 \cdot 10^{+122}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.2 \cdot 10^{-69}:\\ \;\;\;\;\left|ew \cdot \cos t - \frac{t \cdot eh}{ew} \cdot \frac{eh \cdot \sin t}{\mathsf{hypot}\left(1, \frac{t \cdot eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in t around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. mul-1-neg 99.2%

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

5. Simplified 99.2%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot t}}{ew}\right)\right| \]
6. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

7. Applied egg-rr 99.2%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

9. Simplified 99.2%

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

10. Taylor expanded in t around 0 98.3%

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

11. Final simplification 98.3%

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

Alternative 7: 98.2% accurate, 1.8× speedup

Math

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

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 * 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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

   1. add-log-exp 91.4%

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

   2. *-un-lft-identity 91.4%

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

   3. log-prod 91.4%

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

   4. metadata-eval 91.4%

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

   5. add-log-exp 98.6%

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

   6. add-sqr-sqrt 54.5%

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

   7. sqrt-unprod 97.8%

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

   8. sqr-neg 97.8%

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

   9. sqrt-unprod 44.0%

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

   10. add-sqr-sqrt 98.5%

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

9. Applied egg-rr 98.5%

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

    1. +-lft-identity 98.5%

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

11. Simplified 98.5%

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

12. Taylor expanded in t around 0 98.3%

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

13. Final simplification 98.3%

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

Alternative 8: 88.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot eh}{ew}\\ t_2 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq -1.75 \cdot 10^{+122} \lor \neg \left(eh \leq 5.2 \cdot 10^{-69}\right):\\ \;\;\;\;\left|ew - t\_2 \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t - t\_1 \cdot \frac{t\_2}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* t eh) ew)) (t_2 (* eh (sin t))))
   (if (or (<= eh -1.75e+122) (not (<= eh 5.2e-69)))
     (fabs (- ew (* t_2 (sin (atan (/ (* t (- eh)) ew))))))
     (fabs (- (* ew (cos t)) (* t_1 (/ t_2 (hypot 1.0 t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (t * eh) / ew;
	double t_2 = eh * sin(t);
	double tmp;
	if ((eh <= -1.75e+122) || !(eh <= 5.2e-69)) {
		tmp = fabs((ew - (t_2 * sin(atan(((t * -eh) / ew))))));
	} else {
		tmp = fabs(((ew * cos(t)) - (t_1 * (t_2 / hypot(1.0, t_1)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (t * eh) / ew;
	double t_2 = eh * Math.sin(t);
	double tmp;
	if ((eh <= -1.75e+122) || !(eh <= 5.2e-69)) {
		tmp = Math.abs((ew - (t_2 * Math.sin(Math.atan(((t * -eh) / ew))))));
	} else {
		tmp = Math.abs(((ew * Math.cos(t)) - (t_1 * (t_2 / Math.hypot(1.0, t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (t * eh) / ew
	t_2 = eh * math.sin(t)
	tmp = 0
	if (eh <= -1.75e+122) or not (eh <= 5.2e-69):
		tmp = math.fabs((ew - (t_2 * math.sin(math.atan(((t * -eh) / ew))))))
	else:
		tmp = math.fabs(((ew * math.cos(t)) - (t_1 * (t_2 / math.hypot(1.0, t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(t * eh) / ew)
	t_2 = Float64(eh * sin(t))
	tmp = 0.0
	if ((eh <= -1.75e+122) || !(eh <= 5.2e-69))
		tmp = abs(Float64(ew - Float64(t_2 * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))));
	else
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(t_1 * Float64(t_2 / hypot(1.0, t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (t * eh) / ew;
	t_2 = eh * sin(t);
	tmp = 0.0;
	if ((eh <= -1.75e+122) || ~((eh <= 5.2e-69)))
		tmp = abs((ew - (t_2 * sin(atan(((t * -eh) / ew))))));
	else
		tmp = abs(((ew * cos(t)) - (t_1 * (t_2 / hypot(1.0, t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -1.75e+122], N[Not[LessEqual[eh, 5.2e-69]], $MachinePrecision]], N[Abs[N[(ew - N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(t$95$2 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.75000000000000007e122 or 5.2000000000000004e-69 < eh

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 32.9%

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

      5. sqrt-unprod 90.6%

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

      6. sqr-neg 90.6%

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

      7. sqrt-unprod 66.9%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 97.9%

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

   8. Taylor expanded in t around 0 90.6%

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

   9. Taylor expanded in t around 0 90.6%

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

       1. associate-*r* 99.7%

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

       2. mul-1-neg 99.7%

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

   11. Simplified 90.6%

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

   if -1.75000000000000007e122 < eh < 5.2000000000000004e-69

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 70.6%

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

      5. sqrt-unprod 99.6%

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

      6. sqr-neg 99.6%

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

      7. sqrt-unprod 29.3%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 99.0%

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

      1. add-log-exp 92.8%

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

      2. *-un-lft-identity 92.8%

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

      3. log-prod 92.8%

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

      4. metadata-eval 92.8%

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

      5. add-log-exp 99.0%

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

      6. add-sqr-sqrt 70.3%

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

      7. sqrt-unprod 97.8%

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

      8. sqr-neg 97.8%

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

      9. sqrt-unprod 28.7%

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

      10. add-sqr-sqrt 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. +-lft-identity 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in t around 0 98.6%

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

       1. sin-atan 88.2%

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

       2. associate-*r/ 88.1%

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

       3. associate-/l* 83.6%

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

       4. hypot-1-def 83.7%

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

       5. associate-/l* 83.6%

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

   14. Applied egg-rr 83.6%

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

       1. *-commutative 83.6%

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

       2. associate-/l* 84.9%

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

       3. associate-*r/ 89.5%

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

       4. associate-*r/ 89.7%

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

   16. Simplified 89.7%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \color{blue}{\frac{eh \cdot t}{ew} \cdot \frac{eh \cdot \sin t}{\mathsf{hypot}\left(1, \frac{eh \cdot t}{ew}\right)}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.75 \cdot 10^{+122} \lor \neg \left(eh \leq 5.2 \cdot 10^{-69}\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t - \frac{t \cdot eh}{ew} \cdot \frac{eh \cdot \sin t}{\mathsf{hypot}\left(1, \frac{t \cdot eh}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.06 \cdot 10^{+187}:\\ \;\;\;\;\left|ew \cdot \cos t - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.06e+187)
   (fabs (- (* ew (cos t)) (* (* t eh) (sin (atan (/ (* t eh) ew))))))
   (fabs (- ew (* (* eh (sin t)) (sin (atan (/ (* t (- eh)) ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.06e+187) {
		tmp = fabs(((ew * cos(t)) - ((t * eh) * sin(atan(((t * eh) / ew))))));
	} else {
		tmp = fabs((ew - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
	}
	return tmp;
}

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) :: tmp
    if (ew <= (-1.06d+187)) then
        tmp = abs(((ew * cos(t)) - ((t * eh) * sin(atan(((t * eh) / ew))))))
    else
        tmp = abs((ew - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.06e+187) {
		tmp = Math.abs(((ew * Math.cos(t)) - ((t * eh) * Math.sin(Math.atan(((t * eh) / ew))))));
	} else {
		tmp = Math.abs((ew - ((eh * Math.sin(t)) * Math.sin(Math.atan(((t * -eh) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.06e+187:
		tmp = math.fabs(((ew * math.cos(t)) - ((t * eh) * math.sin(math.atan(((t * eh) / ew))))))
	else:
		tmp = math.fabs((ew - ((eh * math.sin(t)) * math.sin(math.atan(((t * -eh) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.06e+187)
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(Float64(t * eh) * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = abs(Float64(ew - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.06e+187)
		tmp = abs(((ew * cos(t)) - ((t * eh) * sin(atan(((t * eh) / ew))))));
	else
		tmp = abs((ew - ((eh * sin(t)) * sin(atan(((t * -eh) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.06e+187], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(N[(t * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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. Split input into 2 regimes

2. if ew < -1.06e187

   1. Initial program 99.7%

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

      1. cos-atan 99.7%

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

      2. hypot-1-def 99.7%

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

      3. associate-/l* 99.7%

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

      4. add-sqr-sqrt 45.7%

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

      5. sqrt-unprod 87.7%

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

      6. sqr-neg 87.7%

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

      7. sqrt-unprod 54.0%

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

      8. add-sqr-sqrt 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 99.7%

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

      1. add-log-exp 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. log-prod 99.7%

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

      4. metadata-eval 99.7%

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

      5. add-log-exp 99.7%

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

      6. add-sqr-sqrt 45.7%

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

      7. sqrt-unprod 99.7%

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

      8. sqr-neg 99.7%

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

      9. sqrt-unprod 54.0%

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

      10. add-sqr-sqrt 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. +-lft-identity 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in t around 0 99.7%

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

   13. Taylor expanded in t around 0 95.9%

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

   if -1.06e187 < ew

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 55.9%

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

      5. sqrt-unprod 96.7%

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

      6. sqr-neg 96.7%

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

      7. sqrt-unprod 43.9%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 98.4%

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

   8. Taylor expanded in t around 0 77.5%

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

   9. Taylor expanded in t around 0 77.5%

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

       1. associate-*r* 99.2%

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

       2. mul-1-neg 99.2%

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

   11. Simplified 77.5%

       \[\leadsto \left|ew \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right) \cdot t}}{ew}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.06 \cdot 10^{+187}:\\ \;\;\;\;\left|ew \cdot \cos t - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% 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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

8. Taylor expanded in t around 0 75.6%

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

9. Taylor expanded in t around 0 75.6%

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

    1. associate-*r* 99.2%

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

    2. mul-1-neg 99.2%

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

11. Simplified 75.6%

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

12. Final simplification 75.6%

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

Alternative 11: 79.6% accurate, 2.2× speedup

Math

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

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 * 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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

   1. add-log-exp 91.4%

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

   2. *-un-lft-identity 91.4%

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

   3. log-prod 91.4%

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

   4. metadata-eval 91.4%

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

   5. add-log-exp 98.6%

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

   6. add-sqr-sqrt 54.5%

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

   7. sqrt-unprod 97.8%

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

   8. sqr-neg 97.8%

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

   9. sqrt-unprod 44.0%

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

   10. add-sqr-sqrt 98.5%

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

9. Applied egg-rr 98.5%

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

    1. +-lft-identity 98.5%

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

11. Simplified 98.5%

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

12. Taylor expanded in t around 0 98.3%

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

13. Taylor expanded in t around 0 75.6%

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

14. Final simplification 75.6%

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

Alternative 12: 55.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - N[(t * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

8. Taylor expanded in t around 0 75.6%

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

9. Taylor expanded in t around 0 51.0%

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

    1. associate-*r* 50.9%

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

    2. *-commutative 50.9%

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

    3. associate-*r* 51.1%

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

    4. mul-1-neg 51.1%

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

    5. associate-*r/ 51.1%

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

    6. distribute-rgt-neg-in 51.1%

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

11. Simplified 51.1%

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

    1. add-sqr-sqrt 31.9%

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

    2. sqrt-unprod 49.5%

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

    3. sqr-neg 49.5%

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

    4. sqrt-unprod 27.8%

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

    5. add-sqr-sqrt 51.0%

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

    6. clear-num 51.0%

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

    7. un-div-inv 51.0%

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

13. Applied egg-rr 51.0%

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

    1. associate-/r/ 51.0%

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

15. Simplified 51.0%

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

16. Final simplification 51.0%

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

Alternative 13: 54.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - N[(N[(t * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(t / (-ew)), $MachinePrecision]), $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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

8. Taylor expanded in t around 0 75.6%

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

9. Taylor expanded in t around 0 51.0%

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

    1. associate-*r* 50.9%

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

    2. *-commutative 50.9%

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

    3. associate-*r* 51.1%

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

    4. mul-1-neg 51.1%

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

    5. associate-*r/ 51.1%

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

    6. distribute-rgt-neg-in 51.1%

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

11. Simplified 51.1%

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

12. Taylor expanded in t around 0 49.8%

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

    1. mul-1-neg 49.8%

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

    2. *-commutative 49.8%

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

    3. associate-*r/ 49.8%

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

    4. distribute-rgt-neg-in 49.8%

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

    5. distribute-frac-neg2 49.8%

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

14. Simplified 49.8%

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

15. Taylor expanded in ew around 0 49.8%

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

    1. *-commutative 49.8%

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

    2. *-commutative 49.8%

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

    3. associate-*l* 49.8%

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

    4. mul-1-neg 49.8%

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

    5. associate-/l* 49.8%

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

    6. distribute-rgt-neg-in 49.8%

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

    7. distribute-frac-neg2 49.8%

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

    8. *-commutative 49.8%

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

17. Simplified 49.8%

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

18. Final simplification 49.8%

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

Alternative 14: 51.6% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (- ew (* t (* eh (* (/ t ew) (/ eh (hypot 1.0 (* eh (/ t ew))))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - N[(t * N[(eh * N[(N[(t / ew), $MachinePrecision] * N[(eh / N[Sqrt[1.0 ^ 2 + N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

8. Taylor expanded in t around 0 75.6%

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

9. Taylor expanded in t around 0 51.0%

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

    1. associate-*r* 50.9%

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

    2. *-commutative 50.9%

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

    3. associate-*r* 51.1%

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

    4. mul-1-neg 51.1%

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

    5. associate-*r/ 51.1%

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

    6. distribute-rgt-neg-in 51.1%

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

11. Simplified 51.1%

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

12. Taylor expanded in t around 0 49.8%

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

    1. mul-1-neg 49.8%

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

    2. *-commutative 49.8%

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

    3. associate-*r/ 49.8%

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

    4. distribute-rgt-neg-in 49.8%

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

    5. distribute-frac-neg2 49.8%

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

14. Simplified 49.8%

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

    1. sin-atan 42.4%

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

    2. associate-*r/ 41.0%

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

    3. associate-*r/ 40.9%

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

    4. add-sqr-sqrt 19.9%

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

    5. sqrt-unprod 32.1%

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

    6. sqr-neg 32.1%

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

    7. sqrt-unprod 21.1%

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

    8. add-sqr-sqrt 40.9%

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

    9. associate-/l* 41.0%

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

    10. hypot-1-def 42.2%

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

    11. associate-*r/ 42.2%

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

    12. add-sqr-sqrt 20.4%

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

    13. sqrt-unprod 35.7%

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

    14. sqr-neg 35.7%

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

    15. sqrt-unprod 21.8%

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

    16. add-sqr-sqrt 42.2%

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

    17. associate-/l* 42.2%

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

16. Applied egg-rr 42.2%

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

    1. associate-/l* 44.1%

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

    2. associate-*r/ 44.2%

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

    3. associate-*l/ 44.1%

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

    4. associate-*r/ 44.3%

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

    5. associate-*r/ 47.4%

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

    6. *-commutative 47.4%

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

    7. associate-/l* 47.3%

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

18. Simplified 47.3%

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

19. Final simplification 47.3%

    \[\leadsto \left|ew - t \cdot \left(eh \cdot \left(\frac{t}{ew} \cdot \frac{eh}{\mathsf{hypot}\left(1, eh \cdot \frac{t}{ew}\right)}\right)\right)\right| \]
20. Add Preprocessing

Alternative 15: 44.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+210} \lor \neg \left(eh \leq 2.5 \cdot 10^{+85} \lor \neg \left(eh \leq 7.8 \cdot 10^{+236}\right) \land eh \leq 5.2 \cdot 10^{+260}\right):\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -8.8e+210)
         (not
          (or (<= eh 2.5e+85) (and (not (<= eh 7.8e+236)) (<= eh 5.2e+260)))))
   (fabs (* t eh))
   (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.8e+210) || !((eh <= 2.5e+85) || (!(eh <= 7.8e+236) && (eh <= 5.2e+260)))) {
		tmp = fabs((t * eh));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

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) :: tmp
    if ((eh <= (-8.8d+210)) .or. (.not. (eh <= 2.5d+85) .or. (.not. (eh <= 7.8d+236)) .and. (eh <= 5.2d+260))) then
        tmp = abs((t * eh))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.8e+210) || !((eh <= 2.5e+85) || (!(eh <= 7.8e+236) && (eh <= 5.2e+260)))) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -8.8e+210) or not ((eh <= 2.5e+85) or (not (eh <= 7.8e+236) and (eh <= 5.2e+260))):
		tmp = math.fabs((t * eh))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -8.8e+210) || !((eh <= 2.5e+85) || (!(eh <= 7.8e+236) && (eh <= 5.2e+260))))
		tmp = abs(Float64(t * eh));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -8.8e+210) || ~(((eh <= 2.5e+85) || (~((eh <= 7.8e+236)) && (eh <= 5.2e+260)))))
		tmp = abs((t * eh));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -8.8e+210], N[Not[Or[LessEqual[eh, 2.5e+85], And[N[Not[LessEqual[eh, 7.8e+236]], $MachinePrecision], LessEqual[eh, 5.2e+260]]]], $MachinePrecision]], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.79999999999999948e210 or 2.5e85 < eh < 7.8000000000000001e236 or 5.1999999999999996e260 < eh

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 34.5%

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

      5. sqrt-unprod 90.3%

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

      6. sqr-neg 90.3%

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

      7. sqrt-unprod 65.3%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 97.6%

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

   8. Taylor expanded in t around 0 94.3%

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

   9. Taylor expanded in t around 0 48.1%

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

       1. associate-*r* 48.1%

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

       2. *-commutative 48.1%

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

       3. associate-*r* 48.1%

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

       4. mul-1-neg 48.1%

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

       5. associate-*r/ 48.1%

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

       6. distribute-rgt-neg-in 48.1%

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

   11. Simplified 48.1%

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

   12. Taylor expanded in t around 0 48.0%

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

       1. mul-1-neg 48.0%

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

       2. *-commutative 48.0%

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

       3. associate-*r/ 48.0%

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

       4. distribute-rgt-neg-in 48.0%

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

       5. distribute-frac-neg2 48.0%

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

   14. Simplified 48.0%

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

   15. Taylor expanded in t around inf 40.8%

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

       1. cancel-sign-sub-inv 40.8%

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

       2. cancel-sign-sub 40.8%

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

       3. remove-double-neg 40.8%

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

       4. mul-1-neg 40.8%

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

       5. *-commutative 40.8%

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

       6. associate-*r/ 40.8%

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

       7. distribute-lft-neg-in 40.8%

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

   17. Simplified 40.8%

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

       1. *-commutative 40.8%

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

       2. sin-atan 17.2%

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

       3. associate-*l/ 10.7%

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

       4. add-sqr-sqrt 4.3%

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

       5. sqrt-unprod 6.2%

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

       6. sqr-neg 6.2%

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

       7. sqrt-unprod 6.3%

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

       8. add-sqr-sqrt 10.7%

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

       9. associate-*r/ 10.7%

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

       10. *-commutative 10.7%

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

       11. associate-/l* 10.7%

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

       12. distribute-lft-neg-out 10.7%

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

       13. distribute-lft-neg-out 10.7%

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

       14. sqr-neg 10.7%

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

   19. Applied egg-rr 14.1%

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

       1. *-commutative 14.1%

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

       2. associate-/l* 30.8%

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

       3. associate-*r/ 30.7%

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

       4. associate-*l/ 19.5%

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

       5. *-commutative 19.5%

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

       6. *-lft-identity 19.5%

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

       7. associate-*l/ 19.6%

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

       8. associate-*l* 30.6%

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

       9. *-commutative 30.6%

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

       10. associate-/r/ 30.7%

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

       11. associate-*l/ 30.6%

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

       12. *-lft-identity 30.6%

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

       13. *-commutative 30.6%

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

       14. *-rgt-identity 30.6%

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

       15. associate-*l/ 30.6%

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

       16. associate-*r/ 30.6%

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

       17. *-commutative 30.6%

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

       18. associate-/r/ 30.6%

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

       19. associate-*l/ 30.8%

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

       20. *-lft-identity 30.8%

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

   21. Simplified 30.8%

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

   22. Taylor expanded in t around -inf 39.5%

       \[\leadsto \left|\color{blue}{eh \cdot t}\right| \]
   23. Step-by-step derivation

       1. *-commutative 39.5%

          \[\leadsto \left|\color{blue}{t \cdot eh}\right| \]

   24. Simplified 39.5%

       \[\leadsto \left|\color{blue}{t \cdot eh}\right| \]

   if -8.79999999999999948e210 < eh < 2.5e85 or 7.8000000000000001e236 < eh < 5.1999999999999996e260

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 60.2%

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

      5. sqrt-unprod 97.3%

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

      6. sqr-neg 97.3%

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

      7. sqrt-unprod 39.6%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 98.8%

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

   8. Taylor expanded in t around 0 70.9%

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

   9. Taylor expanded in t around 0 51.7%

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

       1. associate-*r* 51.7%

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

       2. *-commutative 51.7%

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

       3. associate-*r* 51.8%

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

       4. mul-1-neg 51.8%

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

       5. associate-*r/ 51.8%

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

       6. distribute-rgt-neg-in 51.8%

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

   11. Simplified 51.8%

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

   12. Taylor expanded in t around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. *-commutative 50.3%

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

       3. associate-*r/ 50.3%

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

       4. distribute-rgt-neg-in 50.3%

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

       5. distribute-frac-neg2 50.3%

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

   14. Simplified 50.3%

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

   15. Taylor expanded in ew around inf 47.7%

       \[\leadsto \left|\color{blue}{ew}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+210} \lor \neg \left(eh \leq 2.5 \cdot 10^{+85} \lor \neg \left(eh \leq 7.8 \cdot 10^{+236}\right) \land eh \leq 5.2 \cdot 10^{+260}\right):\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6.2 \cdot 10^{+104} \lor \neg \left(ew \leq 2.4 \cdot 10^{-155}\right):\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(\frac{ew}{t} - eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -6.2e+104) (not (<= ew 2.4e-155)))
   (fabs ew)
   (fabs (* t (- (/ ew t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6.2e+104) || !(ew <= 2.4e-155)) {
		tmp = fabs(ew);
	} else {
		tmp = fabs((t * ((ew / t) - eh)));
	}
	return tmp;
}

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) :: tmp
    if ((ew <= (-6.2d+104)) .or. (.not. (ew <= 2.4d-155))) then
        tmp = abs(ew)
    else
        tmp = abs((t * ((ew / t) - eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6.2e+104) || !(ew <= 2.4e-155)) {
		tmp = Math.abs(ew);
	} else {
		tmp = Math.abs((t * ((ew / t) - eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -6.2e+104) or not (ew <= 2.4e-155):
		tmp = math.fabs(ew)
	else:
		tmp = math.fabs((t * ((ew / t) - eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -6.2e+104) || !(ew <= 2.4e-155))
		tmp = abs(ew);
	else
		tmp = abs(Float64(t * Float64(Float64(ew / t) - eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -6.2e+104) || ~((ew <= 2.4e-155)))
		tmp = abs(ew);
	else
		tmp = abs((t * ((ew / t) - eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -6.2e+104], N[Not[LessEqual[ew, 2.4e-155]], $MachinePrecision]], N[Abs[ew], $MachinePrecision], N[Abs[N[(t * N[(N[(ew / t), $MachinePrecision] - eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.20000000000000033e104 or 2.4e-155 < ew

   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. Add Preprocessing
   3. 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. hypot-1-def 99.8%

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

      3. associate-/l* 99.8%

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

      4. add-sqr-sqrt 54.6%

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

      5. sqrt-unprod 93.8%

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

      6. sqr-neg 93.8%

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

      7. sqrt-unprod 45.2%

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

      8. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 98.9%

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

   8. Taylor expanded in t around 0 70.1%

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

   9. Taylor expanded in t around 0 49.3%

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

       1. associate-*r* 49.3%

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

       2. *-commutative 49.3%

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

       3. associate-*r* 49.4%

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

       4. mul-1-neg 49.4%

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

       5. associate-*r/ 49.4%

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

       6. distribute-rgt-neg-in 49.4%

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

   11. Simplified 49.4%

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

   12. Taylor expanded in t around 0 48.3%

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

       1. mul-1-neg 48.3%

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

       2. *-commutative 48.3%

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

       3. associate-*r/ 48.3%

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

       4. distribute-rgt-neg-in 48.3%

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

       5. distribute-frac-neg2 48.3%

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

   14. Simplified 48.3%

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

   15. Taylor expanded in ew around inf 46.5%

       \[\leadsto \left|\color{blue}{ew}\right| \]

   if -6.20000000000000033e104 < ew < 2.4e-155

   1. Initial program 99.9%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/l* 99.9%

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

      4. add-sqr-sqrt 55.5%

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

      5. sqrt-unprod 98.6%

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

      6. sqr-neg 98.6%

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

      7. sqrt-unprod 44.4%

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

      8. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t around 0 98.1%

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

   8. Taylor expanded in t around 0 83.1%

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

   9. Taylor expanded in t around 0 53.2%

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

       1. associate-*r* 53.2%

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

       2. *-commutative 53.2%

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

       3. associate-*r* 53.3%

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

       4. mul-1-neg 53.3%

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

       5. associate-*r/ 53.3%

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

       6. distribute-rgt-neg-in 53.3%

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

   11. Simplified 53.3%

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

   12. Taylor expanded in t around 0 51.9%

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

       1. mul-1-neg 51.9%

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

       2. *-commutative 51.9%

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

       3. associate-*r/ 51.9%

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

       4. distribute-rgt-neg-in 51.9%

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

       5. distribute-frac-neg2 51.9%

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

   14. Simplified 51.9%

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

   15. Taylor expanded in t around inf 51.0%

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

       1. cancel-sign-sub-inv 51.0%

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

       2. cancel-sign-sub 51.0%

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

       3. remove-double-neg 51.0%

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

       4. mul-1-neg 51.0%

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

       5. *-commutative 51.0%

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

       6. associate-*r/ 51.0%

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

       7. distribute-lft-neg-in 51.0%

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

   17. Simplified 51.0%

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

       1. *-commutative 51.0%

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

       2. sin-atan 35.3%

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

       3. associate-*l/ 33.3%

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

       4. add-sqr-sqrt 15.8%

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

       5. sqrt-unprod 31.4%

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

       6. sqr-neg 31.4%

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

       7. sqrt-unprod 17.4%

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

       8. add-sqr-sqrt 33.3%

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

       9. associate-*r/ 33.4%

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

       10. *-commutative 33.4%

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

       11. associate-/l* 33.0%

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

       12. distribute-lft-neg-out 33.0%

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

       13. distribute-lft-neg-out 33.0%

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

       14. sqr-neg 33.0%

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

   19. Applied egg-rr 37.9%

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

       1. *-commutative 37.9%

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

       2. associate-/l* 45.1%

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

       3. associate-*r/ 45.8%

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

       4. associate-*l/ 38.5%

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

       5. *-commutative 38.5%

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

       6. *-lft-identity 38.5%

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

       7. associate-*l/ 38.6%

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

       8. associate-*l* 45.0%

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

       9. *-commutative 45.0%

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

       10. associate-/r/ 45.1%

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

       11. associate-*l/ 45.4%

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

       12. *-lft-identity 45.4%

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

       13. *-commutative 45.4%

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

       14. *-rgt-identity 45.4%

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

       15. associate-*l/ 45.4%

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

       16. associate-*r/ 45.4%

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

       17. *-commutative 45.4%

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

       18. associate-/r/ 45.4%

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

       19. associate-*l/ 45.3%

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

       20. *-lft-identity 45.3%

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

   21. Simplified 45.3%

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

   22. Taylor expanded in t around inf 51.0%

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

       1. +-commutative 51.0%

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

       2. neg-mul-1 51.0%

          \[\leadsto \left|t \cdot \left(\frac{ew}{t} + \color{blue}{\left(-eh\right)}\right)\right| \]

       3. unsub-neg 51.0%

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

   24. Simplified 51.0%

       \[\leadsto \left|\color{blue}{t \cdot \left(\frac{ew}{t} - eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.2 \cdot 10^{+104} \lor \neg \left(ew \leq 2.4 \cdot 10^{-155}\right):\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(\frac{ew}{t} - eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|ew\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs ew))

C

double code(double eh, double ew, double t) {
	return fabs(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)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(ew);
}

Python

def code(eh, ew, t):
	return math.fabs(ew)

Julia

function code(eh, ew, t)
	return abs(ew)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(ew);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $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. Add Preprocessing
3. 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. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 55.0%

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

   5. sqrt-unprod 95.8%

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

   6. sqr-neg 95.8%

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

   7. sqrt-unprod 44.8%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.6%

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

8. Taylor expanded in t around 0 75.6%

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

9. Taylor expanded in t around 0 51.0%

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

    1. associate-*r* 50.9%

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

    2. *-commutative 50.9%

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

    3. associate-*r* 51.1%

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

    4. mul-1-neg 51.1%

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

    5. associate-*r/ 51.1%

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

    6. distribute-rgt-neg-in 51.1%

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

11. Simplified 51.1%

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

12. Taylor expanded in t around 0 49.8%

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

    1. mul-1-neg 49.8%

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

    2. *-commutative 49.8%

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

    3. associate-*r/ 49.8%

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

    4. distribute-rgt-neg-in 49.8%

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

    5. distribute-frac-neg2 49.8%

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

14. Simplified 49.8%

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

15. Taylor expanded in ew around inf 41.0%

    \[\leadsto \left|\color{blue}{ew}\right| \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))))