Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.9s

Alternatives: 15

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 48.4%

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

   2. sqrt-unprod 93.4%

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

   3. sqr-neg 93.4%

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

   4. sqrt-unprod 51.5%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 44.0%

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

   2. hypot-1-def 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

   3. add-sqr-sqrt 19.0%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   4. sqrt-unprod 38.5%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   5. sqr-neg 38.5%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   6. sqrt-unprod 25.0%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   7. add-sqr-sqrt 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   8. clear-num 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

   9. un-div-inv 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\\ \mathbf{if}\;ew \leq -3.6 \cdot 10^{+114}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 1.1 \cdot 10^{+124}:\\ \;\;\;\;\left|t\_1 \cdot \cos t\_2 - eh \cdot \left(\sin t \cdot \sin t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(eh \cdot \left(-\log \left(e^{\frac{\tan t}{ew}}\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (* (- eh) (/ t ew)))))
   (if (<= ew -3.6e+114)
     (fabs (* t_1 (/ 1.0 (hypot 1.0 (/ eh (/ ew (tan t)))))))
     (if (<= ew 1.1e+124)
       (fabs (- (* t_1 (cos t_2)) (* eh (* (sin t) (sin t_2)))))
       (fabs (* t_1 (cos (atan (* eh (- (log (exp (/ (tan t) ew)))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan((-eh * (t / ew)));
	double tmp;
	if (ew <= -3.6e+114) {
		tmp = fabs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	} else if (ew <= 1.1e+124) {
		tmp = fabs(((t_1 * cos(t_2)) - (eh * (sin(t) * sin(t_2)))));
	} else {
		tmp = fabs((t_1 * cos(atan((eh * -log(exp((tan(t) / ew))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan((-eh * (t / ew)));
	double tmp;
	if (ew <= -3.6e+114) {
		tmp = Math.abs((t_1 * (1.0 / Math.hypot(1.0, (eh / (ew / Math.tan(t)))))));
	} else if (ew <= 1.1e+124) {
		tmp = Math.abs(((t_1 * Math.cos(t_2)) - (eh * (Math.sin(t) * Math.sin(t_2)))));
	} else {
		tmp = Math.abs((t_1 * Math.cos(Math.atan((eh * -Math.log(Math.exp((Math.tan(t) / ew))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan((-eh * (t / ew)))
	tmp = 0
	if ew <= -3.6e+114:
		tmp = math.fabs((t_1 * (1.0 / math.hypot(1.0, (eh / (ew / math.tan(t)))))))
	elif ew <= 1.1e+124:
		tmp = math.fabs(((t_1 * math.cos(t_2)) - (eh * (math.sin(t) * math.sin(t_2)))))
	else:
		tmp = math.fabs((t_1 * math.cos(math.atan((eh * -math.log(math.exp((math.tan(t) / ew))))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(-eh) * Float64(t / ew)))
	tmp = 0.0
	if (ew <= -3.6e+114)
		tmp = abs(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew / tan(t)))))));
	elseif (ew <= 1.1e+124)
		tmp = abs(Float64(Float64(t_1 * cos(t_2)) - Float64(eh * Float64(sin(t) * sin(t_2)))));
	else
		tmp = abs(Float64(t_1 * cos(atan(Float64(eh * Float64(-log(exp(Float64(tan(t) / ew)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan((-eh * (t / ew)));
	tmp = 0.0;
	if (ew <= -3.6e+114)
		tmp = abs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	elseif (ew <= 1.1e+124)
		tmp = abs(((t_1 * cos(t_2)) - (eh * (sin(t) * sin(t_2)))));
	else
		tmp = abs((t_1 * cos(atan((eh * -log(exp((tan(t) / ew))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.6e+114], N[Abs[N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.1e+124], N[Abs[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(eh * (-N[Log[N[Exp[N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.6000000000000001e114

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 99.8%

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

      1. +-inverses 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 99.8%

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

   8. Simplified 99.8%

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

      1. cos-atan 60.1%

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

      2. hypot-1-def 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

      3. add-sqr-sqrt 29.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      4. sqrt-unprod 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      5. sqr-neg 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      6. sqrt-unprod 31.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      7. add-sqr-sqrt 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      8. clear-num 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

      9. un-div-inv 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

   10. Applied egg-rr 99.8%

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

   if -3.6000000000000001e114 < ew < 1.1e124

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 98.7%

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

   6. Taylor expanded in t around 0 94.4%

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

   if 1.1e124 < ew

   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. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. sin-mult 95.9%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 95.9%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 95.9%

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

      1. +-inverses 95.9%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 95.9%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 95.9%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 95.9%

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

   8. Simplified 95.9%

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

      1. add-log-exp 95.9%

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

   10. Applied egg-rr 95.9%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \color{blue}{\log \left(e^{\frac{\tan t}{ew}}\right)}\right) - 0\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.6 \cdot 10^{+114}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 1.1 \cdot 10^{+124}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \left(-\log \left(e^{\frac{\tan t}{ew}}\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \frac{\tan t}{ew}\\ \mathbf{if}\;ew \leq -9.2 \cdot 10^{+113}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 7.5 \cdot 10^{+94}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot t\_2\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(eh \cdot \left(-\log \left(e^{t\_2}\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (/ (tan t) ew)))
   (if (<= ew -9.2e+113)
     (fabs (* t_1 (/ 1.0 (hypot 1.0 (/ eh (/ ew (tan t)))))))
     (if (<= ew 7.5e+94)
       (fabs
        (-
         (* ew (cos (atan (* (- eh) t_2))))
         (* eh (* (sin t) (sin (atan (* (- eh) (/ t ew))))))))
       (fabs (* t_1 (cos (atan (* eh (- (log (exp t_2))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = tan(t) / ew;
	double tmp;
	if (ew <= -9.2e+113) {
		tmp = fabs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	} else if (ew <= 7.5e+94) {
		tmp = fabs(((ew * cos(atan((-eh * t_2)))) - (eh * (sin(t) * sin(atan((-eh * (t / ew))))))));
	} else {
		tmp = fabs((t_1 * cos(atan((eh * -log(exp(t_2)))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.tan(t) / ew;
	double tmp;
	if (ew <= -9.2e+113) {
		tmp = Math.abs((t_1 * (1.0 / Math.hypot(1.0, (eh / (ew / Math.tan(t)))))));
	} else if (ew <= 7.5e+94) {
		tmp = Math.abs(((ew * Math.cos(Math.atan((-eh * t_2)))) - (eh * (Math.sin(t) * Math.sin(Math.atan((-eh * (t / ew))))))));
	} else {
		tmp = Math.abs((t_1 * Math.cos(Math.atan((eh * -Math.log(Math.exp(t_2)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.tan(t) / ew
	tmp = 0
	if ew <= -9.2e+113:
		tmp = math.fabs((t_1 * (1.0 / math.hypot(1.0, (eh / (ew / math.tan(t)))))))
	elif ew <= 7.5e+94:
		tmp = math.fabs(((ew * math.cos(math.atan((-eh * t_2)))) - (eh * (math.sin(t) * math.sin(math.atan((-eh * (t / ew))))))))
	else:
		tmp = math.fabs((t_1 * math.cos(math.atan((eh * -math.log(math.exp(t_2)))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(tan(t) / ew)
	tmp = 0.0
	if (ew <= -9.2e+113)
		tmp = abs(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew / tan(t)))))));
	elseif (ew <= 7.5e+94)
		tmp = abs(Float64(Float64(ew * cos(atan(Float64(Float64(-eh) * t_2)))) - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-eh) * Float64(t / ew))))))));
	else
		tmp = abs(Float64(t_1 * cos(atan(Float64(eh * Float64(-log(exp(t_2))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = tan(t) / ew;
	tmp = 0.0;
	if (ew <= -9.2e+113)
		tmp = abs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	elseif (ew <= 7.5e+94)
		tmp = abs(((ew * cos(atan((-eh * t_2)))) - (eh * (sin(t) * sin(atan((-eh * (t / ew))))))));
	else
		tmp = abs((t_1 * cos(atan((eh * -log(exp(t_2)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[ew, -9.2e+113], N[Abs[N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 7.5e+94], N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[((-eh) * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(eh * (-N[Log[N[Exp[t$95$2], $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -9.19999999999999987e113

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 99.8%

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

      1. +-inverses 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 99.8%

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

   8. Simplified 99.8%

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

      1. cos-atan 60.1%

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

      2. hypot-1-def 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

      3. add-sqr-sqrt 29.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      4. sqrt-unprod 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      5. sqr-neg 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      6. sqrt-unprod 31.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      7. add-sqr-sqrt 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      8. clear-num 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

      9. un-div-inv 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

   10. Applied egg-rr 99.8%

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

   if -9.19999999999999987e113 < ew < 7.49999999999999978e94

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 98.7%

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

   6. Taylor expanded in t around 0 91.4%

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

   if 7.49999999999999978e94 < 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 92.5%

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

      1. +-inverses 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 92.5%

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

   8. Simplified 92.5%

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

      1. add-log-exp 92.5%

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

   10. Applied egg-rr 92.5%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \color{blue}{\log \left(e^{\frac{\tan t}{ew}}\right)}\right) - 0\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9.2 \cdot 10^{+113}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 7.5 \cdot 10^{+94}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \left(-\log \left(e^{\frac{\tan t}{ew}}\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\\ \mathbf{if}\;ew \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\left|ew \cdot \cos t\_2 - eh \cdot \left(\sin t \cdot \sin t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(eh \cdot \left(-\log \left(e^{\frac{\tan t}{ew}}\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (* (- eh) (/ t ew)))))
   (if (<= ew -2e+114)
     (fabs (* t_1 (/ 1.0 (hypot 1.0 (/ eh (/ ew (tan t)))))))
     (if (<= ew 6.8e+90)
       (fabs (- (* ew (cos t_2)) (* eh (* (sin t) (sin t_2)))))
       (fabs (* t_1 (cos (atan (* eh (- (log (exp (/ (tan t) ew)))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan((-eh * (t / ew)));
	double tmp;
	if (ew <= -2e+114) {
		tmp = fabs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	} else if (ew <= 6.8e+90) {
		tmp = fabs(((ew * cos(t_2)) - (eh * (sin(t) * sin(t_2)))));
	} else {
		tmp = fabs((t_1 * cos(atan((eh * -log(exp((tan(t) / ew))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan((-eh * (t / ew)));
	double tmp;
	if (ew <= -2e+114) {
		tmp = Math.abs((t_1 * (1.0 / Math.hypot(1.0, (eh / (ew / Math.tan(t)))))));
	} else if (ew <= 6.8e+90) {
		tmp = Math.abs(((ew * Math.cos(t_2)) - (eh * (Math.sin(t) * Math.sin(t_2)))));
	} else {
		tmp = Math.abs((t_1 * Math.cos(Math.atan((eh * -Math.log(Math.exp((Math.tan(t) / ew))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan((-eh * (t / ew)))
	tmp = 0
	if ew <= -2e+114:
		tmp = math.fabs((t_1 * (1.0 / math.hypot(1.0, (eh / (ew / math.tan(t)))))))
	elif ew <= 6.8e+90:
		tmp = math.fabs(((ew * math.cos(t_2)) - (eh * (math.sin(t) * math.sin(t_2)))))
	else:
		tmp = math.fabs((t_1 * math.cos(math.atan((eh * -math.log(math.exp((math.tan(t) / ew))))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(-eh) * Float64(t / ew)))
	tmp = 0.0
	if (ew <= -2e+114)
		tmp = abs(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew / tan(t)))))));
	elseif (ew <= 6.8e+90)
		tmp = abs(Float64(Float64(ew * cos(t_2)) - Float64(eh * Float64(sin(t) * sin(t_2)))));
	else
		tmp = abs(Float64(t_1 * cos(atan(Float64(eh * Float64(-log(exp(Float64(tan(t) / ew)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan((-eh * (t / ew)));
	tmp = 0.0;
	if (ew <= -2e+114)
		tmp = abs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	elseif (ew <= 6.8e+90)
		tmp = abs(((ew * cos(t_2)) - (eh * (sin(t) * sin(t_2)))));
	else
		tmp = abs((t_1 * cos(atan((eh * -log(exp((tan(t) / ew))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2e+114], N[Abs[N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 6.8e+90], N[Abs[N[(N[(ew * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(eh * (-N[Log[N[Exp[N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2e114

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 99.8%

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

      1. +-inverses 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 99.8%

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

   8. Simplified 99.8%

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

      1. cos-atan 60.1%

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

      2. hypot-1-def 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

      3. add-sqr-sqrt 29.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      4. sqrt-unprod 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      5. sqr-neg 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      6. sqrt-unprod 31.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      7. add-sqr-sqrt 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      8. clear-num 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

      9. un-div-inv 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

   10. Applied egg-rr 99.8%

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

   if -2e114 < ew < 6.80000000000000036e90

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 98.7%

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

   6. Taylor expanded in t around 0 94.2%

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

   7. Taylor expanded in t around 0 90.7%

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

      1. mul-1-neg 90.7%

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

      2. associate-*r/ 90.7%

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

      3. distribute-rgt-neg-in 90.7%

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

      4. remove-double-neg 90.7%

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

      5. distribute-frac-neg2 90.7%

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

      6. remove-double-neg 90.7%

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

   9. Simplified 90.7%

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

   if 6.80000000000000036e90 < 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 92.5%

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

      1. +-inverses 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 92.5%

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

   8. Simplified 92.5%

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

      1. add-log-exp 92.5%

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

   10. Applied egg-rr 92.5%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \color{blue}{\log \left(e^{\frac{\tan t}{ew}}\right)}\right) - 0\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \left(-\log \left(e^{\frac{\tan t}{ew}}\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\\ \mathbf{if}\;ew \leq -8.4 \cdot 10^{+113}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;\left|ew \cdot \cos t\_2 - eh \cdot \left(\sin t \cdot \sin t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (* (- eh) (/ t ew)))))
   (if (<= ew -8.4e+113)
     (fabs (* t_1 (/ 1.0 (hypot 1.0 (/ eh (/ ew (tan t)))))))
     (if (<= ew 2.7e+91)
       (fabs (- (* ew (cos t_2)) (* eh (* (sin t) (sin t_2)))))
       (fabs (* t_1 (cos (atan (* (- eh) (/ (tan t) ew))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan((-eh * (t / ew)));
	double tmp;
	if (ew <= -8.4e+113) {
		tmp = fabs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	} else if (ew <= 2.7e+91) {
		tmp = fabs(((ew * cos(t_2)) - (eh * (sin(t) * sin(t_2)))));
	} else {
		tmp = fabs((t_1 * cos(atan((-eh * (tan(t) / ew))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan((-eh * (t / ew)));
	double tmp;
	if (ew <= -8.4e+113) {
		tmp = Math.abs((t_1 * (1.0 / Math.hypot(1.0, (eh / (ew / Math.tan(t)))))));
	} else if (ew <= 2.7e+91) {
		tmp = Math.abs(((ew * Math.cos(t_2)) - (eh * (Math.sin(t) * Math.sin(t_2)))));
	} else {
		tmp = Math.abs((t_1 * Math.cos(Math.atan((-eh * (Math.tan(t) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan((-eh * (t / ew)))
	tmp = 0
	if ew <= -8.4e+113:
		tmp = math.fabs((t_1 * (1.0 / math.hypot(1.0, (eh / (ew / math.tan(t)))))))
	elif ew <= 2.7e+91:
		tmp = math.fabs(((ew * math.cos(t_2)) - (eh * (math.sin(t) * math.sin(t_2)))))
	else:
		tmp = math.fabs((t_1 * math.cos(math.atan((-eh * (math.tan(t) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(-eh) * Float64(t / ew)))
	tmp = 0.0
	if (ew <= -8.4e+113)
		tmp = abs(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew / tan(t)))))));
	elseif (ew <= 2.7e+91)
		tmp = abs(Float64(Float64(ew * cos(t_2)) - Float64(eh * Float64(sin(t) * sin(t_2)))));
	else
		tmp = abs(Float64(t_1 * cos(atan(Float64(Float64(-eh) * Float64(tan(t) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan((-eh * (t / ew)));
	tmp = 0.0;
	if (ew <= -8.4e+113)
		tmp = abs((t_1 * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))));
	elseif (ew <= 2.7e+91)
		tmp = abs(((ew * cos(t_2)) - (eh * (sin(t) * sin(t_2)))));
	else
		tmp = abs((t_1 * cos(atan((-eh * (tan(t) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -8.4e+113], N[Abs[N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.7e+91], N[Abs[N[(N[(ew * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -8.3999999999999996e113

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 99.8%

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

      1. +-inverses 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 99.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 99.8%

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

   8. Simplified 99.8%

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

      1. cos-atan 60.1%

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

      2. hypot-1-def 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

      3. add-sqr-sqrt 29.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      4. sqrt-unprod 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      5. sqr-neg 49.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      6. sqrt-unprod 31.0%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      7. add-sqr-sqrt 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

      8. clear-num 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

      9. un-div-inv 60.1%

         \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

   10. Applied egg-rr 99.8%

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

   if -8.3999999999999996e113 < ew < 2.7e91

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 98.7%

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

   6. Taylor expanded in t around 0 94.2%

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

   7. Taylor expanded in t around 0 90.7%

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

      1. mul-1-neg 90.7%

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

      2. associate-*r/ 90.7%

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

      3. distribute-rgt-neg-in 90.7%

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

      4. remove-double-neg 90.7%

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

      5. distribute-frac-neg2 90.7%

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

      6. remove-double-neg 90.7%

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

   9. Simplified 90.7%

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

   if 2.7e91 < 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

      2. associate-*r/ 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

   6. Applied egg-rr 92.5%

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

      1. +-inverses 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 92.5%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

      4. mul0-lft 92.5%

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

   8. Simplified 92.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -8.4 \cdot 10^{+113}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \mathbf{elif}\;ew \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Final simplification 62.1%

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

Alternative 9: 61.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

   1. cos-atan 44.0%

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

   2. hypot-1-def 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

   3. add-sqr-sqrt 19.0%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   4. sqrt-unprod 38.5%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   5. sqr-neg 38.5%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   6. sqrt-unprod 25.0%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   7. add-sqr-sqrt 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

   8. clear-num 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

   9. un-div-inv 44.1%

      \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

10. Applied egg-rr 61.8%

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

11. Final simplification 61.8%

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

Alternative 10: 52.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in t around 0 54.0%

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

10. Final simplification 54.0%

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

Alternative 11: 42.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in t around 0 44.4%

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

    1. add-sqr-sqrt 48.4%

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

    2. sqrt-unprod 93.4%

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

    3. sqr-neg 93.4%

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

    4. sqrt-unprod 51.5%

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

    5. add-sqr-sqrt 99.8%

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

    6. associate-*r/ 99.8%

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

11. Applied egg-rr 44.4%

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

12. Final simplification 44.4%

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

Alternative 12: 42.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in t around 0 44.4%

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

    1. cos-atan 44.0%

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

    2. hypot-1-def 44.1%

       \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

    3. add-sqr-sqrt 19.0%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    4. sqrt-unprod 38.5%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    5. sqr-neg 38.5%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    6. sqrt-unprod 25.0%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    7. add-sqr-sqrt 44.1%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    8. clear-num 44.1%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \color{blue}{\frac{1}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

    9. un-div-inv 44.1%

       \[\leadsto \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{\frac{ew}{\tan t}}}\right)} - 0\right| \]

11. Applied egg-rr 44.1%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}} - 0\right| \]

12. Final simplification 44.1%

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

Alternative 13: 42.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in t around 0 44.4%

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

    1. cos-atan 44.0%

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

    2. un-div-inv 44.0%

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

    3. hypot-1-def 44.1%

       \[\leadsto \left|\frac{ew}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{\tan t}{ew}\right)}} - 0\right| \]

    4. add-sqr-sqrt 19.0%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    5. sqrt-unprod 38.5%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    6. sqr-neg 38.5%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    7. sqrt-unprod 25.0%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    8. add-sqr-sqrt 44.1%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{\tan t}{ew}\right)} - 0\right| \]

    9. associate-*r/ 44.1%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh \cdot \tan t}{ew}}\right)} - 0\right| \]

    10. associate-*l/ 44.1%

        \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{ew} \cdot \tan t}\right)} - 0\right| \]

    11. *-commutative 44.1%

        \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\tan t \cdot \frac{eh}{ew}}\right)} - 0\right| \]

11. Applied egg-rr 44.1%

    \[\leadsto \left|\color{blue}{\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}} - 0\right| \]

12. Final simplification 44.1%

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

Alternative 14: 41.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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(atan((eh / (ew / t))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in t around 0 44.4%

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

10. Taylor expanded in t around 0 43.4%

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

    1. clear-num 43.4%

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

    2. un-div-inv 43.4%

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

    3. add-sqr-sqrt 18.7%

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

    4. sqrt-unprod 38.2%

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

    5. sqr-neg 38.2%

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

    6. sqrt-unprod 24.7%

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

    7. add-sqr-sqrt 43.4%

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

12. Applied egg-rr 43.4%

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

13. Final simplification 43.4%

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

Alternative 15: 40.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. sin-mult 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}{2}}\right| \]

   2. associate-*r/ 64.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}{2}}\right| \]

6. Applied egg-rr 62.1%

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

   1. +-inverses 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 62.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) - \color{blue}{0 \cdot \frac{eh}{2}}\right| \]

   4. mul0-lft 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in t around 0 44.4%

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

10. Taylor expanded in t around 0 43.4%

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

    1. cos-atan 42.6%

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

    2. un-div-inv 42.6%

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

    3. hypot-1-def 42.6%

       \[\leadsto \left|\frac{ew}{\color{blue}{\mathsf{hypot}\left(1, \left(-eh\right) \cdot \frac{t}{ew}\right)}} - 0\right| \]

    4. add-sqr-sqrt 18.3%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \frac{t}{ew}\right)} - 0\right| \]

    5. sqrt-unprod 37.2%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \frac{t}{ew}\right)} - 0\right| \]

    6. sqr-neg 37.2%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{eh \cdot eh}} \cdot \frac{t}{ew}\right)} - 0\right| \]

    7. sqrt-unprod 24.3%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \frac{t}{ew}\right)} - 0\right| \]

    8. add-sqr-sqrt 42.6%

       \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, \color{blue}{eh} \cdot \frac{t}{ew}\right)} - 0\right| \]

12. Applied egg-rr 42.6%

    \[\leadsto \left|\color{blue}{\frac{ew}{\mathsf{hypot}\left(1, eh \cdot \frac{t}{ew}\right)}} - 0\right| \]

13. Final simplification 42.6%

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

Reproduce

herbie shell --seed 2024100 
(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)))))))