Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 26.5s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] / N[(ew / eh), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(ew / N[Tan[t], $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. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. clear-num 99.8%

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

   5. add-sqr-sqrt 52.6%

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

   6. sqrt-unprod 94.0%

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

   7. sqr-neg 94.0%

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

   8. sqrt-unprod 47.2%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] / N[(ew / eh), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(ew / N[Tan[t], $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. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 74.3%

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

   2. expm1-udef 58.1%

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

5. Applied egg-rr 59.7%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 75.8%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(ew / N[Tan[t], $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. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. clear-num 99.8%

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

   5. add-sqr-sqrt 52.6%

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

   6. sqrt-unprod 94.0%

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

   7. sqr-neg 94.0%

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

   8. sqrt-unprod 47.2%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ t_2 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 18500000:\\ \;\;\;\;\left|\mathsf{fma}\left(t, eh, ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))) (t_2 (fabs (* (cos t) ew))))
   (if (<= t -1.8e+179)
     t_1
     (if (<= t -1.66e-7)
       t_2
       (if (<= t 18500000.0)
         (fabs (fma t eh (* ew (cos (atan (* (tan t) (/ (- eh) ew)))))))
         (if (<= t 9.8e+69) t_2 t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double t_2 = fabs((cos(t) * ew));
	double tmp;
	if (t <= -1.8e+179) {
		tmp = t_1;
	} else if (t <= -1.66e-7) {
		tmp = t_2;
	} else if (t <= 18500000.0) {
		tmp = fabs(fma(t, eh, (ew * cos(atan((tan(t) * (-eh / ew)))))));
	} else if (t <= 9.8e+69) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	t_2 = abs(Float64(cos(t) * ew))
	tmp = 0.0
	if (t <= -1.8e+179)
		tmp = t_1;
	elseif (t <= -1.66e-7)
		tmp = t_2;
	elseif (t <= 18500000.0)
		tmp = abs(fma(t, eh, Float64(ew * cos(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))))));
	elseif (t <= 9.8e+69)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.8e+179], t$95$1, If[LessEqual[t, -1.66e-7], t$95$2, If[LessEqual[t, 18500000.0], N[Abs[N[(t * eh + N[(ew * N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 9.8e+69], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.7999999999999999e179 or 9.7999999999999999e69 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 65.7%

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

      3. associate-*l/ 65.6%

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

      4. add-sqr-sqrt 35.1%

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

      5. sqrt-unprod 65.6%

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

      6. sqr-neg 65.6%

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

      7. sqrt-unprod 30.3%

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

      8. add-sqr-sqrt 64.2%

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

      9. *-commutative 64.2%

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

      10. associate-/r/ 64.1%

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

      11. div-inv 64.1%

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

      12. clear-num 64.1%

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

      13. hypot-1-def 84.8%

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

      14. *-commutative 84.8%

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

   5. Applied egg-rr 85.0%

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

   6. Taylor expanded in eh around -inf 97.5%

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

      1. mul-1-neg 97.5%

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

   8. Simplified 97.5%

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

   10. Applied egg-rr 46.1%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in ew around 0 64.0%

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

   if -1.7999999999999999e179 < t < -1.66000000000000004e-7 or 1.85e7 < t < 9.7999999999999999e69

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 83.7%

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

      3. associate-*l/ 83.7%

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

      4. add-sqr-sqrt 55.0%

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

      5. sqrt-unprod 83.7%

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

      6. sqr-neg 83.7%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 81.8%

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

      9. *-commutative 81.8%

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

      10. associate-/r/ 81.7%

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

      11. div-inv 81.7%

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

      12. clear-num 81.8%

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

      13. hypot-1-def 89.5%

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

      14. *-commutative 89.5%

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

   5. Applied egg-rr 89.5%

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

   6. Taylor expanded in eh around -inf 96.3%

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

      1. mul-1-neg 96.3%

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

   8. Simplified 96.3%

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

   10. Applied egg-rr 52.0%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in eh around 0 70.8%

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

   if -1.66000000000000004e-7 < t < 1.85e7

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. sin-atan 75.2%

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

      3. associate-*l/ 75.2%

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

      4. add-sqr-sqrt 33.9%

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

      5. sqrt-unprod 73.6%

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

      6. sqr-neg 73.6%

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

      7. sqrt-unprod 41.0%

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

      8. add-sqr-sqrt 74.9%

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

      9. *-commutative 74.9%

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

      10. associate-/r/ 75.4%

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

      11. div-inv 75.4%

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

      12. clear-num 75.4%

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

      13. hypot-1-def 80.1%

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

      14. *-commutative 80.1%

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

   5. Applied egg-rr 93.3%

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

   6. Taylor expanded in eh around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in t around 0 96.7%

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

       1. distribute-lft-out 96.7%

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

       2. mul-1-neg 96.7%

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

       3. *-commutative 96.7%

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

       4. fma-def 96.7%

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

       5. mul-1-neg 96.7%

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

       6. associate-*l/ 96.7%

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

       7. *-commutative 96.7%

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

       8. distribute-rgt-neg-in 96.7%

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

       9. distribute-frac-neg 96.7%

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

   11. Simplified 96.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+179}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;t \leq 18500000:\\ \;\;\;\;\left|\mathsf{fma}\left(t, eh, ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+69}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Cos[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $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. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 73.7%

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

   3. associate-*l/ 73.7%

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

   4. add-sqr-sqrt 38.3%

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

   5. sqrt-unprod 72.9%

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

   6. sqr-neg 72.9%

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

   7. sqrt-unprod 35.2%

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

   8. add-sqr-sqrt 72.8%

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

   9. *-commutative 72.8%

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

   10. associate-/r/ 72.9%

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

   11. div-inv 72.9%

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

   12. clear-num 73.0%

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

   13. hypot-1-def 83.4%

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

   14. *-commutative 83.4%

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

5. Applied egg-rr 89.9%

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

6. Taylor expanded in eh around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

8. Simplified 98.2%

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

   1. fma-udef 98.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

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

Alternative 6: 73.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ t_2 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 18500000:\\ \;\;\;\;\left|t \cdot eh + ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))) (t_2 (fabs (* (cos t) ew))))
   (if (<= t -6.5e+178)
     t_1
     (if (<= t -1.66e-7)
       t_2
       (if (<= t 18500000.0)
         (fabs (+ (* t eh) (* ew (cos (atan (* (tan t) (/ (- eh) ew)))))))
         (if (<= t 7.1e+68) t_2 t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double t_2 = fabs((cos(t) * ew));
	double tmp;
	if (t <= -6.5e+178) {
		tmp = t_1;
	} else if (t <= -1.66e-7) {
		tmp = t_2;
	} else if (t <= 18500000.0) {
		tmp = fabs(((t * eh) + (ew * cos(atan((tan(t) * (-eh / ew)))))));
	} else if (t <= 7.1e+68) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    t_2 = abs((cos(t) * ew))
    if (t <= (-6.5d+178)) then
        tmp = t_1
    else if (t <= (-1.66d-7)) then
        tmp = t_2
    else if (t <= 18500000.0d0) then
        tmp = abs(((t * eh) + (ew * cos(atan((tan(t) * (-eh / ew)))))))
    else if (t <= 7.1d+68) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double t_2 = Math.abs((Math.cos(t) * ew));
	double tmp;
	if (t <= -6.5e+178) {
		tmp = t_1;
	} else if (t <= -1.66e-7) {
		tmp = t_2;
	} else if (t <= 18500000.0) {
		tmp = Math.abs(((t * eh) + (ew * Math.cos(Math.atan((Math.tan(t) * (-eh / ew)))))));
	} else if (t <= 7.1e+68) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	t_2 = math.fabs((math.cos(t) * ew))
	tmp = 0
	if t <= -6.5e+178:
		tmp = t_1
	elif t <= -1.66e-7:
		tmp = t_2
	elif t <= 18500000.0:
		tmp = math.fabs(((t * eh) + (ew * math.cos(math.atan((math.tan(t) * (-eh / ew)))))))
	elif t <= 7.1e+68:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	t_2 = abs(Float64(cos(t) * ew))
	tmp = 0.0
	if (t <= -6.5e+178)
		tmp = t_1;
	elseif (t <= -1.66e-7)
		tmp = t_2;
	elseif (t <= 18500000.0)
		tmp = abs(Float64(Float64(t * eh) + Float64(ew * cos(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))))));
	elseif (t <= 7.1e+68)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	t_2 = abs((cos(t) * ew));
	tmp = 0.0;
	if (t <= -6.5e+178)
		tmp = t_1;
	elseif (t <= -1.66e-7)
		tmp = t_2;
	elseif (t <= 18500000.0)
		tmp = abs(((t * eh) + (ew * cos(atan((tan(t) * (-eh / ew)))))));
	elseif (t <= 7.1e+68)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -6.5e+178], t$95$1, If[LessEqual[t, -1.66e-7], t$95$2, If[LessEqual[t, 18500000.0], N[Abs[N[(N[(t * eh), $MachinePrecision] + N[(ew * N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 7.1e+68], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5000000000000005e178 or 7.1000000000000001e68 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 65.7%

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

      3. associate-*l/ 65.6%

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

      4. add-sqr-sqrt 35.1%

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

      5. sqrt-unprod 65.6%

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

      6. sqr-neg 65.6%

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

      7. sqrt-unprod 30.3%

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

      8. add-sqr-sqrt 64.2%

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

      9. *-commutative 64.2%

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

      10. associate-/r/ 64.1%

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

      11. div-inv 64.1%

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

      12. clear-num 64.1%

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

      13. hypot-1-def 84.8%

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

      14. *-commutative 84.8%

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

   5. Applied egg-rr 85.0%

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

   6. Taylor expanded in eh around -inf 97.5%

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

      1. mul-1-neg 97.5%

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

   8. Simplified 97.5%

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

   10. Applied egg-rr 46.1%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in ew around 0 64.0%

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

   if -6.5000000000000005e178 < t < -1.66000000000000004e-7 or 1.85e7 < t < 7.1000000000000001e68

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 83.7%

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

      3. associate-*l/ 83.7%

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

      4. add-sqr-sqrt 55.0%

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

      5. sqrt-unprod 83.7%

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

      6. sqr-neg 83.7%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 81.8%

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

      9. *-commutative 81.8%

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

      10. associate-/r/ 81.7%

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

      11. div-inv 81.7%

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

      12. clear-num 81.8%

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

      13. hypot-1-def 89.5%

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

      14. *-commutative 89.5%

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

   5. Applied egg-rr 89.5%

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

   6. Taylor expanded in eh around -inf 96.3%

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

      1. mul-1-neg 96.3%

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

   8. Simplified 96.3%

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

   10. Applied egg-rr 52.0%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in eh around 0 70.8%

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

   if -1.66000000000000004e-7 < t < 1.85e7

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. sin-atan 75.2%

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

      3. associate-*l/ 75.2%

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

      4. add-sqr-sqrt 33.9%

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

      5. sqrt-unprod 73.6%

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

      6. sqr-neg 73.6%

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

      7. sqrt-unprod 41.0%

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

      8. add-sqr-sqrt 74.9%

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

      9. *-commutative 74.9%

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

      10. associate-/r/ 75.4%

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

      11. div-inv 75.4%

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

      12. clear-num 75.4%

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

      13. hypot-1-def 80.1%

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

      14. *-commutative 80.1%

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

   5. Applied egg-rr 93.3%

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

   6. Taylor expanded in eh around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in t around 0 96.7%

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

       1. distribute-lft-out 96.7%

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

       2. *-commutative 96.7%

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

       3. mul-1-neg 96.7%

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

       4. associate-*l/ 96.7%

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

       5. distribute-rgt-neg-in 96.7%

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

   11. Simplified 96.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+178}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;t \leq 18500000:\\ \;\;\;\;\left|t \cdot eh + ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{+68}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]

Alternative 7: 73.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ t_2 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 18500000:\\ \;\;\;\;\left|ew + t \cdot eh\right|\\ \mathbf{elif}\;t \leq 1.46 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))) (t_2 (fabs (* (cos t) ew))))
   (if (<= t -1.02e+180)
     t_1
     (if (<= t -1.66e-7)
       t_2
       (if (<= t 18500000.0)
         (fabs (+ ew (* t eh)))
         (if (<= t 1.46e+72) t_2 t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double t_2 = fabs((cos(t) * ew));
	double tmp;
	if (t <= -1.02e+180) {
		tmp = t_1;
	} else if (t <= -1.66e-7) {
		tmp = t_2;
	} else if (t <= 18500000.0) {
		tmp = fabs((ew + (t * eh)));
	} else if (t <= 1.46e+72) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    t_2 = abs((cos(t) * ew))
    if (t <= (-1.02d+180)) then
        tmp = t_1
    else if (t <= (-1.66d-7)) then
        tmp = t_2
    else if (t <= 18500000.0d0) then
        tmp = abs((ew + (t * eh)))
    else if (t <= 1.46d+72) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double t_2 = Math.abs((Math.cos(t) * ew));
	double tmp;
	if (t <= -1.02e+180) {
		tmp = t_1;
	} else if (t <= -1.66e-7) {
		tmp = t_2;
	} else if (t <= 18500000.0) {
		tmp = Math.abs((ew + (t * eh)));
	} else if (t <= 1.46e+72) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	t_2 = math.fabs((math.cos(t) * ew))
	tmp = 0
	if t <= -1.02e+180:
		tmp = t_1
	elif t <= -1.66e-7:
		tmp = t_2
	elif t <= 18500000.0:
		tmp = math.fabs((ew + (t * eh)))
	elif t <= 1.46e+72:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	t_2 = abs(Float64(cos(t) * ew))
	tmp = 0.0
	if (t <= -1.02e+180)
		tmp = t_1;
	elseif (t <= -1.66e-7)
		tmp = t_2;
	elseif (t <= 18500000.0)
		tmp = abs(Float64(ew + Float64(t * eh)));
	elseif (t <= 1.46e+72)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	t_2 = abs((cos(t) * ew));
	tmp = 0.0;
	if (t <= -1.02e+180)
		tmp = t_1;
	elseif (t <= -1.66e-7)
		tmp = t_2;
	elseif (t <= 18500000.0)
		tmp = abs((ew + (t * eh)));
	elseif (t <= 1.46e+72)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.02e+180], t$95$1, If[LessEqual[t, -1.66e-7], t$95$2, If[LessEqual[t, 18500000.0], N[Abs[N[(ew + N[(t * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.46e+72], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.02e180 or 1.45999999999999999e72 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 65.7%

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

      3. associate-*l/ 65.6%

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

      4. add-sqr-sqrt 35.1%

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

      5. sqrt-unprod 65.6%

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

      6. sqr-neg 65.6%

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

      7. sqrt-unprod 30.3%

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

      8. add-sqr-sqrt 64.2%

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

      9. *-commutative 64.2%

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

      10. associate-/r/ 64.1%

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

      11. div-inv 64.1%

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

      12. clear-num 64.1%

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

      13. hypot-1-def 84.8%

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

      14. *-commutative 84.8%

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

   5. Applied egg-rr 85.0%

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

   6. Taylor expanded in eh around -inf 97.5%

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

      1. mul-1-neg 97.5%

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

   8. Simplified 97.5%

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

   10. Applied egg-rr 46.1%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in ew around 0 64.0%

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

   if -1.02e180 < t < -1.66000000000000004e-7 or 1.85e7 < t < 1.45999999999999999e72

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 83.7%

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

      3. associate-*l/ 83.7%

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

      4. add-sqr-sqrt 55.0%

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

      5. sqrt-unprod 83.7%

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

      6. sqr-neg 83.7%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 81.8%

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

      9. *-commutative 81.8%

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

      10. associate-/r/ 81.7%

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

      11. div-inv 81.7%

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

      12. clear-num 81.8%

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

      13. hypot-1-def 89.5%

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

      14. *-commutative 89.5%

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

   5. Applied egg-rr 89.5%

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

   6. Taylor expanded in eh around -inf 96.3%

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

      1. mul-1-neg 96.3%

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

   8. Simplified 96.3%

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

   10. Applied egg-rr 52.0%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in eh around 0 70.8%

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

   if -1.66000000000000004e-7 < t < 1.85e7

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. sin-atan 75.2%

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

      3. associate-*l/ 75.2%

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

      4. add-sqr-sqrt 33.9%

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

      5. sqrt-unprod 73.6%

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

      6. sqr-neg 73.6%

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

      7. sqrt-unprod 41.0%

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

      8. add-sqr-sqrt 74.9%

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

      9. *-commutative 74.9%

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

      10. associate-/r/ 75.4%

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

      11. div-inv 75.4%

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

      12. clear-num 75.4%

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

      13. hypot-1-def 80.1%

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

      14. *-commutative 80.1%

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

   5. Applied egg-rr 93.3%

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

   6. Taylor expanded in eh around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

   8. Simplified 99.5%

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

   10. Applied egg-rr 47.5%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in t around 0 96.4%

       \[\leadsto \left|\color{blue}{ew + eh \cdot t}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+180}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;t \leq 18500000:\\ \;\;\;\;\left|ew + t \cdot eh\right|\\ \mathbf{elif}\;t \leq 1.46 \cdot 10^{+72}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]

Alternative 8: 74.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.45 \lor \neg \left(t \leq 8.2 \cdot 10^{-13}\right):\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.45) (not (<= t 8.2e-13)))
   (fabs (* eh (sin t)))
   (fabs (+ ew (* t eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.45) || !(t <= 8.2e-13)) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = fabs((ew + (t * eh)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.45d0)) .or. (.not. (t <= 8.2d-13))) then
        tmp = abs((eh * sin(t)))
    else
        tmp = abs((ew + (t * eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.45) || !(t <= 8.2e-13)) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = Math.abs((ew + (t * eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.45) or not (t <= 8.2e-13):
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = math.fabs((ew + (t * eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.45) || !(t <= 8.2e-13))
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = abs(Float64(ew + Float64(t * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.45) || ~((t <= 8.2e-13)))
		tmp = abs((eh * sin(t)));
	else
		tmp = abs((ew + (t * eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.45], N[Not[LessEqual[t, 8.2e-13]], $MachinePrecision]], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(t * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.450000000000000011 or 8.2000000000000004e-13 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. sin-atan 69.6%

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

      3. associate-*l/ 69.5%

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

      4. add-sqr-sqrt 39.8%

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

      5. sqrt-unprod 69.5%

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

      6. sqr-neg 69.5%

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

      7. sqrt-unprod 29.6%

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

      8. add-sqr-sqrt 68.0%

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

      9. *-commutative 68.0%

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

      10. associate-/r/ 67.9%

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

      11. div-inv 67.9%

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

      12. clear-num 67.9%

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

      13. hypot-1-def 85.5%

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

      14. *-commutative 85.5%

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

   5. Applied egg-rr 85.6%

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

   6. Taylor expanded in eh around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

   8. Simplified 97.1%

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

   10. Applied egg-rr 49.1%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in ew around 0 55.0%

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

   if -0.450000000000000011 < t < 8.2000000000000004e-13

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. sin-atan 78.4%

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

      3. associate-*l/ 78.4%

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

      4. add-sqr-sqrt 36.7%

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

      5. sqrt-unprod 76.8%

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

      6. sqr-neg 76.8%

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

      7. sqrt-unprod 41.5%

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

      8. add-sqr-sqrt 78.2%

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

      9. *-commutative 78.2%

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

      10. associate-/r/ 78.7%

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

      11. div-inv 78.7%

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

      12. clear-num 78.7%

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

      13. hypot-1-def 81.1%

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

      14. *-commutative 81.1%

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

   5. Applied egg-rr 94.7%

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

   6. Taylor expanded in eh around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

   8. Simplified 99.5%

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

   10. Applied egg-rr 46.6%

       \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

   11. Taylor expanded in t around 0 98.5%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.45 \lor \neg \left(t \leq 8.2 \cdot 10^{-13}\right):\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot eh\right|\\ \end{array} \]

Alternative 9: 53.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \left|ew + t \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (+ ew (* t eh))))

C

double code(double eh, double ew, double t) {
	return fabs((ew + (t * eh)));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew + (t * eh)))
end function

Java

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

Python

def code(eh, ew, t):
	return math.fabs((ew + (t * eh)))

Julia

function code(eh, ew, t)
	return abs(Float64(ew + Float64(t * eh)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew + (t * eh)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew + N[(t * eh), $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. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 73.7%

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

   3. associate-*l/ 73.7%

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

   4. add-sqr-sqrt 38.3%

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

   5. sqrt-unprod 72.9%

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

   6. sqr-neg 72.9%

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

   7. sqrt-unprod 35.2%

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

   8. add-sqr-sqrt 72.8%

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

   9. *-commutative 72.8%

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

   10. associate-/r/ 72.9%

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

   11. div-inv 72.9%

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

   12. clear-num 73.0%

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

   13. hypot-1-def 83.4%

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

   14. *-commutative 83.4%

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

5. Applied egg-rr 89.9%

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

6. Taylor expanded in eh around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

8. Simplified 98.2%

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

10. Applied egg-rr 47.9%

    \[\leadsto \left|\color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}, eh \cdot \sin t\right)}\right)}^{2}}\right| \]

11. Taylor expanded in t around 0 52.8%

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

12. Final simplification 52.8%

    \[\leadsto \left|ew + t \cdot eh\right| \]

Alternative 10: 42.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|ew\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs ew))

C

double code(double eh, double ew, double t) {
	return fabs(ew);
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(ew)
end function

Java

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

Python

def code(eh, ew, t):
	return math.fabs(ew)

Julia

function code(eh, ew, t)
	return abs(ew)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(ew);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 57.8%

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

   1. +-inverses 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-/l* 57.8%

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

   4. div0 57.8%

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

7. Simplified 57.8%

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

8. Taylor expanded in t around 0 38.6%

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

   1. associate-*r* 38.6%

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

   2. neg-mul-1 38.6%

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

   3. mul-1-neg 38.6%

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

   4. associate-*l/ 38.6%

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

   5. *-commutative 38.6%

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

   6. distribute-rgt-neg-in 38.6%

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

10. Simplified 38.6%

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

    1. expm1-log1p-u 25.7%

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

    2. expm1-udef 15.0%

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

12. Applied egg-rr 12.8%

    \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right)} - 1}\right| \]
13. Step-by-step derivation

    1. expm1-def 23.3%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right)\right)}\right| \]

    2. expm1-log1p 38.3%

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

    3. associate-*l/ 38.3%

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

    4. associate-*r/ 38.3%

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

14. Simplified 38.3%

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

15. Taylor expanded in ew around inf 38.8%

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

16. Final simplification 38.8%

    \[\leadsto \left|ew\right| \]

Reproduce

herbie shell --seed 2023279 
(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)))))))