Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.9s

Alternatives: 13

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \left|\mathsf{fma}\left(ew, \cos t\_1 \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin t\_1\right)\right)\right| \end{array} \end{array} \]

FPCore

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

Julia

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

Wolfram

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

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

   4. add-sqr-sqrt 47.2%

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

   5. sqrt-unprod 93.5%

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

   6. sqr-neg 93.5%

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

   7. sqrt-unprod 52.6%

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

   8. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

   4. add-sqr-sqrt 47.2%

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

   5. sqrt-unprod 93.5%

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

   6. sqr-neg 93.5%

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

   7. sqrt-unprod 52.6%

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

   8. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in eh around 0 98.0%

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

    1. mul-1-neg 98.0%

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

    2. +-commutative 98.0%

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

    3. unsub-neg 98.0%

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

    4. associate-*r* 98.0%

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

    5. mul-1-neg 98.0%

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

    6. distribute-frac-neg2 98.0%

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

11. Simplified 98.0%

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

Alternative 4: 78.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;ew \leq -2 \cdot 10^{-94}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 2.1 \cdot 10^{-215}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;t\_1 + eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (fabs t_1)))
   (if (<= ew -2e-94)
     t_2
     (if (<= ew 2.1e-215)
       (fabs (* (* eh (sin t)) (sin (atan (* eh (/ (tan t) (- ew)))))))
       (if (<= ew 1.55e+63)
         (+ t_1 (* eh (* (sin t) (sin (atan (/ (* eh (tan t)) ew))))))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = fabs(t_1);
	double tmp;
	if (ew <= -2e-94) {
		tmp = t_2;
	} else if (ew <= 2.1e-215) {
		tmp = fabs(((eh * sin(t)) * sin(atan((eh * (tan(t) / -ew))))));
	} else if (ew <= 1.55e+63) {
		tmp = t_1 + (eh * (sin(t) * sin(atan(((eh * tan(t)) / ew)))));
	} else {
		tmp = t_2;
	}
	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 = ew * cos(t)
    t_2 = abs(t_1)
    if (ew <= (-2d-94)) then
        tmp = t_2
    else if (ew <= 2.1d-215) then
        tmp = abs(((eh * sin(t)) * sin(atan((eh * (tan(t) / -ew))))))
    else if (ew <= 1.55d+63) then
        tmp = t_1 + (eh * (sin(t) * sin(atan(((eh * tan(t)) / ew)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.abs(t_1);
	double tmp;
	if (ew <= -2e-94) {
		tmp = t_2;
	} else if (ew <= 2.1e-215) {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((eh * (Math.tan(t) / -ew))))));
	} else if (ew <= 1.55e+63) {
		tmp = t_1 + (eh * (Math.sin(t) * Math.sin(Math.atan(((eh * Math.tan(t)) / ew)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.fabs(t_1)
	tmp = 0
	if ew <= -2e-94:
		tmp = t_2
	elif ew <= 2.1e-215:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((eh * (math.tan(t) / -ew))))))
	elif ew <= 1.55e+63:
		tmp = t_1 + (eh * (math.sin(t) * math.sin(math.atan(((eh * math.tan(t)) / ew)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = abs(t_1)
	tmp = 0.0
	if (ew <= -2e-94)
		tmp = t_2;
	elseif (ew <= 2.1e-215)
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	elseif (ew <= 1.55e+63)
		tmp = Float64(t_1 + Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(eh * tan(t)) / ew))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = abs(t_1);
	tmp = 0.0;
	if (ew <= -2e-94)
		tmp = t_2;
	elseif (ew <= 2.1e-215)
		tmp = abs(((eh * sin(t)) * sin(atan((eh * (tan(t) / -ew))))));
	elseif (ew <= 1.55e+63)
		tmp = t_1 + (eh * (sin(t) * sin(atan(((eh * tan(t)) / ew)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[ew, -2e-94], t$95$2, If[LessEqual[ew, 2.1e-215], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.55e+63], N[(t$95$1 + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.9999999999999999e-94 or 1.55e63 < ew

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. add-sqr-sqrt 45.8%

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

      5. sqrt-unprod 90.1%

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

      6. sqr-neg 90.1%

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

      7. sqrt-unprod 54.0%

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

      8. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in ew around inf 87.6%

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

   if -1.9999999999999999e-94 < ew < 2.1e-215

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ew around 0 80.1%

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

      1. associate-*r* 80.1%

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

      2. mul-1-neg 80.1%

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

      3. distribute-frac-neg2 80.1%

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

      4. associate-/l* 80.1%

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

   7. Simplified 80.1%

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

   if 2.1e-215 < ew < 1.55e63

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-out 99.7%

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

      6. distribute-rgt-neg-out 99.7%

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

   6. Applied egg-rr 83.5%

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

   7. Taylor expanded in eh around 0 81.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 2.1 \cdot 10^{-215}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.5 \cdot 10^{-98} \lor \neg \left(ew \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -4.5e-98) (not (<= ew 3e-60)))
   (fabs (* ew (cos t)))
   (fabs (* (* eh (sin t)) (sin (atan (* eh (/ (tan t) (- ew)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -4.5e-98) || !(ew <= 3e-60)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((eh * sin(t)) * sin(atan((eh * (tan(t) / -ew))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-4.5d-98)) .or. (.not. (ew <= 3d-60))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs(((eh * sin(t)) * sin(atan((eh * (tan(t) / -ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -4.5e-98) || !(ew <= 3e-60)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((eh * (Math.tan(t) / -ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -4.5e-98) or not (ew <= 3e-60):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((eh * (math.tan(t) / -ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -4.5e-98) || !(ew <= 3e-60))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -4.5e-98) || ~((ew <= 3e-60)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((eh * sin(t)) * sin(atan((eh * (tan(t) / -ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -4.5e-98], N[Not[LessEqual[ew, 3e-60]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.49999999999999997e-98 or 3.00000000000000019e-60 < ew

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. add-sqr-sqrt 46.2%

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

      5. sqrt-unprod 91.2%

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

      6. sqr-neg 91.2%

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

      7. sqrt-unprod 53.6%

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

      8. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in ew around inf 84.8%

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

   if -4.49999999999999997e-98 < ew < 3.00000000000000019e-60

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ew around 0 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. distribute-frac-neg2 74.2%

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

      4. associate-/l* 74.2%

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

   7. Simplified 74.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.5 \cdot 10^{-98} \lor \neg \left(ew \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ t_2 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -3 \cdot 10^{-142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq -5.2 \cdot 10^{-256}:\\ \;\;\;\;eh \cdot \left(t\_1 \cdot \left(-\sin t\right)\right)\\ \mathbf{elif}\;ew \leq 3.7 \cdot 10^{-257}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot t\_1\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;ew + \left(eh \cdot t\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (* eh (/ (tan t) ew))))) (t_2 (fabs (* ew (cos t)))))
   (if (<= ew -3e-142)
     t_2
     (if (<= ew -5.2e-256)
       (* eh (* t_1 (- (sin t))))
       (if (<= ew 3.7e-257)
         (* (* eh (sin t)) t_1)
         (if (<= ew 8.5e+38) (+ ew (* (* eh t) t_1)) t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan((eh * (tan(t) / ew))));
	double t_2 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -3e-142) {
		tmp = t_2;
	} else if (ew <= -5.2e-256) {
		tmp = eh * (t_1 * -sin(t));
	} else if (ew <= 3.7e-257) {
		tmp = (eh * sin(t)) * t_1;
	} else if (ew <= 8.5e+38) {
		tmp = ew + ((eh * t) * t_1);
	} else {
		tmp = t_2;
	}
	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 = sin(atan((eh * (tan(t) / ew))))
    t_2 = abs((ew * cos(t)))
    if (ew <= (-3d-142)) then
        tmp = t_2
    else if (ew <= (-5.2d-256)) then
        tmp = eh * (t_1 * -sin(t))
    else if (ew <= 3.7d-257) then
        tmp = (eh * sin(t)) * t_1
    else if (ew <= 8.5d+38) then
        tmp = ew + ((eh * t) * t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan((eh * (Math.tan(t) / ew))));
	double t_2 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -3e-142) {
		tmp = t_2;
	} else if (ew <= -5.2e-256) {
		tmp = eh * (t_1 * -Math.sin(t));
	} else if (ew <= 3.7e-257) {
		tmp = (eh * Math.sin(t)) * t_1;
	} else if (ew <= 8.5e+38) {
		tmp = ew + ((eh * t) * t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan((eh * (math.tan(t) / ew))))
	t_2 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -3e-142:
		tmp = t_2
	elif ew <= -5.2e-256:
		tmp = eh * (t_1 * -math.sin(t))
	elif ew <= 3.7e-257:
		tmp = (eh * math.sin(t)) * t_1
	elif ew <= 8.5e+38:
		tmp = ew + ((eh * t) * t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(eh * Float64(tan(t) / ew))))
	t_2 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -3e-142)
		tmp = t_2;
	elseif (ew <= -5.2e-256)
		tmp = Float64(eh * Float64(t_1 * Float64(-sin(t))));
	elseif (ew <= 3.7e-257)
		tmp = Float64(Float64(eh * sin(t)) * t_1);
	elseif (ew <= 8.5e+38)
		tmp = Float64(ew + Float64(Float64(eh * t) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan((eh * (tan(t) / ew))));
	t_2 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -3e-142)
		tmp = t_2;
	elseif (ew <= -5.2e-256)
		tmp = eh * (t_1 * -sin(t));
	elseif (ew <= 3.7e-257)
		tmp = (eh * sin(t)) * t_1;
	elseif (ew <= 8.5e+38)
		tmp = ew + ((eh * t) * t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3e-142], t$95$2, If[LessEqual[ew, -5.2e-256], N[(eh * N[(t$95$1 * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 3.7e-257], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[ew, 8.5e+38], N[(ew + N[(N[(eh * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if ew < -3.0000000000000001e-142 or 8.4999999999999997e38 < ew

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. add-sqr-sqrt 46.9%

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

      5. sqrt-unprod 90.5%

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

      6. sqr-neg 90.5%

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

      7. sqrt-unprod 52.9%

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

      8. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in ew around inf 84.1%

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

   if -3.0000000000000001e-142 < ew < -5.2000000000000002e-256

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right), ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 78.4%

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

      2. associate-*r/ 78.4%

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

      3. associate-*l/ 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r/ 78.4%

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

      6. associate-*r/ 78.4%

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

      7. associate-*l/ 78.4%

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

      8. *-commutative 78.4%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in eh around inf 61.3%

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

       1. mul-1-neg 61.3%

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

       2. associate-*r* 61.3%

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

       3. *-commutative 61.3%

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

       4. associate-*r/ 61.3%

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

       5. associate-*r* 61.3%

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

       6. distribute-rgt-neg-in 61.3%

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

       7. distribute-rgt-neg-in 61.3%

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

       8. associate-*r/ 61.3%

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

       9. *-commutative 61.3%

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

       10. associate-/l* 61.3%

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

   11. Simplified 61.3%

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

   if -5.2000000000000002e-256 < ew < 3.69999999999999984e-257

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-out 99.7%

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

      6. distribute-rgt-neg-out 99.7%

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

   6. Applied egg-rr 75.3%

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

   7. Taylor expanded in ew around 0 62.3%

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

      1. associate-*r* 62.3%

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

      2. associate-/l* 62.3%

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

   9. Simplified 62.3%

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

   if 3.69999999999999984e-257 < ew < 8.4999999999999997e38

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. distribute-rgt-neg-out 99.8%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in t around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. *-commutative 67.1%

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

      3. associate-/l* 67.1%

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

   9. Simplified 67.1%

      \[\leadsto \color{blue}{ew + \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3 \cdot 10^{-142}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq -5.2 \cdot 10^{-256}:\\ \;\;\;\;eh \cdot \left(\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot \left(-\sin t\right)\right)\\ \mathbf{elif}\;ew \leq 3.7 \cdot 10^{-257}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;ew + \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-8} \lor \neg \left(t \leq 0.082\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \left(eh \cdot t\right) - ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -9e-8) (not (<= t 0.082)))
   (fabs (* ew (cos t)))
   (fabs (- (* (sin (atan (/ (* eh (tan t)) (- ew)))) (* eh t)) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -9e-8) || !(t <= 0.082)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((sin(atan(((eh * tan(t)) / -ew))) * (eh * t)) - ew));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-9d-8)) .or. (.not. (t <= 0.082d0))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs(((sin(atan(((eh * tan(t)) / -ew))) * (eh * t)) - ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -9e-8) || !(t <= 0.082)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs(((Math.sin(Math.atan(((eh * Math.tan(t)) / -ew))) * (eh * t)) - ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -9e-8) or not (t <= 0.082):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((math.sin(math.atan(((eh * math.tan(t)) / -ew))) * (eh * t)) - ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -9e-8) || !(t <= 0.082))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))) * Float64(eh * t)) - ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -9e-8) || ~((t <= 0.082)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((sin(atan(((eh * tan(t)) / -ew))) * (eh * t)) - ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -9e-8], N[Not[LessEqual[t, 0.082]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * t), $MachinePrecision]), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.99999999999999986e-8 or 0.0820000000000000034 < 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. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

      1. cos-atan 99.6%

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

      2. un-div-inv 99.6%

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

      3. hypot-1-def 99.6%

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

      4. add-sqr-sqrt 45.7%

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

      5. sqrt-unprod 96.2%

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

      6. sqr-neg 96.2%

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

      7. sqrt-unprod 53.9%

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

      8. add-sqr-sqrt 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in ew around inf 57.2%

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

   if -8.99999999999999986e-8 < t < 0.0820000000000000034

   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. sub-neg 100.0%

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

      2. associate-*l* 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. cos-atan 100.0%

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

      2. un-div-inv 100.0%

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

      3. hypot-1-def 100.0%

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

      4. add-sqr-sqrt 48.8%

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

      5. sqrt-unprod 90.6%

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

      6. sqr-neg 90.6%

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

      7. sqrt-unprod 51.2%

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

      8. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 97.8%

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

       1. mul-1-neg 97.8%

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

       2. unsub-neg 97.8%

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

       3. associate-*r* 97.8%

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

       4. *-commutative 97.8%

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

       5. mul-1-neg 97.8%

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

       6. distribute-frac-neg2 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-8} \lor \neg \left(t \leq 0.082\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \left(eh \cdot t\right) - ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ t_2 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -4.15 \cdot 10^{-222}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 3.15 \cdot 10^{-257}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot t\_1\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;ew + \left(eh \cdot t\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (* eh (/ (tan t) ew))))) (t_2 (fabs (* ew (cos t)))))
   (if (<= ew -4.15e-222)
     t_2
     (if (<= ew 3.15e-257)
       (* (* eh (sin t)) t_1)
       (if (<= ew 8.5e+38) (+ ew (* (* eh t) t_1)) t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan((eh * (tan(t) / ew))));
	double t_2 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -4.15e-222) {
		tmp = t_2;
	} else if (ew <= 3.15e-257) {
		tmp = (eh * sin(t)) * t_1;
	} else if (ew <= 8.5e+38) {
		tmp = ew + ((eh * t) * t_1);
	} else {
		tmp = t_2;
	}
	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 = sin(atan((eh * (tan(t) / ew))))
    t_2 = abs((ew * cos(t)))
    if (ew <= (-4.15d-222)) then
        tmp = t_2
    else if (ew <= 3.15d-257) then
        tmp = (eh * sin(t)) * t_1
    else if (ew <= 8.5d+38) then
        tmp = ew + ((eh * t) * t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan((eh * (Math.tan(t) / ew))));
	double t_2 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -4.15e-222) {
		tmp = t_2;
	} else if (ew <= 3.15e-257) {
		tmp = (eh * Math.sin(t)) * t_1;
	} else if (ew <= 8.5e+38) {
		tmp = ew + ((eh * t) * t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan((eh * (math.tan(t) / ew))))
	t_2 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -4.15e-222:
		tmp = t_2
	elif ew <= 3.15e-257:
		tmp = (eh * math.sin(t)) * t_1
	elif ew <= 8.5e+38:
		tmp = ew + ((eh * t) * t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(eh * Float64(tan(t) / ew))))
	t_2 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -4.15e-222)
		tmp = t_2;
	elseif (ew <= 3.15e-257)
		tmp = Float64(Float64(eh * sin(t)) * t_1);
	elseif (ew <= 8.5e+38)
		tmp = Float64(ew + Float64(Float64(eh * t) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan((eh * (tan(t) / ew))));
	t_2 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -4.15e-222)
		tmp = t_2;
	elseif (ew <= 3.15e-257)
		tmp = (eh * sin(t)) * t_1;
	elseif (ew <= 8.5e+38)
		tmp = ew + ((eh * t) * t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -4.15e-222], t$95$2, If[LessEqual[ew, 3.15e-257], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[ew, 8.5e+38], N[(ew + N[(N[(eh * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.15000000000000005e-222 or 8.4999999999999997e38 < ew

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. add-sqr-sqrt 46.3%

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

      5. sqrt-unprod 91.3%

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

      6. sqr-neg 91.3%

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

      7. sqrt-unprod 53.5%

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

      8. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in ew around inf 78.3%

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

   if -4.15000000000000005e-222 < ew < 3.14999999999999997e-257

   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. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-*l* 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-out 99.6%

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

      6. distribute-rgt-neg-out 99.6%

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

   6. Applied egg-rr 68.0%

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

   7. Taylor expanded in ew around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. associate-/l* 57.4%

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

   9. Simplified 57.4%

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

   if 3.14999999999999997e-257 < ew < 8.4999999999999997e38

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. distribute-rgt-neg-out 99.8%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in t around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. *-commutative 67.1%

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

      3. associate-/l* 67.1%

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

   9. Simplified 67.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.15 \cdot 10^{-222}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 3.15 \cdot 10^{-257}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;ew + \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -6 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-ew\right) - eh \cdot \left(t \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right)\right)\\ \mathbf{elif}\;ew \leq 9 \cdot 10^{+38}:\\ \;\;\;\;ew + \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -6e-142)
     t_1
     (if (<= ew -5e-310)
       (- (- ew) (* eh (* t (sin (atan (* eh (/ t ew)))))))
       (if (<= ew 9e+38)
         (+ ew (* (* eh t) (sin (atan (* eh (/ (tan t) ew))))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -6e-142) {
		tmp = t_1;
	} else if (ew <= -5e-310) {
		tmp = -ew - (eh * (t * sin(atan((eh * (t / ew))))));
	} else if (ew <= 9e+38) {
		tmp = ew + ((eh * t) * sin(atan((eh * (tan(t) / ew)))));
	} 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) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-6d-142)) then
        tmp = t_1
    else if (ew <= (-5d-310)) then
        tmp = -ew - (eh * (t * sin(atan((eh * (t / ew))))))
    else if (ew <= 9d+38) then
        tmp = ew + ((eh * t) * sin(atan((eh * (tan(t) / ew)))))
    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((ew * Math.cos(t)));
	double tmp;
	if (ew <= -6e-142) {
		tmp = t_1;
	} else if (ew <= -5e-310) {
		tmp = -ew - (eh * (t * Math.sin(Math.atan((eh * (t / ew))))));
	} else if (ew <= 9e+38) {
		tmp = ew + ((eh * t) * Math.sin(Math.atan((eh * (Math.tan(t) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -6e-142:
		tmp = t_1
	elif ew <= -5e-310:
		tmp = -ew - (eh * (t * math.sin(math.atan((eh * (t / ew))))))
	elif ew <= 9e+38:
		tmp = ew + ((eh * t) * math.sin(math.atan((eh * (math.tan(t) / ew)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -6e-142)
		tmp = t_1;
	elseif (ew <= -5e-310)
		tmp = Float64(Float64(-ew) - Float64(eh * Float64(t * sin(atan(Float64(eh * Float64(t / ew)))))));
	elseif (ew <= 9e+38)
		tmp = Float64(ew + Float64(Float64(eh * t) * sin(atan(Float64(eh * Float64(tan(t) / ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -6e-142)
		tmp = t_1;
	elseif (ew <= -5e-310)
		tmp = -ew - (eh * (t * sin(atan((eh * (t / ew))))));
	elseif (ew <= 9e+38)
		tmp = ew + ((eh * t) * sin(atan((eh * (tan(t) / ew)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6e-142], t$95$1, If[LessEqual[ew, -5e-310], N[((-ew) - N[(eh * N[(t * N[Sin[N[ArcTan[N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 9e+38], N[(ew + N[(N[(eh * t), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -6.0000000000000002e-142 or 8.99999999999999961e38 < ew

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. add-sqr-sqrt 46.9%

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

      5. sqrt-unprod 90.5%

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

      6. sqr-neg 90.5%

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

      7. sqrt-unprod 52.9%

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

      8. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in ew around inf 84.1%

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

   if -6.0000000000000002e-142 < ew < -4.999999999999985e-310

   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. sub-neg 99.6%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right), ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 66.1%

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

      2. associate-*r/ 66.1%

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

      3. associate-*l/ 66.1%

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

      4. *-commutative 66.1%

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

      5. associate-*r/ 66.1%

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

      6. associate-*r/ 66.1%

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

      7. associate-*l/ 66.1%

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

      8. *-commutative 66.1%

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

   8. Simplified 66.1%

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

   9. Taylor expanded in t around 0 30.9%

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

       1. associate-*r* 30.9%

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

       2. neg-mul-1 30.9%

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

       3. associate-/l* 30.9%

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

   11. Simplified 30.9%

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

   12. Taylor expanded in t around 0 32.4%

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

   if -4.999999999999985e-310 < ew < 8.99999999999999961e38

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. distribute-rgt-neg-out 99.8%

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

   6. Applied egg-rr 78.9%

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

   7. Taylor expanded in t around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-/l* 64.0%

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

   9. Simplified 64.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6 \cdot 10^{-142}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-ew\right) - eh \cdot \left(t \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right)\right)\\ \mathbf{elif}\;ew \leq 9 \cdot 10^{+38}:\\ \;\;\;\;ew + \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

   4. add-sqr-sqrt 47.2%

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

   5. sqrt-unprod 93.5%

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

   6. sqr-neg 93.5%

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

   7. sqrt-unprod 52.6%

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

   8. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in ew around inf 64.8%

   \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
10. Add Preprocessing

Alternative 11: 49.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00028 \lor \neg \left(t \leq 1.1 \cdot 10^{+83}\right):\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.00028) (not (<= t 1.1e+83))) (* (cos t) (- ew)) (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.00028) || !(t <= 1.1e+83)) {
		tmp = cos(t) * -ew;
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.00028d0)) .or. (.not. (t <= 1.1d+83))) then
        tmp = cos(t) * -ew
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.00028) || !(t <= 1.1e+83)) {
		tmp = Math.cos(t) * -ew;
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.00028) or not (t <= 1.1e+83):
		tmp = math.cos(t) * -ew
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.00028) || !(t <= 1.1e+83))
		tmp = Float64(cos(t) * Float64(-ew));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.00028) || ~((t <= 1.1e+83)))
		tmp = cos(t) * -ew;
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.00028], N[Not[LessEqual[t, 1.1e+83]], $MachinePrecision]], N[(N[Cos[t], $MachinePrecision] * (-ew)), $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.7999999999999998e-4 or 1.09999999999999999e83 < 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. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right), ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 47.9%

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

      2. associate-*r/ 47.9%

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

      3. associate-*l/ 47.9%

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

      4. *-commutative 47.9%

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

      5. associate-*r/ 47.9%

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

      6. associate-*r/ 47.9%

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

      7. associate-*l/ 47.9%

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

      8. *-commutative 47.9%

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

   8. Simplified 47.9%

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

   9. Taylor expanded in eh around 0 28.8%

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

       1. associate-*r* 28.8%

          \[\leadsto \color{blue}{\left(-1 \cdot ew\right) \cdot \cos t} \]

       2. neg-mul-1 28.8%

          \[\leadsto \color{blue}{\left(-ew\right)} \cdot \cos t \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{\left(-ew\right) \cdot \cos t} \]

   if -2.7999999999999998e-4 < t < 1.09999999999999999e83

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. cos-atan 99.9%

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

      2. un-div-inv 99.9%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt 49.2%

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

      5. sqrt-unprod 91.5%

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

      6. sqr-neg 91.5%

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

      7. sqrt-unprod 50.7%

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

      8. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in t around 0 66.9%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00028 \lor \neg \left(t \leq 1.1 \cdot 10^{+83}\right):\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.3% 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. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

   4. add-sqr-sqrt 47.2%

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

   5. sqrt-unprod 93.5%

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

   6. sqr-neg 93.5%

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

   7. sqrt-unprod 52.6%

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

   8. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 42.8%

   \[\leadsto \left|\color{blue}{ew}\right| \]
10. Add Preprocessing

Alternative 13: 22.1% accurate, 460.5× speedup

Math

\[\begin{array}{l} \\ -ew \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return -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 = -ew
end function

Java

public static double code(double eh, double ew, double t) {
	return -ew;
}

Python

def code(eh, ew, t):
	return -ew

Julia

function code(eh, ew, t)
	return Float64(-ew)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := (-ew)

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 46.0%

   \[\leadsto \color{blue}{0 - \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right), ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)} \]
7. Step-by-step derivation

   1. neg-sub0 46.0%

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

   2. associate-*r/ 46.0%

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

   3. associate-*l/ 46.0%

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

   4. *-commutative 46.0%

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

   5. associate-*r/ 46.0%

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

   6. associate-*r/ 46.0%

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

   7. associate-*l/ 46.0%

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

   8. *-commutative 46.0%

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

8. Simplified 46.0%

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

9. Taylor expanded in t around 0 21.9%

   \[\leadsto \color{blue}{-1 \cdot ew} \]
10. Step-by-step derivation

    1. neg-mul-1 21.9%

       \[\leadsto \color{blue}{-ew} \]

11. Simplified 21.9%

    \[\leadsto \color{blue}{-ew} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(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)))))))