Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.9s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- 0.0 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)) / (0.0 - 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)) / (0.0d0 - 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)) / (0.0 - 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)) / (0.0 - 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(eh * tan(t)) / Float64(0.0 - 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)) / (0.0 - 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] / N[(0.0 - ew), $MachinePrecision]), $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]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \]
4. Add Preprocessing

Alternative 2: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (cos (atan (/ (* eh (tan t)) (- 0.0 ew)))))
   (* (* eh (sin t)) (sin (atan (/ (* t (- 0.0 eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * cos(atan(((eh * tan(t)) / (0.0 - ew))))) - ((eh * sin(t)) * sin(atan(((t * (0.0 - eh)) / ew))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((((ew * cos(t)) * cos(atan(((eh * tan(t)) / (0.0d0 - ew))))) - ((eh * sin(t)) * sin(atan(((t * (0.0d0 - eh)) / ew))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) * Math.cos(Math.atan(((eh * Math.tan(t)) / (0.0 - ew))))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((t * (0.0 - eh)) / ew))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) * math.cos(math.atan(((eh * math.tan(t)) / (0.0 - ew))))) - ((eh * math.sin(t)) * math.sin(math.atan(((t * (0.0 - eh)) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(0.0 - eh)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * cos(atan(((eh * tan(t)) / (0.0 - ew))))) - ((eh * sin(t)) * sin(atan(((t * (0.0 - eh)) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * N[(0.0 - eh), $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. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \color{blue}{t}\right), ew\right)\right)\right)\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 99.5%

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

6. Final simplification 99.5%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)\right| \]
7. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \tan t\\ \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \frac{t\_1}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{0 - ew}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (tan t))))
   (fabs
    (-
     (/ (* ew (cos t)) (hypot 1.0 (/ t_1 ew)))
     (* (* eh (sin t)) (sin (atan (/ t_1 (- 0.0 ew)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * tan(t);
	return fabs((((ew * cos(t)) / hypot(1.0, (t_1 / ew))) - ((eh * sin(t)) * sin(atan((t_1 / (0.0 - ew)))))));
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.tan(t);
	return Math.abs((((ew * Math.cos(t)) / Math.hypot(1.0, (t_1 / ew))) - ((eh * Math.sin(t)) * Math.sin(Math.atan((t_1 / (0.0 - ew)))))));
}

Python

def code(eh, ew, t):
	t_1 = eh * math.tan(t)
	return math.fabs((((ew * math.cos(t)) / math.hypot(1.0, (t_1 / ew))) - ((eh * math.sin(t)) * math.sin(math.atan((t_1 / (0.0 - ew)))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * tan(t))
	return abs(Float64(Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(t_1 / ew))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(t_1 / Float64(0.0 - ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh * tan(t);
	tmp = abs((((ew * cos(t)) / hypot(1.0, (t_1 / ew))) - ((eh * sin(t)) * sin(atan((t_1 / (0.0 - ew)))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(t$95$1 / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t$95$1 / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(ew \cdot \cos t\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}} \cdot \frac{1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left(ew \cdot \cos t\right) \cdot 1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}} \cdot {\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}} \cdot {\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. pow-prod-up N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}\right)}^{\frac{1}{2}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh \cdot \tan t}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \]
8. Add Preprocessing

Alternative 4: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \tan t\\ \left|\frac{ew}{\frac{\mathsf{hypot}\left(1, \frac{t\_1}{ew}\right)}{\cos t}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{0 - ew}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (tan t))))
   (fabs
    (-
     (/ ew (/ (hypot 1.0 (/ t_1 ew)) (cos t)))
     (* (* eh (sin t)) (sin (atan (/ t_1 (- 0.0 ew)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * tan(t);
	return fabs(((ew / (hypot(1.0, (t_1 / ew)) / cos(t))) - ((eh * sin(t)) * sin(atan((t_1 / (0.0 - ew)))))));
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.tan(t);
	return Math.abs(((ew / (Math.hypot(1.0, (t_1 / ew)) / Math.cos(t))) - ((eh * Math.sin(t)) * Math.sin(Math.atan((t_1 / (0.0 - ew)))))));
}

Python

def code(eh, ew, t):
	t_1 = eh * math.tan(t)
	return math.fabs(((ew / (math.hypot(1.0, (t_1 / ew)) / math.cos(t))) - ((eh * math.sin(t)) * math.sin(math.atan((t_1 / (0.0 - ew)))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * tan(t))
	return abs(Float64(Float64(ew / Float64(hypot(1.0, Float64(t_1 / ew)) / cos(t))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(t_1 / Float64(0.0 - ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh * tan(t);
	tmp = abs(((ew / (hypot(1.0, (t_1 / ew)) / cos(t))) - ((eh * sin(t)) * sin(atan((t_1 / (0.0 - ew)))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(ew / N[(N[Sqrt[1.0 ^ 2 + N[(t$95$1 / ew), $MachinePrecision] ^ 2], $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t$95$1 / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(ew \cdot \cos t\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}} \cdot \frac{1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left(ew \cdot \cos t\right) \cdot 1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}} \cdot {\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}} \cdot {\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. pow-prod-up N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}\right)}^{\frac{1}{2}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. clear-num N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew \cdot \cos t}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left({\left(\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew \cdot \cos t}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left({\left(\frac{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}}{\cos t}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left({\left(\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew} \cdot \frac{1}{\cos t}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left({\left(\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}\right)}^{-1} \cdot {\left(\frac{1}{\cos t}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}} \cdot {\left(\frac{1}{\cos t}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}}\right), \left({\left(\frac{1}{\cos t}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.7%

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

   1. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}} \cdot \frac{1}{\frac{1}{\cos t}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. remove-double-div N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}} \cdot \cos t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 \cdot \cos t}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{ew}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 \cdot \cos t}{\frac{1}{\frac{ew}{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. associate-/r/ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 \cdot \cos t}{\frac{1}{ew} \cdot \sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. frac-times N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\frac{1}{ew}} \cdot \frac{\cos t}{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\frac{1}{ew}} \cdot \frac{1}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{\cos t}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{\frac{1}{ew}}}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{\cos t}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. remove-double-div N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew}{\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{\cos t}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(ew, \left(\frac{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}{\cos t}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \left|\frac{ew}{\frac{\mathsf{hypot}\left(1, \frac{eh \cdot \tan t}{ew}\right)}{\cos t}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \]
12. Add Preprocessing

Alternative 5: 91.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\\ t_2 := \left(eh \cdot \sin t\right) \cdot t\_1\\ \mathbf{if}\;ew \leq -1.75 \cdot 10^{-195}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \sin t \cdot \frac{eh \cdot t\_1}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 7.6 \cdot 10^{-199}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{t\_2}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (* eh (tan t)) (- 0.0 ew)))))
        (t_2 (* (* eh (sin t)) t_1)))
   (if (<= ew -1.75e-195)
     (fabs (* ew (- (cos t) (* (sin t) (/ (* eh t_1) ew)))))
     (if (<= ew 7.6e-199) (fabs t_2) (fabs (* ew (- (cos t) (/ t_2 ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((eh * tan(t)) / (0.0 - ew))));
	double t_2 = (eh * sin(t)) * t_1;
	double tmp;
	if (ew <= -1.75e-195) {
		tmp = fabs((ew * (cos(t) - (sin(t) * ((eh * t_1) / ew)))));
	} else if (ew <= 7.6e-199) {
		tmp = fabs(t_2);
	} else {
		tmp = fabs((ew * (cos(t) - (t_2 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(atan(((eh * tan(t)) / (0.0d0 - ew))))
    t_2 = (eh * sin(t)) * t_1
    if (ew <= (-1.75d-195)) then
        tmp = abs((ew * (cos(t) - (sin(t) * ((eh * t_1) / ew)))))
    else if (ew <= 7.6d-199) then
        tmp = abs(t_2)
    else
        tmp = abs((ew * (cos(t) - (t_2 / ew))))
    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)) / (0.0 - ew))));
	double t_2 = (eh * Math.sin(t)) * t_1;
	double tmp;
	if (ew <= -1.75e-195) {
		tmp = Math.abs((ew * (Math.cos(t) - (Math.sin(t) * ((eh * t_1) / ew)))));
	} else if (ew <= 7.6e-199) {
		tmp = Math.abs(t_2);
	} else {
		tmp = Math.abs((ew * (Math.cos(t) - (t_2 / ew))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew))))
	t_2 = (eh * math.sin(t)) * t_1
	tmp = 0
	if ew <= -1.75e-195:
		tmp = math.fabs((ew * (math.cos(t) - (math.sin(t) * ((eh * t_1) / ew)))))
	elif ew <= 7.6e-199:
		tmp = math.fabs(t_2)
	else:
		tmp = math.fabs((ew * (math.cos(t) - (t_2 / ew))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew))))
	t_2 = Float64(Float64(eh * sin(t)) * t_1)
	tmp = 0.0
	if (ew <= -1.75e-195)
		tmp = abs(Float64(ew * Float64(cos(t) - Float64(sin(t) * Float64(Float64(eh * t_1) / ew)))));
	elseif (ew <= 7.6e-199)
		tmp = abs(t_2);
	else
		tmp = abs(Float64(ew * Float64(cos(t) - Float64(t_2 / ew))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((eh * tan(t)) / (0.0 - ew))));
	t_2 = (eh * sin(t)) * t_1;
	tmp = 0.0;
	if (ew <= -1.75e-195)
		tmp = abs((ew * (cos(t) - (sin(t) * ((eh * t_1) / ew)))));
	elseif (ew <= 7.6e-199)
		tmp = abs(t_2);
	else
		tmp = abs((ew * (cos(t) - (t_2 / ew))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[ew, -1.75e-195], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(N[(eh * t$95$1), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 7.6e-199], N[Abs[t$95$2], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(t$95$2 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.75000000000000007e-195

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 95.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)}{ew}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\sin t \cdot \frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)}{ew}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\sin t, \left(\frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)}{ew}\right)\right)\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)}{ew}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right), ew\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 95.9%

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

   if -1.75000000000000007e-195 < ew < 7.5999999999999996e-199

   1. Initial program 99.9%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\sin t \cdot eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\sin t, eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      11. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      17. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

      19. neg-lowering-neg.f64 84.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right) \]

   7. Simplified 84.2%

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

   if 7.5999999999999996e-199 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 97.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.75 \cdot 10^{-195}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \sin t \cdot \frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 7.6 \cdot 10^{-199}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\\ t_2 := \left|ew \cdot \left(\cos t - \frac{t\_1}{ew}\right)\right|\\ \mathbf{if}\;ew \leq -5.6 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{-200}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) (- 0.0 ew))))))
        (t_2 (fabs (* ew (- (cos t) (/ t_1 ew))))))
   (if (<= ew -5.6e-196) t_2 (if (<= ew 2.9e-200) (fabs t_1) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - ew))));
	double t_2 = fabs((ew * (cos(t) - (t_1 / ew))));
	double tmp;
	if (ew <= -5.6e-196) {
		tmp = t_2;
	} else if (ew <= 2.9e-200) {
		tmp = fabs(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 = (eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0d0 - ew))))
    t_2 = abs((ew * (cos(t) - (t_1 / ew))))
    if (ew <= (-5.6d-196)) then
        tmp = t_2
    else if (ew <= 2.9d-200) then
        tmp = abs(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 = (eh * Math.sin(t)) * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - ew))));
	double t_2 = Math.abs((ew * (Math.cos(t) - (t_1 / ew))));
	double tmp;
	if (ew <= -5.6e-196) {
		tmp = t_2;
	} else if (ew <= 2.9e-200) {
		tmp = Math.abs(t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh * math.sin(t)) * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew))))
	t_2 = math.fabs((ew * (math.cos(t) - (t_1 / ew))))
	tmp = 0
	if ew <= -5.6e-196:
		tmp = t_2
	elif ew <= 2.9e-200:
		tmp = math.fabs(t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew)))))
	t_2 = abs(Float64(ew * Float64(cos(t) - Float64(t_1 / ew))))
	tmp = 0.0
	if (ew <= -5.6e-196)
		tmp = t_2;
	elseif (ew <= 2.9e-200)
		tmp = abs(t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - ew))));
	t_2 = abs((ew * (cos(t) - (t_1 / ew))));
	tmp = 0.0;
	if (ew <= -5.6e-196)
		tmp = t_2;
	elseif (ew <= 2.9e-200)
		tmp = abs(t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(t$95$1 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.6e-196], t$95$2, If[LessEqual[ew, 2.9e-200], N[Abs[t$95$1], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.5999999999999997e-196 or 2.9e-200 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 96.6%

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

   if -5.5999999999999997e-196 < ew < 2.9e-200

   1. Initial program 99.9%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\sin t \cdot eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\sin t, eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      11. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      17. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

      19. neg-lowering-neg.f64 84.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right) \]

   7. Simplified 84.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.6 \cdot 10^{-196}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{-200}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\frac{\frac{1}{ew}}{\cos t}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (/ 1.0 (/ (/ 1.0 ew) (cos t)))
   (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) (- 0.0 ew))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((1.0 / ((1.0 / ew) / cos(t))) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - 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(((1.0d0 / ((1.0d0 / ew) / cos(t))) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0d0 - ew)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((1.0 / ((1.0 / ew) / Math.cos(t))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - ew)))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((1.0 / ((1.0 / ew) / math.cos(t))) - ((eh * math.sin(t)) * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(1.0 / Float64(Float64(1.0 / ew) / cos(t))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((1.0 / ((1.0 / ew) / cos(t))) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - ew)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(1.0 / N[(N[(1.0 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(ew \cdot \cos t\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right) \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{-1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. unpow-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}} \cdot \frac{1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left(ew \cdot \cos t\right) \cdot 1}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}} \cdot {\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}} \cdot {\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\frac{1}{4}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. pow-prod-up N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{{\left(1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}\right)}^{\frac{1}{2}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 \cdot 1 + \frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

7. Taylor expanded in eh around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{ew \cdot \cos t}\right)}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\frac{1}{ew}}{\cos t}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \cos t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \cos t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. cos-lowering-cos.f64 98.7%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{cos.f64}\left(t\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

9. Simplified 98.7%

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

10. Final simplification 98.7%

    \[\leadsto \left|\frac{1}{\frac{\frac{1}{ew}}{\cos t}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \]
11. Add Preprocessing

Alternative 8: 91.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)\\ t_2 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -1.4 \cdot 10^{-194}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \sin t \cdot \frac{eh \cdot t\_1}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 2.55 \cdot 10^{-200}:\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{t\_2 \cdot t\_1}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (* t (- 0.0 eh)) ew)))) (t_2 (* eh (sin t))))
   (if (<= ew -1.4e-194)
     (fabs (* ew (- (cos t) (* (sin t) (/ (* eh t_1) ew)))))
     (if (<= ew 2.55e-200)
       (fabs (* t_2 (sin (atan (/ (* eh (tan t)) (- 0.0 ew))))))
       (fabs (* ew (- (cos t) (/ (* t_2 t_1) ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((t * (0.0 - eh)) / ew)));
	double t_2 = eh * sin(t);
	double tmp;
	if (ew <= -1.4e-194) {
		tmp = fabs((ew * (cos(t) - (sin(t) * ((eh * t_1) / ew)))));
	} else if (ew <= 2.55e-200) {
		tmp = fabs((t_2 * sin(atan(((eh * tan(t)) / (0.0 - ew))))));
	} else {
		tmp = fabs((ew * (cos(t) - ((t_2 * t_1) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(atan(((t * (0.0d0 - eh)) / ew)))
    t_2 = eh * sin(t)
    if (ew <= (-1.4d-194)) then
        tmp = abs((ew * (cos(t) - (sin(t) * ((eh * t_1) / ew)))))
    else if (ew <= 2.55d-200) then
        tmp = abs((t_2 * sin(atan(((eh * tan(t)) / (0.0d0 - ew))))))
    else
        tmp = abs((ew * (cos(t) - ((t_2 * t_1) / ew))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan(((t * (0.0 - eh)) / ew)));
	double t_2 = eh * Math.sin(t);
	double tmp;
	if (ew <= -1.4e-194) {
		tmp = Math.abs((ew * (Math.cos(t) - (Math.sin(t) * ((eh * t_1) / ew)))));
	} else if (ew <= 2.55e-200) {
		tmp = Math.abs((t_2 * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - ew))))));
	} else {
		tmp = Math.abs((ew * (Math.cos(t) - ((t_2 * t_1) / ew))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((t * (0.0 - eh)) / ew)))
	t_2 = eh * math.sin(t)
	tmp = 0
	if ew <= -1.4e-194:
		tmp = math.fabs((ew * (math.cos(t) - (math.sin(t) * ((eh * t_1) / ew)))))
	elif ew <= 2.55e-200:
		tmp = math.fabs((t_2 * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew))))))
	else:
		tmp = math.fabs((ew * (math.cos(t) - ((t_2 * t_1) / ew))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(t * Float64(0.0 - eh)) / ew)))
	t_2 = Float64(eh * sin(t))
	tmp = 0.0
	if (ew <= -1.4e-194)
		tmp = abs(Float64(ew * Float64(cos(t) - Float64(sin(t) * Float64(Float64(eh * t_1) / ew)))));
	elseif (ew <= 2.55e-200)
		tmp = abs(Float64(t_2 * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew))))));
	else
		tmp = abs(Float64(ew * Float64(cos(t) - Float64(Float64(t_2 * t_1) / ew))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((t * (0.0 - eh)) / ew)));
	t_2 = eh * sin(t);
	tmp = 0.0;
	if (ew <= -1.4e-194)
		tmp = abs((ew * (cos(t) - (sin(t) * ((eh * t_1) / ew)))));
	elseif (ew <= 2.55e-200)
		tmp = abs((t_2 * sin(atan(((eh * tan(t)) / (0.0 - ew))))));
	else
		tmp = abs((ew * (cos(t) - ((t_2 * t_1) / ew))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(t * N[(0.0 - eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.4e-194], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(N[(eh * t$95$1), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.55e-200], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(N[(t$95$2 * t$95$1), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.40000000000000006e-194

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 95.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)}{ew}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\sin t \cdot \frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)}{ew}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\sin t, \left(\frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)}{ew}\right)\right)\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\frac{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)}{ew}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right), ew\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 95.9%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right)\right)\right), ew\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot t}{ew}\right)\right)\right)\right)\right), ew\right)\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(0 - \frac{eh \cdot t}{ew}\right)\right)\right)\right), ew\right)\right)\right)\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{eh \cdot t}{ew}\right)\right)\right)\right)\right), ew\right)\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(eh \cdot t\right), ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(t \cdot eh\right), ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 95.6%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, eh\right), ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right)\right) \]

   12. Simplified 95.6%

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

   if -1.40000000000000006e-194 < ew < 2.5499999999999999e-200

   1. Initial program 99.9%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\sin t \cdot eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\sin t, eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      11. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      17. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

      19. neg-lowering-neg.f64 84.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right) \]

   7. Simplified 84.2%

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

   if 2.5499999999999999e-200 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 97.4%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(eh \cdot t\right)}, \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(t \cdot eh\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right) \]

      2. *-lowering-*.f64 97.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right) \]

   10. Simplified 97.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.4 \cdot 10^{-194}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \sin t \cdot \frac{eh \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 2.55 \cdot 10^{-200}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \left|ew \cdot \left(\cos t - \frac{t\_1 \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)}{ew}\right)\right|\\ \mathbf{if}\;ew \leq -1.4 \cdot 10^{-195}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 1.9 \cdot 10^{-198}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2
         (fabs
          (*
           ew
           (- (cos t) (/ (* t_1 (sin (atan (/ (* t (- 0.0 eh)) ew)))) ew))))))
   (if (<= ew -1.4e-195)
     t_2
     (if (<= ew 1.9e-198)
       (fabs (* t_1 (sin (atan (/ (* eh (tan t)) (- 0.0 ew))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = fabs((ew * (cos(t) - ((t_1 * sin(atan(((t * (0.0 - eh)) / ew)))) / ew))));
	double tmp;
	if (ew <= -1.4e-195) {
		tmp = t_2;
	} else if (ew <= 1.9e-198) {
		tmp = fabs((t_1 * sin(atan(((eh * tan(t)) / (0.0 - 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 = eh * sin(t)
    t_2 = abs((ew * (cos(t) - ((t_1 * sin(atan(((t * (0.0d0 - eh)) / ew)))) / ew))))
    if (ew <= (-1.4d-195)) then
        tmp = t_2
    else if (ew <= 1.9d-198) then
        tmp = abs((t_1 * sin(atan(((eh * tan(t)) / (0.0d0 - 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 = eh * Math.sin(t);
	double t_2 = Math.abs((ew * (Math.cos(t) - ((t_1 * Math.sin(Math.atan(((t * (0.0 - eh)) / ew)))) / ew))));
	double tmp;
	if (ew <= -1.4e-195) {
		tmp = t_2;
	} else if (ew <= 1.9e-198) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - ew))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	t_2 = math.fabs((ew * (math.cos(t) - ((t_1 * math.sin(math.atan(((t * (0.0 - eh)) / ew)))) / ew))))
	tmp = 0
	if ew <= -1.4e-195:
		tmp = t_2
	elif ew <= 1.9e-198:
		tmp = math.fabs((t_1 * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = abs(Float64(ew * Float64(cos(t) - Float64(Float64(t_1 * sin(atan(Float64(Float64(t * Float64(0.0 - eh)) / ew)))) / ew))))
	tmp = 0.0
	if (ew <= -1.4e-195)
		tmp = t_2;
	elseif (ew <= 1.9e-198)
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	t_2 = abs((ew * (cos(t) - ((t_1 * sin(atan(((t * (0.0 - eh)) / ew)))) / ew))));
	tmp = 0.0;
	if (ew <= -1.4e-195)
		tmp = t_2;
	elseif (ew <= 1.9e-198)
		tmp = abs((t_1 * sin(atan(((eh * tan(t)) / (0.0 - ew))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(t * N[(0.0 - eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.4e-195], t$95$2, If[LessEqual[ew, 1.9e-198], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.40000000000000002e-195 or 1.9000000000000001e-198 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(eh \cdot t\right)}, \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(t \cdot eh\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right) \]

      2. *-lowering-*.f64 96.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right), ew\right)\right)\right)\right) \]

   10. Simplified 96.2%

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

   if -1.40000000000000002e-195 < ew < 1.9000000000000001e-198

   1. Initial program 99.9%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\sin t \cdot eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\sin t, eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      11. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      17. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

      19. neg-lowering-neg.f64 84.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right) \]

   7. Simplified 84.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.4 \cdot 10^{-195}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.9 \cdot 10^{-198}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - \frac{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -54000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.7 \cdot 10^{-102}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -54000.0)
     t_1
     (if (<= ew 2.7e-102)
       (fabs (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) (- 0.0 ew))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -54000.0) {
		tmp = t_1;
	} else if (ew <= 2.7e-102) {
		tmp = fabs(((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - 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 <= (-54000.0d0)) then
        tmp = t_1
    else if (ew <= 2.7d-102) then
        tmp = abs(((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0d0 - 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 <= -54000.0) {
		tmp = t_1;
	} else if (ew <= 2.7e-102) {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - 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 <= -54000.0:
		tmp = t_1
	elif ew <= 2.7e-102:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - 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 <= -54000.0)
		tmp = t_1;
	elseif (ew <= 2.7e-102)
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - 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 <= -54000.0)
		tmp = t_1;
	elseif (ew <= 2.7e-102)
		tmp = abs(((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - 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, -54000.0], t$95$1, If[LessEqual[ew, 2.7e-102], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -54000 or 2.7e-102 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \cos t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\cos t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\cos t, ew\right)\right) \]

      3. cos-lowering-cos.f64 88.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right)\right) \]

   7. Simplified 88.3%

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

   if -54000 < ew < 2.7e-102

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(eh \cdot \sin t\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\sin t \cdot eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\sin t, eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      11. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      17. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

      19. neg-lowering-neg.f64 71.4%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), eh\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right) \]

   7. Simplified 71.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -54000:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 2.7 \cdot 10^{-102}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;t \leq -0.0078:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.0085:\\ \;\;\;\;\left|ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= t -0.0078)
     t_1
     (if (<= t 0.0085)
       (fabs
        (+
         ew
         (*
          t
          (-
           (* t (* ew -0.5))
           (* eh (sin (atan (/ (* eh (tan t)) (- 0.0 ew)))))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (t <= -0.0078) {
		tmp = t_1;
	} else if (t <= 0.0085) {
		tmp = fabs((ew + (t * ((t * (ew * -0.5)) - (eh * sin(atan(((eh * tan(t)) / (0.0 - 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 (t <= (-0.0078d0)) then
        tmp = t_1
    else if (t <= 0.0085d0) then
        tmp = abs((ew + (t * ((t * (ew * (-0.5d0))) - (eh * sin(atan(((eh * tan(t)) / (0.0d0 - 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 (t <= -0.0078) {
		tmp = t_1;
	} else if (t <= 0.0085) {
		tmp = Math.abs((ew + (t * ((t * (ew * -0.5)) - (eh * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - ew)))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if t <= -0.0078:
		tmp = t_1
	elif t <= 0.0085:
		tmp = math.fabs((ew + (t * ((t * (ew * -0.5)) - (eh * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew)))))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (t <= -0.0078)
		tmp = t_1;
	elseif (t <= 0.0085)
		tmp = abs(Float64(ew + Float64(t * Float64(Float64(t * Float64(ew * -0.5)) - Float64(eh * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - 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 (t <= -0.0078)
		tmp = t_1;
	elseif (t <= 0.0085)
		tmp = abs((ew + (t * ((t * (ew * -0.5)) - (eh * sin(atan(((eh * tan(t)) / (0.0 - 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[t, -0.0078], t$95$1, If[LessEqual[t, 0.0085], N[Abs[N[(ew + N[(t * N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0077999999999999996 or 0.0085000000000000006 < t

   1. Initial program 99.7%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \cos t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\cos t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\cos t, ew\right)\right) \]

      3. cos-lowering-cos.f64 51.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right)\right) \]

   7. Simplified 51.6%

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

   if -0.0077999999999999996 < t < 0.0085000000000000006

   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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + -1 \cdot \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t + \left(\mathsf{neg}\left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \left(\cos t - \frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\cos t, \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{ew}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{/.f64}\left(\left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right), ew\right)\right)\right)\right) \]

   7. Simplified 95.8%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew + t \cdot \left(-1 \cdot \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + \frac{-1}{2} \cdot \left(ew \cdot t\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \left(t \cdot \left(-1 \cdot \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + \frac{-1}{2} \cdot \left(ew \cdot t\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \left(-1 \cdot \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + \frac{-1}{2} \cdot \left(ew \cdot t\right)\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \left(\left(\mathsf{neg}\left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) + \frac{-1}{2} \cdot \left(ew \cdot t\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \left(ew \cdot t\right) + \left(\mathsf{neg}\left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \left(ew \cdot t\right)\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(\left(ew \cdot t\right) \cdot \frac{-1}{2}\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(\left(t \cdot ew\right) \cdot \frac{-1}{2}\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(t \cdot \left(ew \cdot \frac{-1}{2}\right)\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot ew\right)\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot ew\right)\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \left(ew \cdot \frac{-1}{2}\right)\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(ew, \frac{-1}{2}\right)\right), \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(ew, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(ew, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

      16. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(ew, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 97.5%

       \[\leadsto \left|\color{blue}{ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0078:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.0085:\\ \;\;\;\;\left|ew + t \cdot \left(t \cdot \left(ew \cdot -0.5\right) - eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;t \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.000118:\\ \;\;\;\;\left|ew - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= t -0.005)
     t_1
     (if (<= t 0.000118)
       (fabs (- ew (* (* t eh) (sin (atan (/ (* t (- 0.0 eh)) ew))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (t <= -0.005) {
		tmp = t_1;
	} else if (t <= 0.000118) {
		tmp = fabs((ew - ((t * eh) * sin(atan(((t * (0.0 - eh)) / 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 (t <= (-0.005d0)) then
        tmp = t_1
    else if (t <= 0.000118d0) then
        tmp = abs((ew - ((t * eh) * sin(atan(((t * (0.0d0 - eh)) / 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 (t <= -0.005) {
		tmp = t_1;
	} else if (t <= 0.000118) {
		tmp = Math.abs((ew - ((t * eh) * Math.sin(Math.atan(((t * (0.0 - eh)) / ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if t <= -0.005:
		tmp = t_1
	elif t <= 0.000118:
		tmp = math.fabs((ew - ((t * eh) * math.sin(math.atan(((t * (0.0 - eh)) / ew))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (t <= -0.005)
		tmp = t_1;
	elseif (t <= 0.000118)
		tmp = abs(Float64(ew - Float64(Float64(t * eh) * sin(atan(Float64(Float64(t * Float64(0.0 - eh)) / 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 (t <= -0.005)
		tmp = t_1;
	elseif (t <= 0.000118)
		tmp = abs((ew - ((t * eh) * sin(atan(((t * (0.0 - eh)) / 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[t, -0.005], t$95$1, If[LessEqual[t, 0.000118], N[Abs[N[(ew - N[(N[(t * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * N[(0.0 - eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0050000000000000001 or 1.18e-4 < t

   1. Initial program 99.7%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \cos t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\cos t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\cos t, ew\right)\right) \]

      3. cos-lowering-cos.f64 51.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right)\right) \]

   7. Simplified 51.6%

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

   if -0.0050000000000000001 < t < 1.18e-4

   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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(ew - eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(eh \cdot t\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      9. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. neg-lowering-neg.f64 97.3%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 97.3%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(eh \cdot t\right)}, \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(t \cdot eh\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 97.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.005:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.000118:\\ \;\;\;\;\left|ew - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(0 - eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;t \leq -0.0009:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.0022:\\ \;\;\;\;\left|t \cdot eh - ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= t -0.0009) t_1 (if (<= t 0.0022) (fabs (- (* t eh) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (t <= -0.0009) {
		tmp = t_1;
	} else if (t <= 0.0022) {
		tmp = fabs(((t * eh) - 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 (t <= (-0.0009d0)) then
        tmp = t_1
    else if (t <= 0.0022d0) then
        tmp = abs(((t * eh) - 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 (t <= -0.0009) {
		tmp = t_1;
	} else if (t <= 0.0022) {
		tmp = Math.abs(((t * eh) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if t <= -0.0009:
		tmp = t_1
	elif t <= 0.0022:
		tmp = math.fabs(((t * eh) - ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (t <= -0.0009)
		tmp = t_1;
	elseif (t <= 0.0022)
		tmp = abs(Float64(Float64(t * eh) - 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 (t <= -0.0009)
		tmp = t_1;
	elseif (t <= 0.0022)
		tmp = abs(((t * eh) - 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[t, -0.0009], t$95$1, If[LessEqual[t, 0.0022], N[Abs[N[(N[(t * eh), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.9999999999999998e-4 or 0.00220000000000000013 < t

   1. Initial program 99.7%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \cos t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\cos t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\cos t, ew\right)\right) \]

      3. cos-lowering-cos.f64 51.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), ew\right)\right) \]

   7. Simplified 51.6%

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

   if -8.9999999999999998e-4 < t < 0.00220000000000000013

   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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(ew - eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(eh \cdot t\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      9. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. neg-lowering-neg.f64 97.3%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 97.3%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\mathsf{neg}\left(eh \cdot \tan t\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(eh\right)\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(0 - eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} + 1}}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 95.1%

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

   10. Taylor expanded in eh around -inf

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \color{blue}{\left(eh \cdot t\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(t \cdot eh\right)\right)\right) \]

       2. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(t, eh\right)\right)\right) \]

   12. Simplified 96.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0009:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.0022:\\ \;\;\;\;\left|t \cdot eh - ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.2 \cdot 10^{-195}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.5 \cdot 10^{-262}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -5.2e-195)
   (fabs ew)
   (if (<= ew 1.5e-262) (fabs (* t eh)) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -5.2e-195) {
		tmp = fabs(ew);
	} else if (ew <= 1.5e-262) {
		tmp = fabs((t * eh));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-5.2d-195)) then
        tmp = abs(ew)
    else if (ew <= 1.5d-262) then
        tmp = abs((t * eh))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -5.2e-195) {
		tmp = Math.abs(ew);
	} else if (ew <= 1.5e-262) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -5.2e-195:
		tmp = math.fabs(ew)
	elif ew <= 1.5e-262:
		tmp = math.fabs((t * eh))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -5.2e-195)
		tmp = abs(ew);
	elseif (ew <= 1.5e-262)
		tmp = abs(Float64(t * eh));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -5.2e-195)
		tmp = abs(ew);
	elseif (ew <= 1.5e-262)
		tmp = abs((t * eh));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -5.2e-195], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 1.5e-262], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.2000000000000003e-195 or 1.50000000000000009e-262 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
   6. Step-by-step derivation

   7. Simplified 50.2%

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

   if -5.2000000000000003e-195 < ew < 1.50000000000000009e-262

   1. Initial program 99.9%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(ew - eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(eh \cdot t\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

      9. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. neg-lowering-neg.f64 38.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 38.0%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\mathsf{neg}\left(eh \cdot \tan t\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(eh\right)\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(0 - eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} + 1}}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 32.4%

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

   10. Taylor expanded in ew around 0

       \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot t\right)}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(t \cdot eh\right)\right) \]

       2. *-lowering-*.f64 32.7%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, eh\right)\right) \]

   12. Simplified 32.7%

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

Alternative 15: 54.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(ew - eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(eh \cdot t\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   9. atan-lowering-atan.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. neg-lowering-neg.f64 54.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 54.0%

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

   1. frac-2neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\mathsf{neg}\left(eh \cdot \tan t\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

   2. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

   3. remove-double-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right) \]

   4. sin-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(eh\right)\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(0 - eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} + 1}}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 52.4%

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

10. Taylor expanded in eh around -inf

    \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \color{blue}{\left(eh \cdot t\right)}\right)\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(t \cdot eh\right)\right)\right) \]

    2. *-lowering-*.f64 53.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(t, eh\right)\right)\right) \]

12. Simplified 53.0%

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

13. Final simplification 53.0%

    \[\leadsto \left|t \cdot eh - ew\right| \]
14. Add Preprocessing

Alternative 16: 54.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(ew - eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \left(\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(eh \cdot t\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right) \]

   9. atan-lowering-atan.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{\mathsf{neg}\left(ew\right)}\right)\right)\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh \cdot \tan t}{-1 \cdot ew}\right)\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(eh \cdot \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \tan t\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(-1 \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{neg}\left(ew\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. neg-lowering-neg.f64 54.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right), \mathsf{neg.f64}\left(ew\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 54.0%

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

   1. frac-2neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\mathsf{neg}\left(eh \cdot \tan t\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

   2. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{\mathsf{neg}\left(\left(\mathsf{neg}\left(ew\right)\right)\right)}\right)\right)\right)\right) \]

   3. remove-double-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right) \]

   4. sin-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(eh\right)\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\left(0 - eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, eh\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, eh\right), \left(\frac{\frac{\tan t}{ew}}{\sqrt{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} + 1}}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 52.4%

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

10. Taylor expanded in ew around 0

    \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew - -1 \cdot \left(eh \cdot t\right)\right)}\right) \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(-1 \cdot \left(eh \cdot t\right)\right)\right)\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(ew + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot t\right)\right)\right)\right)\right)\right) \]

    3. remove-double-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(ew + eh \cdot t\right)\right) \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \left(eh \cdot t\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \left(t \cdot eh\right)\right)\right) \]

    6. *-lowering-*.f64 53.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(t, eh\right)\right)\right) \]

12. Simplified 53.0%

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

Alternative 17: 42.5% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\frac{1}{2}}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1} \cdot {\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right), \left({\left({\left(1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left({\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1} \cdot {\left({\left(1 + {\left(\frac{\tan t \cdot eh}{ew}\right)}^{2}\right)}^{0.25}\right)}^{-1}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
6. Step-by-step derivation

7. Simplified 43.6%

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

Reproduce

herbie shell --seed 2024155 
(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)))))))