Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 12.3s

Alternatives: 6

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

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

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]]

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%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * cos(atan(-((eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan((eh * (t / -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] / ew), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(t / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.5

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 90.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* eh (/ 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 * (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 * (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 * (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 * (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(eh * Float64(t / Float64(-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 * (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[(eh * N[(t / (-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. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.5

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

5. Simplified 99.5%

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

6. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 91.4

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

8. Simplified 91.4%

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

9. Final simplification 91.4%

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

Alternative 4: 80.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.5

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

5. Simplified 99.5%

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

6. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 58.3

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-eh \cdot \frac{t}{ew}\right)\right| \]

8. Simplified 58.3%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, ew \cdot -0.5, ew\right)} \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-eh \cdot \frac{t}{ew}\right)\right| \]

9. Taylor expanded in t around 0

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-atan.f64 N/A

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

    4. mul-1-neg N/A

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

    5. distribute-neg-frac2 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-tan.f64 N/A

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

    9. lower-neg.f64 79.1

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

11. Simplified 79.1%

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

12. Final simplification 79.1%

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

Alternative 5: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.5

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

5. Simplified 99.5%

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

6. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 91.4

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

8. Simplified 91.4%

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

9. Taylor expanded in t around 0

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/l* N/A

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

    5. distribute-rgt-neg-out N/A

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

    6. mul-1-neg N/A

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

    7. lower-atan.f64 N/A

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

    8. associate-*r/ N/A

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

    9. associate-*r/ N/A

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

    10. mul-1-neg N/A

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

    11. distribute-rgt-neg-in N/A

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

    12. *-commutative N/A

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

    13. lower-/.f64 N/A

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

    14. distribute-rgt-neg-in N/A

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

    15. lower-*.f64 N/A

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

    16. lower-neg.f64 78.0

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

11. Simplified 78.0%

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

12. Final simplification 78.0%

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

Alternative 6: 54.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.5

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

5. Simplified 99.5%

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

6. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 91.4

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

8. Simplified 91.4%

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

9. Taylor expanded in t around 0

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/l* N/A

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

    5. distribute-rgt-neg-out N/A

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

    6. mul-1-neg N/A

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

    7. lower-atan.f64 N/A

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

    8. associate-*r/ N/A

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

    9. associate-*r/ N/A

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

    10. mul-1-neg N/A

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

    11. distribute-rgt-neg-in N/A

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

    12. *-commutative N/A

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

    13. lower-/.f64 N/A

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

    14. distribute-rgt-neg-in N/A

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

    15. lower-*.f64 N/A

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

    16. lower-neg.f64 78.0

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

11. Simplified 78.0%

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

12. Taylor expanded in t around 0

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

    1. lower-*.f64 58.2

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

14. Simplified 58.2%

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

15. Final simplification 58.2%

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

Reproduce

herbie shell --seed 2024215 
(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)))))))