Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{{\left(\frac{\tan t}{ew} \cdot eh\right)}^{2} + 1}} \cdot \left(ew \cdot \cos t\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (/ 1.0 (sqrt (+ (pow (* (/ (tan t) ew) eh) 2.0) 1.0))) (* ew (cos t)))
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) (- eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((1.0 / sqrt((pow(((tan(t) / ew) * eh), 2.0) + 1.0))) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan(((tan(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((((1.0d0 / sqrt(((((tan(t) / ew) * eh) ** 2.0d0) + 1.0d0))) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((1.0 / Math.sqrt((Math.pow(((Math.tan(t) / ew) * eh), 2.0) + 1.0))) * (ew * Math.cos(t))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((Math.tan(t) * -eh) / ew))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((1.0 / math.sqrt((math.pow(((math.tan(t) / ew) * eh), 2.0) + 1.0))) * (ew * math.cos(t))) - ((eh * math.sin(t)) * math.sin(math.atan(((math.tan(t) * -eh) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(1.0 / sqrt(Float64((Float64(Float64(tan(t) / ew) * eh) ^ 2.0) + 1.0))) * Float64(ew * cos(t))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * Float64(-eh)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((1.0 / sqrt(((((tan(t) / ew) * eh) ^ 2.0) + 1.0))) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(1.0 / N[Sqrt[N[(N[Power[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $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. Step-by-step derivation

   1. lift-tan.f64 N/A

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

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

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

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

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

   4. +-lft-identity N/A

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

   5. +-lft-identity N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. cos-atan N/A

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

   14. lower-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-tan.f64 N/A

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

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

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

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

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

   4. +-lft-identity N/A

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

   5. +-lft-identity N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. cos-atan N/A

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

   14. lower-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 65.7

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

7. Simplified 65.7%

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

8. Taylor expanded in ew around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 4: 98.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-tan.f64 N/A

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

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

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

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

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

   4. +-lft-identity N/A

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

   5. +-lft-identity N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. cos-atan N/A

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

   14. lower-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 65.7

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

7. Simplified 65.7%

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

8. Taylor expanded in ew around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

10. Simplified 99.0%

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

11. Taylor expanded in t around 0

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

    1. mul-1-neg N/A

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

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

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

    3. mul-1-neg N/A

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

    4. lower-*.f64 N/A

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

    5. mul-1-neg N/A

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

    6. lower-neg.f64 98.8

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

13. Simplified 98.8%

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

14. Final simplification 98.8%

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

Alternative 5: 69.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := eh \cdot \sin t\\ t_3 := \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\\ t_4 := \left|t\_1 - t\_3 \cdot \left(t \cdot eh\right)\right|\\ \mathbf{if}\;ew \leq -3.9 \cdot 10^{+105}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;ew \leq -1.35 \cdot 10^{-160}:\\ \;\;\;\;\left|t\_1 + t\_2 \cdot \frac{t \cdot eh}{ew \cdot \sqrt{\mathsf{fma}\left(eh, \frac{eh \cdot \left(t \cdot t\right)}{ew \cdot ew}, 1\right)}}\right|\\ \mathbf{elif}\;ew \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot -0.5, ew\right) - t\_2 \cdot t\_3\right|\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (* eh (sin t)))
        (t_3 (sin (atan (/ (* t (- eh)) ew))))
        (t_4 (fabs (- t_1 (* t_3 (* t eh))))))
   (if (<= ew -3.9e+105)
     t_4
     (if (<= ew -1.35e-160)
       (fabs
        (+
         t_1
         (*
          t_2
          (/
           (* t eh)
           (* ew (sqrt (fma eh (/ (* eh (* t t)) (* ew ew)) 1.0)))))))
       (if (<= ew 6.2e-188)
         (fabs (- (fma ew (* (* t t) -0.5) ew) (* t_2 t_3)))
         t_4)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = eh * sin(t);
	double t_3 = sin(atan(((t * -eh) / ew)));
	double t_4 = fabs((t_1 - (t_3 * (t * eh))));
	double tmp;
	if (ew <= -3.9e+105) {
		tmp = t_4;
	} else if (ew <= -1.35e-160) {
		tmp = fabs((t_1 + (t_2 * ((t * eh) / (ew * sqrt(fma(eh, ((eh * (t * t)) / (ew * ew)), 1.0)))))));
	} else if (ew <= 6.2e-188) {
		tmp = fabs((fma(ew, ((t * t) * -0.5), ew) - (t_2 * t_3)));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(eh * sin(t))
	t_3 = sin(atan(Float64(Float64(t * Float64(-eh)) / ew)))
	t_4 = abs(Float64(t_1 - Float64(t_3 * Float64(t * eh))))
	tmp = 0.0
	if (ew <= -3.9e+105)
		tmp = t_4;
	elseif (ew <= -1.35e-160)
		tmp = abs(Float64(t_1 + Float64(t_2 * Float64(Float64(t * eh) / Float64(ew * sqrt(fma(eh, Float64(Float64(eh * Float64(t * t)) / Float64(ew * ew)), 1.0)))))));
	elseif (ew <= 6.2e-188)
		tmp = abs(Float64(fma(ew, Float64(Float64(t * t) * -0.5), ew) - Float64(t_2 * t_3)));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(t$95$1 - N[(t$95$3 * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.9e+105], t$95$4, If[LessEqual[ew, -1.35e-160], N[Abs[N[(t$95$1 + N[(t$95$2 * N[(N[(t * eh), $MachinePrecision] / N[(ew * N[Sqrt[N[(eh * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 6.2e-188], N[Abs[N[(N[(ew * N[(N[(t * t), $MachinePrecision] * -0.5), $MachinePrecision] + ew), $MachinePrecision] - N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.89999999999999978e105 or 6.2000000000000004e-188 < 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. lift-tan.f64 N/A

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

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

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

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

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

      4. +-lft-identity N/A

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

      5. +-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. cos-atan N/A

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

      14. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 99.3

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

   10. Simplified 99.3%

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

   11. Taylor expanded in t around 0

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

       1. lower-*.f64 83.3

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

   13. Simplified 83.3%

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

   if -3.89999999999999978e105 < ew < -1.35000000000000005e-160

   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. lift-tan.f64 N/A

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

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

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

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

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

      4. +-lft-identity N/A

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

      5. +-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. cos-atan N/A

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

      14. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 98.9%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 98.9

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

   10. Simplified 98.9%

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

   12. Applied egg-rr 70.8%

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

   if -1.35000000000000005e-160 < ew < 6.2000000000000004e-188

   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. lift-tan.f64 N/A

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

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

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

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

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

      4. +-lft-identity N/A

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

      5. +-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. cos-atan N/A

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

      14. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 97.4%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.4

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

   10. Simplified 97.4%

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

   11. Taylor expanded in t around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 66.1

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

   13. Simplified 66.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.9 \cdot 10^{+105}:\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq -1.35 \cdot 10^{-160}:\\ \;\;\;\;\left|ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \frac{t \cdot eh}{ew \cdot \sqrt{\mathsf{fma}\left(eh, \frac{eh \cdot \left(t \cdot t\right)}{ew \cdot ew}, 1\right)}}\right|\\ \mathbf{elif}\;ew \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot -0.5, ew\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \left|t\_1 - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{if}\;ew \leq -3.9 \cdot 10^{+105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq -5.8 \cdot 10^{-142}:\\ \;\;\;\;\left|t\_1 + \left(eh \cdot \sin t\right) \cdot \frac{t \cdot eh}{ew \cdot \sqrt{\mathsf{fma}\left(eh, \frac{eh \cdot \left(t \cdot t\right)}{ew \cdot ew}, 1\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (fabs (- t_1 (* (sin (atan (/ (* t (- eh)) ew))) (* t eh))))))
   (if (<= ew -3.9e+105)
     t_2
     (if (<= ew -5.8e-142)
       (fabs
        (+
         t_1
         (*
          (* eh (sin t))
          (/
           (* t eh)
           (* ew (sqrt (fma eh (/ (* eh (* t t)) (* ew ew)) 1.0)))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = fabs((t_1 - (sin(atan(((t * -eh) / ew))) * (t * eh))));
	double tmp;
	if (ew <= -3.9e+105) {
		tmp = t_2;
	} else if (ew <= -5.8e-142) {
		tmp = fabs((t_1 + ((eh * sin(t)) * ((t * eh) / (ew * sqrt(fma(eh, ((eh * (t * t)) / (ew * ew)), 1.0)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = abs(Float64(t_1 - Float64(sin(atan(Float64(Float64(t * Float64(-eh)) / ew))) * Float64(t * eh))))
	tmp = 0.0
	if (ew <= -3.9e+105)
		tmp = t_2;
	elseif (ew <= -5.8e-142)
		tmp = abs(Float64(t_1 + Float64(Float64(eh * sin(t)) * Float64(Float64(t * eh) / Float64(ew * sqrt(fma(eh, Float64(Float64(eh * Float64(t * t)) / Float64(ew * ew)), 1.0)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 - N[(N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.9e+105], t$95$2, If[LessEqual[ew, -5.8e-142], N[Abs[N[(t$95$1 + N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(N[(t * eh), $MachinePrecision] / N[(ew * N[Sqrt[N[(eh * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.89999999999999978e105 or -5.7999999999999998e-142 < 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. lift-tan.f64 N/A

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

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

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

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

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

      4. +-lft-identity N/A

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

      5. +-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. cos-atan N/A

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

      14. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 99.0%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 98.8

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

   10. Simplified 98.8%

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

   11. Taylor expanded in t around 0

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

       1. lower-*.f64 75.3

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

   13. Simplified 75.3%

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

   if -3.89999999999999978e105 < ew < -5.7999999999999998e-142

   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. lift-tan.f64 N/A

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

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

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

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

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

      4. +-lft-identity N/A

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

      5. +-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. cos-atan N/A

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

      14. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 98.8%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 98.8

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

   10. Simplified 98.8%

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

   12. Applied egg-rr 71.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.9 \cdot 10^{+105}:\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq -5.8 \cdot 10^{-142}:\\ \;\;\;\;\left|ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \frac{t \cdot eh}{ew \cdot \sqrt{\mathsf{fma}\left(eh, \frac{eh \cdot \left(t \cdot t\right)}{ew \cdot ew}, 1\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(N[(t * eh), $MachinePrecision] / N[(ew * N[Sqrt[N[(eh * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $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. lift-tan.f64 N/A

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

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

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

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

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

   4. +-lft-identity N/A

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

   5. +-lft-identity N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. cos-atan N/A

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

   14. lower-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

7. Simplified 99.0%

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

8. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. lower-/.f64 N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 98.8

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

10. Simplified 98.8%

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

12. Applied egg-rr 56.8%

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

13. Final simplification 56.8%

    \[\leadsto \left|ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \frac{t \cdot eh}{ew \cdot \sqrt{\mathsf{fma}\left(eh, \frac{eh \cdot \left(t \cdot t\right)}{ew \cdot ew}, 1\right)}}\right| \]
14. Add Preprocessing

Alternative 8: 50.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * N[(N[(eh * N[(t * eh), $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sqrt[N[(eh * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $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. lift-tan.f64 N/A

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

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

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

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

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

   4. +-lft-identity N/A

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

   5. +-lft-identity N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. cos-atan N/A

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

   14. lower-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

7. Simplified 99.0%

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

8. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. lower-/.f64 N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 98.8

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

10. Simplified 98.8%

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

12. Applied egg-rr 54.8%

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

13. Final simplification 54.8%

    \[\leadsto \left|ew \cdot \cos t + \sin t \cdot \frac{eh \cdot \left(t \cdot eh\right)}{ew \cdot \sqrt{\mathsf{fma}\left(eh, \frac{eh \cdot \left(t \cdot t\right)}{ew \cdot ew}, 1\right)}}\right| \]
14. 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)))))))