Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	return abs(Float64(Float64(cos(t_1) * Float64(ew * cos(t))) - 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(((cos(t_1) * (ew * cos(t))) - ((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[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 93.2%

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

   6. sqr-neg 93.2%

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

   7. sqrt-unprod 49.5%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	return math.fabs(((math.cos(math.atan(((eh * math.tan(t)) / -ew))) * (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(cos(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))) * Float64(ew * cos(t))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(t * Float64(eh / Float64(-ew))))))))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $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[(t * N[(eh / (-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 99.4%

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

   1. mul-1-neg 99.4%

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

   2. distribute-neg-frac2 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*r/ 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 93.2%

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

   6. sqr-neg 93.2%

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

   7. sqrt-unprod 49.5%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 93.2%

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

   6. sqr-neg 93.2%

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

   7. sqrt-unprod 49.5%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 99.1%

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

   1. add-cube-cbrt 99.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(\sqrt[3]{\left(-eh\right) \cdot \tan t} \cdot \sqrt[3]{\left(-eh\right) \cdot \tan t}\right) \cdot \sqrt[3]{\left(-eh\right) \cdot \tan t}}}{ew}\right)\right| \]

   2. pow3 99.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{{\left(\sqrt[3]{\left(-eh\right) \cdot \tan t}\right)}^{3}}}{ew}\right)\right| \]

   3. add-sqr-sqrt 50.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{{\left(\sqrt[3]{\color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \tan t}\right)}^{3}}{ew}\right)\right| \]

   4. sqrt-unprod 98.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{{\left(\sqrt[3]{\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \tan t}\right)}^{3}}{ew}\right)\right| \]

   5. sqr-neg 98.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{{\left(\sqrt[3]{\sqrt{\color{blue}{eh \cdot eh}} \cdot \tan t}\right)}^{3}}{ew}\right)\right| \]

   6. sqrt-unprod 49.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{{\left(\sqrt[3]{\color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \tan t}\right)}^{3}}{ew}\right)\right| \]

   7. add-sqr-sqrt 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{{\left(\sqrt[3]{\color{blue}{eh} \cdot \tan t}\right)}^{3}}{ew}\right)\right| \]

   8. *-commutative 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{{\left(\sqrt[3]{\color{blue}{\tan t \cdot eh}}\right)}^{3}}{ew}\right)\right| \]

9. Applied egg-rr 99.0%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{{\left(\sqrt[3]{\tan t \cdot eh}\right)}^{3}}}{ew}\right)\right| \]
10. Step-by-step derivation

    1. rem-cube-cbrt 99.0%

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

    2. *-commutative 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 6: 98.3% accurate, 1.8× speedup

Math

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

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 * eh) / Float64(-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. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 93.2%

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

   6. sqr-neg 93.2%

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

   7. sqrt-unprod 49.5%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 99.1%

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

8. Taylor expanded in t around 0 98.8%

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

   1. associate-*r* 78.9%

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

   2. mul-1-neg 78.9%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

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

Alternative 7: 56.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 93.2%

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

   6. sqr-neg 93.2%

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

   7. sqrt-unprod 49.5%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 99.1%

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

8. Taylor expanded in t around 0 78.9%

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

9. Taylor expanded in t around 0 54.5%

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

    1. associate-*r* 54.4%

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

    2. *-commutative 54.4%

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

    3. associate-*r* 54.6%

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

    4. mul-1-neg 54.6%

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

    5. distribute-neg-frac2 54.6%

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

11. Simplified 54.6%

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

12. Final simplification 54.6%

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

Alternative 8: 79.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - 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. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 93.2%

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

   6. sqr-neg 93.2%

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

   7. sqrt-unprod 49.5%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 99.1%

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

8. Taylor expanded in t around 0 78.9%

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

9. Taylor expanded in t around 0 78.9%

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

    1. associate-*r* 78.9%

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

    2. mul-1-neg 78.9%

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

11. Simplified 78.9%

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

12. Final simplification 78.9%

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

Reproduce

herbie shell --seed 2024096 
(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)))))))