Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.9s

Alternatives: 10

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (log1p (expm1 (cos (atan (* (/ eh ew) (tan t)))))))
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) (- eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * log1p(expm1(cos(atan(((eh / ew) * tan(t))))))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) * Math.log1p(Math.expm1(Math.cos(Math.atan(((eh / ew) * Math.tan(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)) * math.log1p(math.expm1(math.cos(math.atan(((eh / ew) * math.tan(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(ew * cos(t)) * log1p(expm1(cos(atan(Float64(Float64(eh / ew) * tan(t))))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * Float64(-eh)) / ew))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. log1p-expm1-u 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   2. associate-/l* 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \color{blue}{\left(\frac{-eh}{\frac{ew}{\tan t}}\right)}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   3. associate-/r/ 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot \tan t\right)}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. add-sqr-sqrt 44.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\color{blue}{\sqrt{-eh} \cdot \sqrt{-eh}}}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. sqrt-unprod 94.3%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. sqr-neg 94.3%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\sqrt{\color{blue}{eh \cdot eh}}}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   7. sqrt-unprod 55.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\color{blue}{\sqrt{eh} \cdot \sqrt{eh}}}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   8. add-sqr-sqrt 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{\color{blue}{eh}}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

3. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

4. Final simplification 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right| \]

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

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

   4. associate-/r/ 99.8%

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

   5. add-sqr-sqrt 44.4%

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

   6. sqrt-unprod 93.9%

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

   7. sqr-neg 93.9%

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

   8. sqrt-unprod 55.4%

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

   9. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((eh * -t) / ew)))) + ((ew * cos(t)) * (-1.0 / hypot(1.0, ((eh * tan(t)) / 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 * (-t)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $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. 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%

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

   4. associate-/r/ 99.8%

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

   5. add-sqr-sqrt 44.4%

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

   6. sqrt-unprod 93.9%

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

   7. sqr-neg 93.9%

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

   8. sqrt-unprod 55.4%

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

   9. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 98.9%

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

   1. mul-1-neg 75.9%

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

   2. distribute-rgt-neg-in 75.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. associate-/r/ 99.8%

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

   5. add-sqr-sqrt 44.4%

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

   6. sqrt-unprod 93.9%

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

   7. sqr-neg 93.9%

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

   8. sqrt-unprod 55.4%

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

   9. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

5. Simplified 99.8%

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

   1. expm1-log1p-u 83.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-eh\right) \cdot \tan t\right)\right)}}{ew}\right)\right| \]

   2. expm1-udef 72.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-eh\right) \cdot \tan t\right)} - 1}}{ew}\right)\right| \]

   3. add-sqr-sqrt 30.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   4. sqrt-unprod 72.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   5. sqr-neg 72.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{eh \cdot eh}} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   6. sqrt-unprod 41.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   7. add-sqr-sqrt 72.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{eh} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

7. Applied egg-rr 72.1%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(eh \cdot \tan t\right)} - 1}}{ew}\right)\right| \]
8. Step-by-step derivation

   1. expm1-def 81.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(eh \cdot \tan t\right)\right)}}{ew}\right)\right| \]

   2. expm1-log1p 98.1%

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

   3. *-commutative 98.1%

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

9. Simplified 98.1%

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

10. Taylor expanded in t around 0 97.9%

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

11. Final simplification 97.9%

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

Alternative 5: 79.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $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. Taylor expanded in t around 0 77.4%

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

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

   4. associate-/r/ 99.8%

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

   5. add-sqr-sqrt 44.4%

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

   6. sqrt-unprod 93.9%

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

   7. sqr-neg 93.9%

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

   8. sqrt-unprod 55.4%

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

   9. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 77.4%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

6. Simplified 77.4%

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

7. Final simplification 77.4%

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

Alternative 6: 79.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * (-t)), $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. Taylor expanded in t around 0 77.4%

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

3. Taylor expanded in t around 0 77.4%

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

   1. mul-1-neg 75.9%

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

   2. distribute-rgt-neg-in 75.9%

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

5. Simplified 77.4%

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

6. Final simplification 77.4%

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

Alternative 7: 78.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. associate-/r/ 99.8%

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

   5. add-sqr-sqrt 44.4%

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

   6. sqrt-unprod 93.9%

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

   7. sqr-neg 93.9%

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

   8. sqrt-unprod 55.4%

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

   9. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

5. Simplified 99.8%

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

   1. expm1-log1p-u 83.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-eh\right) \cdot \tan t\right)\right)}}{ew}\right)\right| \]

   2. expm1-udef 72.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-eh\right) \cdot \tan t\right)} - 1}}{ew}\right)\right| \]

   3. add-sqr-sqrt 30.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   4. sqrt-unprod 72.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   5. sqr-neg 72.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{eh \cdot eh}} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   6. sqrt-unprod 41.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

   7. add-sqr-sqrt 72.1%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{eh} \cdot \tan t\right)} - 1}{ew}\right)\right| \]

7. Applied egg-rr 72.1%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(eh \cdot \tan t\right)} - 1}}{ew}\right)\right| \]
8. Step-by-step derivation

   1. expm1-def 81.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\tan t \cdot eh}{ew}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(eh \cdot \tan t\right)\right)}}{ew}\right)\right| \]

   2. expm1-log1p 98.1%

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

   3. *-commutative 98.1%

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

9. Simplified 98.1%

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

10. Taylor expanded in t around 0 76.4%

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

11. Taylor expanded in t around 0 76.4%

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

12. Final simplification 76.4%

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

Alternative 8: 77.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * (-t)), $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. Taylor expanded in t around 0 77.4%

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

3. Taylor expanded in t around 0 75.9%

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

   1. mul-1-neg 75.9%

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

   2. distribute-rgt-neg-in 75.9%

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

5. Simplified 75.9%

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

6. Taylor expanded in t around 0 76.0%

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

   1. associate-*r/ 76.0%

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

   2. mul-1-neg 76.0%

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

8. Simplified 76.0%

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

   1. associate-/l* 76.0%

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

   2. associate-/r/ 76.0%

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

   3. add-sqr-sqrt 33.6%

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

   4. sqrt-unprod 69.1%

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

   5. sqr-neg 69.1%

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

   6. sqrt-unprod 42.4%

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

   7. add-sqr-sqrt 76.0%

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

10. Applied egg-rr 76.0%

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

11. Final simplification 76.0%

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

Alternative 9: 54.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * (-t)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(ew * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(t * eh), $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. Taylor expanded in t around 0 77.4%

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

3. Taylor expanded in t around 0 75.9%

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

   1. mul-1-neg 75.9%

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

   2. distribute-rgt-neg-in 75.9%

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

5. Simplified 75.9%

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

6. Taylor expanded in t around 0 76.0%

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

   1. associate-*r/ 76.0%

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

   2. mul-1-neg 76.0%

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

8. Simplified 76.0%

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

9. Taylor expanded in t around 0 55.9%

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

10. Final simplification 55.9%

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

Alternative 10: 77.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew / N[Sqrt[1.0 ^ 2 + N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * (-t)), $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. Taylor expanded in t around 0 77.4%

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

3. Taylor expanded in t around 0 75.9%

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

   1. mul-1-neg 75.9%

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

   2. distribute-rgt-neg-in 75.9%

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

5. Simplified 75.9%

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

6. Taylor expanded in t around 0 76.0%

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

   1. associate-*r/ 76.0%

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

   2. mul-1-neg 76.0%

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

8. Simplified 76.0%

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

   1. expm1-log1p-u 56.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t \cdot eh}{ew}\right)\right| \]

   2. expm1-udef 47.3%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t \cdot eh}{ew}\right)\right| \]

10. Applied egg-rr 49.5%

    \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \frac{t}{\frac{ew}{eh}}\right)}\right)} - 1\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t \cdot eh}{ew}\right)\right| \]
11. Step-by-step derivation

    1. expm1-def 58.8%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \frac{t}{\frac{ew}{eh}}\right)}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t \cdot eh}{ew}\right)\right| \]

    2. expm1-log1p 75.7%

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

    3. associate-/l* 75.7%

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

    4. associate-*r/ 75.7%

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

12. Simplified 75.7%

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

13. Final simplification 75.7%

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

Reproduce

herbie shell --seed 2023181 
(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)))))))