Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.5s

Alternatives: 14

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

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

FPCore

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

C

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

Java

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

Python

def code(eh, ew, t):
	return math.fabs(((math.expm1(math.log1p(math.cos(math.atan(((eh / ew) * math.tan(t)))))) * (ew * math.cos(t))) - ((eh * math.sin(t)) * math.sin(math.atan(((math.tan(t) * -eh) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(expm1(log1p(cos(atan(Float64(Float64(eh / ew) * tan(t)))))) * Float64(ew * cos(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[(Exp[N[Log[1 + N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. associate-/l* 99.8%

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

   3. associate-/r/ 99.8%

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

   4. add-sqr-sqrt 52.6%

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

   5. sqrt-unprod 94.1%

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

   6. sqr-neg 94.1%

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

   7. sqrt-unprod 47.2%

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

   8. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt[3]{{\left(\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)\right)}^{-3}} \cdot \left(ew \cdot \cos t\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (cbrt (pow (hypot 1.0 (* (/ eh ew) (tan t))) -3.0)) (* ew (cos t)))
   (* (* eh (sin t)) (sin (atan (/ (* (tan t) (- eh)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((cbrt(pow(hypot(1.0, ((eh / ew) * tan(t))), -3.0)) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.cbrt(Math.pow(Math.hypot(1.0, ((eh / ew) * Math.tan(t))), -3.0)) * (ew * Math.cos(t))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((Math.tan(t) * -eh) / ew))))));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(cbrt((hypot(1.0, Float64(Float64(eh / ew) * tan(t))) ^ -3.0)) * Float64(ew * cos(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[Power[N[Power[N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. add-cbrt-cube 99.8%

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

   2. pow3 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/r/ 99.8%

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

   5. add-sqr-sqrt 52.6%

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

   6. sqrt-unprod 94.1%

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

   7. sqr-neg 94.1%

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

   8. sqrt-unprod 47.2%

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

   9. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

5. Applied egg-rr 99.8%

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

   1. pow1/3 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{{\left({\left(\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right)}^{3}\right)}^{0.3333333333333333}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   2. inv-pow 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left({\color{blue}{\left({\left(\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   3. pow-pow 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\color{blue}{\left({\left(\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. metadata-eval 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left({\left(\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

7. Applied egg-rr 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{{\left({\left(\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)\right)}^{-3}\right)}^{0.3333333333333333}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
8. Step-by-step derivation

   1. unpow1/3 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 35.0%

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

   2. hypot-1-def 35.0%

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

   3. associate-/l* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. add-sqr-sqrt 20.2%

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

   6. sqrt-unprod 35.1%

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

   7. sqr-neg 35.1%

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

   8. sqrt-unprod 14.8%

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

   9. add-sqr-sqrt 35.0%

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

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. Final simplification 99.8%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. cos-atan 35.0%

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

   2. hypot-1-def 35.0%

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

   3. associate-/l* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. add-sqr-sqrt 20.2%

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

   6. sqrt-unprod 35.1%

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

   7. sqr-neg 35.1%

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

   8. sqrt-unprod 14.8%

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

   9. add-sqr-sqrt 35.0%

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

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. Taylor expanded in t around 0 99.1%

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

   1. mul-1-neg 34.9%

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

   2. distribute-rgt-neg-in 34.9%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 5: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 35.0%

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

   2. hypot-1-def 35.0%

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

   3. associate-/l* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. add-sqr-sqrt 20.2%

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

   6. sqrt-unprod 35.1%

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

   7. sqr-neg 35.1%

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

   8. sqrt-unprod 14.8%

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

   9. add-sqr-sqrt 35.0%

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

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. Taylor expanded in eh around 0 98.8%

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

5. Final simplification 98.8%

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

Alternative 6: 98.3% 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. Step-by-step derivation

   1. cos-atan 35.0%

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

   2. hypot-1-def 35.0%

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

   3. associate-/l* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. add-sqr-sqrt 20.2%

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

   6. sqrt-unprod 35.1%

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

   7. sqr-neg 35.1%

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

   8. sqrt-unprod 14.8%

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

   9. add-sqr-sqrt 35.0%

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

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. Taylor expanded in eh around 0 98.8%

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

   1. add-log-exp 91.1%

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

   2. *-un-lft-identity 91.1%

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

   3. log-prod 91.1%

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

   4. metadata-eval 91.1%

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

   5. add-log-exp 98.8%

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

   6. add-sqr-sqrt 52.4%

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

   7. sqrt-unprod 97.2%

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

   8. sqr-neg 97.2%

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

   9. sqrt-unprod 46.5%

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

   10. add-sqr-sqrt 98.8%

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

6. Applied egg-rr 98.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot 1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{0 + eh \cdot \tan t}}{ew}\right)\right| \]
7. Step-by-step derivation

   1. +-lft-identity 98.8%

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

   2. *-commutative 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 7: 98.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - 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(t)) - ((eh * sin(t)) * sin(atan(((eh * -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 * (-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. Step-by-step derivation

   1. cos-atan 35.0%

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

   2. hypot-1-def 35.0%

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

   3. associate-/l* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. add-sqr-sqrt 20.2%

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

   6. sqrt-unprod 35.1%

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

   7. sqr-neg 35.1%

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

   8. sqrt-unprod 14.8%

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

   9. add-sqr-sqrt 35.0%

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

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. Taylor expanded in eh around 0 98.8%

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

5. Taylor expanded in t around 0 98.5%

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

   1. mul-1-neg 34.9%

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

   2. distribute-rgt-neg-in 34.9%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 8: 78.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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))))
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))));
}

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(atan(((tan(t) * -eh) / ew)))) - (eh * sin(t))));
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[(eh * N[Sin[t], $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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

   1. tan-quot 61.0%

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

   2. associate-*r/ 60.9%

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

6. Applied egg-rr 60.9%

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

7. Taylor expanded in eh around inf 81.4%

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

8. Final simplification 81.4%

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

Alternative 9: 78.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(ew * N[Cos[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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eh around -inf 81.4%

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

   1. mul-1-neg 81.4%

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

   2. distribute-rgt-neg-in 81.4%

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

7. Simplified 81.4%

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

8. Final simplification 81.4%

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

Alternative 10: 37.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((ew * cos(atan(((tan(t) * -eh) / ew)))) - ((t * t) / ((1.0d0 / eh) * (ew / eh)))))
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)))) - ((t * t) / ((1.0 / eh) * (ew / eh)))));
}

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(atan(((tan(t) * -eh) / ew)))) - ((t * t) / ((1.0 / eh) * (ew / eh)))));
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[(t * t), $MachinePrecision] / N[(N[(1.0 / eh), $MachinePrecision] * N[(ew / eh), $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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in t around 0 35.1%

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

   1. associate-/l* 35.1%

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

   2. unpow2 35.1%

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

   3. unpow2 35.1%

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

7. Simplified 35.1%

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

   1. *-un-lft-identity 35.1%

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

   2. times-frac 39.6%

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

9. Applied egg-rr 39.6%

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

10. Final simplification 39.6%

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

Alternative 11: 36.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(t * N[(N[(eh * eh), $MachinePrecision] * N[(t / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(ew * N[Cos[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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in t around 0 35.1%

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

   1. associate-/l* 35.1%

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

   2. unpow2 35.1%

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

   3. unpow2 35.1%

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

7. Simplified 35.1%

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

8. Taylor expanded in t around 0 35.1%

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

   1. unpow2 35.1%

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

   2. associate-/l* 35.1%

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

   3. unpow2 35.1%

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

   4. associate-*r/ 36.5%

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

   5. unpow2 36.5%

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

   6. associate-/r/ 35.9%

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

   7. unpow2 35.9%

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

10. Simplified 35.9%

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

11. Final simplification 35.9%

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

Alternative 12: 36.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(atan(((tan(t) * -eh) / ew)))) - (t * (t / (ew / (eh * eh))))));
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[(t * N[(t / N[(ew / N[(eh * eh), $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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in t around 0 35.1%

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

   1. associate-/l* 35.1%

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

   2. unpow2 35.1%

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

   3. unpow2 35.1%

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

7. Simplified 35.1%

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

   1. *-un-lft-identity 35.1%

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

   2. times-frac 36.5%

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

9. Applied egg-rr 36.5%

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

10. Final simplification 36.5%

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

Alternative 13: 34.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t * t), $MachinePrecision] / N[(ew / N[(eh * eh), $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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in t around 0 35.1%

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

   1. associate-/l* 35.1%

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

   2. unpow2 35.1%

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

   3. unpow2 35.1%

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

7. Simplified 35.1%

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

   1. cos-atan 35.0%

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

   2. hypot-1-def 35.0%

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

   3. associate-/l* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. add-sqr-sqrt 20.2%

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

   6. sqrt-unprod 35.1%

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

   7. sqr-neg 35.1%

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

   8. sqrt-unprod 14.8%

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

   9. add-sqr-sqrt 35.0%

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

9. Applied egg-rr 35.0%

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

10. Final simplification 35.0%

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

Alternative 14: 34.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[(eh * (-t)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(t * t), $MachinePrecision] / N[(ew / N[(eh * eh), $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 81.6%

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

   1. *-commutative 81.6%

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

   2. sin-atan 62.5%

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

   3. associate-*l/ 60.4%

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

   4. associate-/l* 60.4%

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

   5. add-sqr-sqrt 33.4%

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

   6. sqrt-unprod 46.0%

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

   7. sqr-neg 46.0%

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

   8. sqrt-unprod 26.8%

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

   9. add-sqr-sqrt 60.3%

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

   10. associate-/l* 60.3%

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

   11. associate-*l/ 58.7%

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

   12. hypot-1-def 61.0%

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

   13. associate-/l* 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in t around 0 35.1%

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

   1. associate-/l* 35.1%

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

   2. unpow2 35.1%

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

   3. unpow2 35.1%

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

7. Simplified 35.1%

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

8. Taylor expanded in t around 0 34.9%

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

   1. mul-1-neg 34.9%

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

   2. distribute-rgt-neg-in 34.9%

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

10. Simplified 34.9%

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

11. Final simplification 34.9%

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

Reproduce

herbie shell --seed 2023240 
(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)))))))