Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.4s

Alternatives: 9

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

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

FPCore

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

C

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

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.cbrt(Math.pow(Math.cos(Math.atan(((eh / ew) * Math.tan(t)))), 3.0)) * (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(cbrt((cos(atan(Float64(Float64(eh / ew) * tan(t)))) ^ 3.0)) * Float64(ew * cos(t))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(-Float64(eh * tan(t))) / ew))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Power[N[Power[N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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[(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. 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 53.4%

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

   6. sqrt-unprod 94.2%

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

   7. sqr-neg 94.2%

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

   8. sqrt-unprod 46.4%

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

   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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Log[1 + N[(Exp[N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $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 * 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. log1p-expm1-u 99.8%

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

   2. associate-/l* 99.8%

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

   3. associate-/r/ 99.8%

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

   4. add-sqr-sqrt 53.4%

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

   5. sqrt-unprod 94.2%

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

   6. sqr-neg 94.2%

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

   7. sqrt-unprod 46.4%

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

   8. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 0.9× speedup

Math

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

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 (/ (- (* eh (tan t))) 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((-(eh * tan(t)) / 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((-(eh * Math.tan(t)) / 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(-Float64(eh * tan(t))) / 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[(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 99.5%

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

   2. hypot-1-def 99.5%

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

   3. associate-/l* 99.5%

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

   4. associate-/r/ 99.5%

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

   5. add-sqr-sqrt 53.4%

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

   6. sqrt-unprod 93.7%

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

   7. sqr-neg 93.7%

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

   8. sqrt-unprod 46.1%

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

   9. add-sqr-sqrt 99.5%

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

3. Applied egg-rr 99.8%

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

   1. rem-cbrt-cube 99.8%

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

   2. inv-pow 99.8%

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

   3. pow-pow 99.8%

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

   4. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

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 (tan 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 * tan(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 * Math.tan(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 * math.tan(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(-Float64(eh * tan(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 * tan(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 * 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 99.5%

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

   2. hypot-1-def 99.5%

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

   3. associate-/l* 99.5%

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

   4. associate-/r/ 99.5%

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

   5. add-sqr-sqrt 53.4%

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

   6. sqrt-unprod 93.7%

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

   7. sqr-neg 93.7%

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

   8. sqrt-unprod 46.1%

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

   9. add-sqr-sqrt 99.5%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 5: 99.1% accurate, 1.3× speedup

Math

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

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) (/ ew t))))))))

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 / (ew / t)))))));
}

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 / (ew / t)))))));
}

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 / (ew / t)))))))

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(ew / t)))))))
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 / (ew / t)))))));
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[((-eh) / N[(ew / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. associate-/l* 99.5%

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

   3. distribute-neg-frac 99.5%

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

4. Simplified 99.5%

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

   1. cos-atan 99.5%

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

   2. hypot-1-def 99.5%

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

   3. associate-/l* 99.5%

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

   4. associate-/r/ 99.5%

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

   5. add-sqr-sqrt 53.4%

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

   6. sqrt-unprod 93.7%

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

   7. sqr-neg 93.7%

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

   8. sqrt-unprod 46.1%

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

   9. add-sqr-sqrt 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(-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. 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 53.4%

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

   6. sqrt-unprod 94.2%

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

   7. sqr-neg 94.2%

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

   8. sqrt-unprod 46.4%

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

   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. Taylor expanded in eh around 0 98.8%

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

7. 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: 98.4% 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. 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 53.4%

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

   6. sqrt-unprod 94.2%

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

   7. sqr-neg 94.2%

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

   8. sqrt-unprod 46.4%

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

   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. Taylor expanded in eh around 0 98.8%

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

   1. expm1-log1p-u 83.3%

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

   2. expm1-udef 70.4%

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

   3. add-sqr-sqrt 37.3%

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

   4. sqrt-unprod 64.6%

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

   5. sqr-neg 64.6%

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

   6. sqrt-unprod 27.4%

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

   7. add-sqr-sqrt 65.1%

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

8. Applied egg-rr 65.1%

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

   1. expm1-def 78.0%

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

   2. expm1-log1p 98.8%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

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

Alternative 8: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(ew * N[Cos[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. 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 53.4%

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

   6. sqrt-unprod 94.2%

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

   7. sqr-neg 94.2%

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

   8. sqrt-unprod 46.4%

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

   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. Taylor expanded in eh around 0 98.8%

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

7. Taylor expanded in t around 0 98.7%

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

   1. mul-1-neg 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 9: 79.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[((-N[(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. 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 53.4%

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

   6. sqrt-unprod 94.2%

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

   7. sqr-neg 94.2%

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

   8. sqrt-unprod 46.4%

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

   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. Taylor expanded in eh around 0 98.8%

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

7. Taylor expanded in t around 0 80.1%

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

8. Final simplification 80.1%

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

Reproduce

herbie shell --seed 2023332 
(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)))))))