Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.5s

Alternatives: 13

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, 0.9× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (log (exp (/ 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(((log(exp((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(((Math.log(Math.exp((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(((math.log(math.exp((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(log(exp(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(-tan(t))) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((log(exp((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[Log[N[Exp[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(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-log-exp 99.8%

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

   2. associate-/l* 99.8%

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

   3. associate-/r/ 99.8%

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

   4. add-sqr-sqrt 50.7%

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

   5. sqrt-unprod 92.8%

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

   6. sqr-neg 92.8%

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

   7. sqrt-unprod 49.1%

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

   8. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

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 (- (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(eh * Float64(-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 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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 99.2%

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

   1. associate-*r/ 31.6%

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

   2. mul-1-neg 31.6%

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

   3. *-commutative 31.6%

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

   4. distribute-rgt-neg-in 31.6%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 4: 98.1% accurate, 1.5× speedup

Math

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

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 * Float64(-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 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.0% 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 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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 98.7%

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

   1. add-log-exp 88.3%

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

   2. *-un-lft-identity 88.3%

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

   3. log-prod 88.3%

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

   4. metadata-eval 88.3%

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

   5. add-log-exp 98.7%

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

   6. add-sqr-sqrt 50.2%

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

   7. sqrt-unprod 97.6%

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

   8. sqr-neg 97.6%

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

   9. sqrt-unprod 48.5%

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

   10. add-sqr-sqrt 98.7%

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

6. Applied egg-rr 98.7%

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

   1. +-lft-identity 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 6: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.5 \cdot 10^{+96} \lor \neg \left(ew \leq 4.4 \cdot 10^{+77}\right):\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right) - eh \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.5e+96) (not (<= ew 4.4e+77)))
   (fabs (- (* ew (cos t)) (* (sin (atan (/ (* t (- eh)) ew))) (* t eh))))
   (fabs (- (* ew (cos (atan (/ (* eh (- (tan t))) ew)))) (* eh (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.5e+96) || !(ew <= 4.4e+77)) {
		tmp = fabs(((ew * cos(t)) - (sin(atan(((t * -eh) / ew))) * (t * eh))));
	} else {
		tmp = fabs(((ew * cos(atan(((eh * -tan(t)) / ew)))) - (eh * sin(t))));
	}
	return tmp;
}

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) :: tmp
    if ((ew <= (-1.5d+96)) .or. (.not. (ew <= 4.4d+77))) then
        tmp = abs(((ew * cos(t)) - (sin(atan(((t * -eh) / ew))) * (t * eh))))
    else
        tmp = abs(((ew * cos(atan(((eh * -tan(t)) / ew)))) - (eh * sin(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.5e+96) || !(ew <= 4.4e+77)) {
		tmp = Math.abs(((ew * Math.cos(t)) - (Math.sin(Math.atan(((t * -eh) / ew))) * (t * eh))));
	} else {
		tmp = Math.abs(((ew * Math.cos(Math.atan(((eh * -Math.tan(t)) / ew)))) - (eh * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.5e+96) or not (ew <= 4.4e+77):
		tmp = math.fabs(((ew * math.cos(t)) - (math.sin(math.atan(((t * -eh) / ew))) * (t * eh))))
	else:
		tmp = math.fabs(((ew * math.cos(math.atan(((eh * -math.tan(t)) / ew)))) - (eh * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.5e+96) || !(ew <= 4.4e+77))
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(sin(atan(Float64(Float64(t * Float64(-eh)) / ew))) * Float64(t * eh))));
	else
		tmp = abs(Float64(Float64(ew * cos(atan(Float64(Float64(eh * Float64(-tan(t))) / ew)))) - Float64(eh * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.5e+96) || ~((ew <= 4.4e+77)))
		tmp = abs(((ew * cos(t)) - (sin(atan(((t * -eh) / ew))) * (t * eh))));
	else
		tmp = abs(((ew * cos(atan(((eh * -tan(t)) / ew)))) - (eh * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.5e+96], N[Not[LessEqual[ew, 4.4e+77]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.5e96 or 4.4000000000000001e77 < ew

   1. Initial program 99.7%

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

      1. cos-atan 39.3%

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

      2. hypot-1-def 39.3%

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

      3. associate-/l* 39.3%

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

      4. associate-/r/ 39.3%

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

      5. add-sqr-sqrt 20.6%

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

      6. sqrt-unprod 39.3%

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

      7. sqr-neg 39.3%

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

      8. sqrt-unprod 18.7%

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

      9. add-sqr-sqrt 39.3%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in eh around 0 98.9%

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

   5. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 39.3%

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

      2. mul-1-neg 39.3%

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

      3. *-commutative 39.3%

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

      4. distribute-rgt-neg-in 39.3%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in t around 0 84.5%

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

   if -1.5e96 < ew < 4.4000000000000001e77

   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 88.1%

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

      1. sin-atan 53.6%

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

      2. associate-*r/ 53.3%

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

      3. associate-/l* 53.1%

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

      4. add-sqr-sqrt 27.6%

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

      5. sqrt-unprod 44.5%

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

      6. sqr-neg 44.5%

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

      7. sqrt-unprod 25.4%

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

      8. add-sqr-sqrt 52.7%

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

      9. associate-/l* 52.9%

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

      10. associate-*l/ 49.7%

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

      11. hypot-1-def 56.3%

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

      12. associate-/l* 56.3%

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

      13. add-sqr-sqrt 28.7%

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

   4. Applied egg-rr 56.2%

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

      1. associate-*l* 56.1%

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

      2. associate-*r* 56.0%

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

      3. *-commutative 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in eh around inf 87.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.5 \cdot 10^{+96} \lor \neg \left(ew \leq 4.4 \cdot 10^{+77}\right):\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right) - eh \cdot \sin t\right|\\ \end{array} \]

Alternative 7: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.14 \cdot 10^{+95} \lor \neg \left(ew \leq 4.05 \cdot 10^{+77}\right):\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.14e+95) (not (<= ew 4.05e+77)))
   (fabs (- (* ew (cos t)) (* (sin (atan (/ (* t (- eh)) ew))) (* t eh))))
   (fabs (+ (* eh (sin t)) (* ew (cos (atan (/ (* eh (- (tan t))) ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.14e+95) || !(ew <= 4.05e+77)) {
		tmp = fabs(((ew * cos(t)) - (sin(atan(((t * -eh) / ew))) * (t * eh))));
	} else {
		tmp = fabs(((eh * sin(t)) + (ew * cos(atan(((eh * -tan(t)) / ew))))));
	}
	return tmp;
}

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) :: tmp
    if ((ew <= (-1.14d+95)) .or. (.not. (ew <= 4.05d+77))) then
        tmp = abs(((ew * cos(t)) - (sin(atan(((t * -eh) / ew))) * (t * eh))))
    else
        tmp = abs(((eh * sin(t)) + (ew * cos(atan(((eh * -tan(t)) / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.14e+95) || !(ew <= 4.05e+77)) {
		tmp = Math.abs(((ew * Math.cos(t)) - (Math.sin(Math.atan(((t * -eh) / ew))) * (t * eh))));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) + (ew * Math.cos(Math.atan(((eh * -Math.tan(t)) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.14e+95) or not (ew <= 4.05e+77):
		tmp = math.fabs(((ew * math.cos(t)) - (math.sin(math.atan(((t * -eh) / ew))) * (t * eh))))
	else:
		tmp = math.fabs(((eh * math.sin(t)) + (ew * math.cos(math.atan(((eh * -math.tan(t)) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.14e+95) || !(ew <= 4.05e+77))
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(sin(atan(Float64(Float64(t * Float64(-eh)) / ew))) * Float64(t * eh))));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) + Float64(ew * cos(atan(Float64(Float64(eh * Float64(-tan(t))) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.14e+95) || ~((ew <= 4.05e+77)))
		tmp = abs(((ew * cos(t)) - (sin(atan(((t * -eh) / ew))) * (t * eh))));
	else
		tmp = abs(((eh * sin(t)) + (ew * cos(atan(((eh * -tan(t)) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.14e+95], N[Not[LessEqual[ew, 4.05e+77]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(ew * N[Cos[N[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.14e95 or 4.04999999999999976e77 < ew

   1. Initial program 99.7%

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

      1. cos-atan 39.3%

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

      2. hypot-1-def 39.3%

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

      3. associate-/l* 39.3%

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

      4. associate-/r/ 39.3%

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

      5. add-sqr-sqrt 20.6%

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

      6. sqrt-unprod 39.3%

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

      7. sqr-neg 39.3%

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

      8. sqrt-unprod 18.7%

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

      9. add-sqr-sqrt 39.3%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in eh around 0 98.9%

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

   5. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 39.3%

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

      2. mul-1-neg 39.3%

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

      3. *-commutative 39.3%

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

      4. distribute-rgt-neg-in 39.3%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in t around 0 84.5%

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

   if -1.14e95 < ew < 4.04999999999999976e77

   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 88.1%

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

      1. sin-atan 53.6%

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

      2. associate-*r/ 53.3%

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

      3. associate-/l* 53.1%

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

      4. add-sqr-sqrt 27.6%

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

      5. sqrt-unprod 44.5%

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

      6. sqr-neg 44.5%

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

      7. sqrt-unprod 25.4%

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

      8. add-sqr-sqrt 52.7%

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

      9. associate-/l* 52.9%

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

      10. associate-*l/ 49.7%

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

      11. hypot-1-def 56.3%

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

      12. associate-/l* 56.3%

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

      13. add-sqr-sqrt 28.7%

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

   4. Applied egg-rr 56.2%

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

      1. associate-*l* 56.1%

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

      2. associate-*r* 56.0%

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

      3. *-commutative 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in eh around -inf 87.5%

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

      1. mul-1-neg 87.5%

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

      2. distribute-rgt-neg-in 87.5%

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

   9. Simplified 87.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.14 \cdot 10^{+95} \lor \neg \left(ew \leq 4.05 \cdot 10^{+77}\right):\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right|\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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 98.7%

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

5. Taylor expanded in t around 0 98.4%

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

   1. associate-*r/ 31.6%

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

   2. mul-1-neg 31.6%

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

   3. *-commutative 31.6%

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

   4. distribute-rgt-neg-in 31.6%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 9: 97.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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 98.7%

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

5. Taylor expanded in t around 0 98.4%

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

   1. associate-*r/ 31.6%

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

   2. mul-1-neg 31.6%

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

   3. *-commutative 31.6%

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

   4. distribute-rgt-neg-in 31.6%

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

7. Simplified 98.4%

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

   1. expm1-log1p-u 79.6%

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

   2. expm1-udef 68.9%

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

   3. associate-/l* 69.3%

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

   4. add-sqr-sqrt 37.3%

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

   5. sqrt-unprod 69.0%

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

   6. sqr-neg 69.0%

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

   7. sqrt-unprod 32.9%

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

   8. add-sqr-sqrt 70.8%

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

9. Applied egg-rr 70.8%

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

    1. expm1-def 81.6%

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

    2. expm1-log1p 98.4%

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

    3. *-commutative 98.4%

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

    4. associate-/r/ 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 10: 83.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\\ \mathbf{if}\;ew \leq -3.6 \cdot 10^{+95} \lor \neg \left(ew \leq 7.2 \cdot 10^{+76}\right):\\ \;\;\;\;\left|ew \cdot \cos t - t_1 \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot t_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (* t (- eh)) ew)))))
   (if (or (<= ew -3.6e+95) (not (<= ew 7.2e+76)))
     (fabs (- (* ew (cos t)) (* t_1 (* t eh))))
     (fabs (- ew (* (* eh (sin t)) t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((t * -eh) / ew)));
	double tmp;
	if ((ew <= -3.6e+95) || !(ew <= 7.2e+76)) {
		tmp = fabs(((ew * cos(t)) - (t_1 * (t * eh))));
	} else {
		tmp = fabs((ew - ((eh * sin(t)) * t_1)));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_1 = sin(atan(((t * -eh) / ew)))
    if ((ew <= (-3.6d+95)) .or. (.not. (ew <= 7.2d+76))) then
        tmp = abs(((ew * cos(t)) - (t_1 * (t * eh))))
    else
        tmp = abs((ew - ((eh * sin(t)) * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan(((t * -eh) / ew)));
	double tmp;
	if ((ew <= -3.6e+95) || !(ew <= 7.2e+76)) {
		tmp = Math.abs(((ew * Math.cos(t)) - (t_1 * (t * eh))));
	} else {
		tmp = Math.abs((ew - ((eh * Math.sin(t)) * t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((t * -eh) / ew)))
	tmp = 0
	if (ew <= -3.6e+95) or not (ew <= 7.2e+76):
		tmp = math.fabs(((ew * math.cos(t)) - (t_1 * (t * eh))))
	else:
		tmp = math.fabs((ew - ((eh * math.sin(t)) * t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(t * Float64(-eh)) / ew)))
	tmp = 0.0
	if ((ew <= -3.6e+95) || !(ew <= 7.2e+76))
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(t_1 * Float64(t * eh))));
	else
		tmp = abs(Float64(ew - Float64(Float64(eh * sin(t)) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((t * -eh) / ew)));
	tmp = 0.0;
	if ((ew <= -3.6e+95) || ~((ew <= 7.2e+76)))
		tmp = abs(((ew * cos(t)) - (t_1 * (t * eh))));
	else
		tmp = abs((ew - ((eh * sin(t)) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -3.6e+95], N[Not[LessEqual[ew, 7.2e+76]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.59999999999999978e95 or 7.2000000000000006e76 < ew

   1. Initial program 99.7%

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

      1. cos-atan 39.3%

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

      2. hypot-1-def 39.3%

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

      3. associate-/l* 39.3%

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

      4. associate-/r/ 39.3%

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

      5. add-sqr-sqrt 20.6%

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

      6. sqrt-unprod 39.3%

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

      7. sqr-neg 39.3%

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

      8. sqrt-unprod 18.7%

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

      9. add-sqr-sqrt 39.3%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in eh around 0 98.9%

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

   5. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 39.3%

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

      2. mul-1-neg 39.3%

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

      3. *-commutative 39.3%

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

      4. distribute-rgt-neg-in 39.3%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in t around 0 84.5%

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

   if -3.59999999999999978e95 < ew < 7.2000000000000006e76

   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 27.3%

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

      2. hypot-1-def 27.3%

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

      3. associate-/l* 27.3%

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

      4. associate-/r/ 27.3%

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

      5. add-sqr-sqrt 13.2%

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

      6. sqrt-unprod 27.3%

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

      7. sqr-neg 27.3%

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

      8. sqrt-unprod 14.1%

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

      9. add-sqr-sqrt 27.3%

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

   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 98.5%

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

   5. Taylor expanded in t around 0 98.3%

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

      1. associate-*r/ 27.2%

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

      2. mul-1-neg 27.2%

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

      3. *-commutative 27.2%

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

      4. distribute-rgt-neg-in 27.2%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in t around 0 87.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.6 \cdot 10^{+95} \lor \neg \left(ew \leq 7.2 \cdot 10^{+76}\right):\\ \;\;\;\;\left|ew \cdot \cos t - \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \end{array} \]

Alternative 11: 78.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 31.7%

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

   2. hypot-1-def 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-/r/ 31.7%

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

   5. add-sqr-sqrt 15.9%

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

   6. sqrt-unprod 31.7%

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

   7. sqr-neg 31.7%

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

   8. sqrt-unprod 15.8%

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

   9. add-sqr-sqrt 31.7%

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

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 98.7%

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

5. Taylor expanded in t around 0 98.4%

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

   1. associate-*r/ 31.6%

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

   2. mul-1-neg 31.6%

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

   3. *-commutative 31.6%

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

   4. distribute-rgt-neg-in 31.6%

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

7. Simplified 98.4%

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

8. Taylor expanded in t around 0 77.0%

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

9. Final simplification 77.0%

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

Alternative 12: 34.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 77.7%

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

   1. sin-atan 55.9%

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

   2. associate-*r/ 54.6%

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

   3. associate-/l* 54.5%

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

   4. add-sqr-sqrt 27.5%

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

   5. sqrt-unprod 44.0%

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

   6. sqr-neg 44.0%

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

   7. sqrt-unprod 26.9%

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

   8. add-sqr-sqrt 54.3%

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

   9. associate-/l* 54.4%

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

   10. associate-*l/ 52.4%

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

   11. hypot-1-def 56.5%

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

   12. associate-/l* 56.5%

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

   13. add-sqr-sqrt 28.2%

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

4. Applied egg-rr 56.5%

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

   1. associate-*l* 56.4%

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

   2. associate-*r* 56.4%

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

   3. *-commutative 56.4%

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

6. Simplified 56.4%

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

7. Taylor expanded in t around 0 31.8%

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

   1. associate-/l* 31.7%

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

   2. unpow2 31.7%

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

   3. unpow2 31.7%

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

9. Simplified 31.7%

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

    1. cos-atan 31.7%

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

    2. hypot-1-def 31.7%

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

    3. associate-/l* 31.7%

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

    4. associate-/r/ 31.7%

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

    5. add-sqr-sqrt 15.9%

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

    6. sqrt-unprod 31.7%

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

    7. sqr-neg 31.7%

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

    8. sqrt-unprod 15.8%

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

    9. add-sqr-sqrt 31.7%

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

11. Applied egg-rr 31.7%

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

12. Final simplification 31.7%

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

Alternative 13: 34.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(atan(((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[(t * (-eh)), $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 77.7%

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

   1. sin-atan 55.9%

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

   2. associate-*r/ 54.6%

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

   3. associate-/l* 54.5%

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

   4. add-sqr-sqrt 27.5%

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

   5. sqrt-unprod 44.0%

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

   6. sqr-neg 44.0%

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

   7. sqrt-unprod 26.9%

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

   8. add-sqr-sqrt 54.3%

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

   9. associate-/l* 54.4%

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

   10. associate-*l/ 52.4%

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

   11. hypot-1-def 56.5%

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

   12. associate-/l* 56.5%

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

   13. add-sqr-sqrt 28.2%

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

4. Applied egg-rr 56.5%

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

   1. associate-*l* 56.4%

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

   2. associate-*r* 56.4%

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

   3. *-commutative 56.4%

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

6. Simplified 56.4%

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

7. Taylor expanded in t around 0 31.8%

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

   1. associate-/l* 31.7%

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

   2. unpow2 31.7%

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

   3. unpow2 31.7%

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

9. Simplified 31.7%

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

10. Taylor expanded in t around 0 31.6%

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

    1. associate-*r/ 31.6%

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

    2. mul-1-neg 31.6%

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

    3. *-commutative 31.6%

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

    4. distribute-rgt-neg-in 31.6%

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

12. Simplified 31.6%

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

13. Final simplification 31.6%

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

Reproduce

herbie shell --seed 2023207 
(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)))))))