Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(ew * N[(N[Cos[t$95$1], $MachinePrecision] * (-N[Cos[t], $MachinePrecision])), $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $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. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. distribute-neg-frac 99.8%

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

4. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-rgt-neg-out 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. associate-*r/ 99.8%

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

   5. cos-atan 99.8%

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

   6. hypot-1-def 99.8%

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

   7. add-sqr-sqrt 52.6%

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

   8. sqrt-unprod 96.3%

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

   9. sqr-neg 96.3%

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

   10. sqrt-unprod 47.2%

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

   11. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. distribute-neg-frac 99.8%

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

4. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-rgt-neg-out 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. associate-*r/ 99.8%

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

   5. cos-atan 99.8%

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

   6. hypot-1-def 99.8%

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

   7. add-sqr-sqrt 52.6%

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

   8. sqrt-unprod 96.3%

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

   9. sqr-neg 96.3%

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

   10. sqrt-unprod 47.2%

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

   11. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. add-sqr-sqrt 52.6%

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

   5. sqrt-unprod 94.3%

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

   6. sqr-neg 94.3%

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

   7. sqrt-unprod 47.1%

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

   8. add-sqr-sqrt 98.8%

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

   9. associate-*r/ 98.8%

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

   10. frac-2neg 98.8%

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

   11. distribute-lft-neg-out 98.8%

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

   12. add-sqr-sqrt 51.7%

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

   13. sqrt-unprod 93.7%

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

   14. sqr-neg 93.7%

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

   15. sqrt-unprod 47.2%

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

   16. add-sqr-sqrt 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-/l* 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(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. Taylor expanded in t around 0 98.9%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

   3. associate-*r* 59.5%

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

   4. neg-mul-1 59.5%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan t \cdot \frac{eh}{ew}\\ \mathbf{if}\;eh \leq -1.28 \cdot 10^{+50}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} t_1 - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 42000000000:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot eh\right)}^{3}}}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, t_1\right)} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tan t) (/ eh ew))))
   (if (<= eh -1.28e+50)
     (fabs
      (-
       (* ew (cos (atan t_1)))
       (* (* eh (sin t)) (sin (atan (/ (* (tan t) (- eh)) ew))))))
     (if (<= eh 42000000000.0)
       (fabs
        (* (* ew (cos t)) (cos (atan (/ (cbrt (pow (* (tan t) eh) 3.0)) ew)))))
       (fabs
        (-
         (* ew (/ 1.0 (hypot 1.0 t_1)))
         (* eh (* (sin t) (sin (atan (* (tan t) (/ (- eh) ew))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) * (eh / ew);
	double tmp;
	if (eh <= -1.28e+50) {
		tmp = fabs(((ew * cos(atan(t_1))) - ((eh * sin(t)) * sin(atan(((tan(t) * -eh) / ew))))));
	} else if (eh <= 42000000000.0) {
		tmp = fabs(((ew * cos(t)) * cos(atan((cbrt(pow((tan(t) * eh), 3.0)) / ew)))));
	} else {
		tmp = fabs(((ew * (1.0 / hypot(1.0, t_1))) - (eh * (sin(t) * sin(atan((tan(t) * (-eh / ew))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.tan(t) * (eh / ew);
	double tmp;
	if (eh <= -1.28e+50) {
		tmp = Math.abs(((ew * Math.cos(Math.atan(t_1))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((Math.tan(t) * -eh) / ew))))));
	} else if (eh <= 42000000000.0) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(Math.atan((Math.cbrt(Math.pow((Math.tan(t) * eh), 3.0)) / ew)))));
	} else {
		tmp = Math.abs(((ew * (1.0 / Math.hypot(1.0, t_1))) - (eh * (Math.sin(t) * Math.sin(Math.atan((Math.tan(t) * (-eh / ew))))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) * Float64(eh / ew))
	tmp = 0.0
	if (eh <= -1.28e+50)
		tmp = abs(Float64(Float64(ew * cos(atan(t_1))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(tan(t) * Float64(-eh)) / ew))))));
	elseif (eh <= 42000000000.0)
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(atan(Float64(cbrt((Float64(tan(t) * eh) ^ 3.0)) / ew)))));
	else
		tmp = abs(Float64(Float64(ew * Float64(1.0 / hypot(1.0, t_1))) - Float64(eh * Float64(sin(t) * sin(atan(Float64(tan(t) * Float64(Float64(-eh) / ew))))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -1.28e+50], N[Abs[N[(N[(ew * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 42000000000.0], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[Power[N[Power[N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.28000000000000006e50

   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. distribute-lft-neg-out 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. associate-*l/ 99.8%

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

      4. add-sqr-sqrt 57.2%

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

      5. sqrt-unprod 90.5%

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

      6. sqr-neg 90.5%

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

      7. sqrt-unprod 42.5%

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

      8. add-sqr-sqrt 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in t around 0 91.5%

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

   if -1.28000000000000006e50 < eh < 4.2e10

   1. Initial program 99.8%

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

   3. Applied egg-rr 87.8%

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

      1. +-inverses 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

      4. div0 87.8%

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

   5. Simplified 87.8%

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

      1. add-cbrt-cube 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{\sqrt[3]{\left(\left(\left(-eh\right) \cdot \tan t\right) \cdot \left(\left(-eh\right) \cdot \tan t\right)\right) \cdot \left(\left(-eh\right) \cdot \tan t\right)}}}{ew}\right) - 0\right| \]

      2. pow3 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{\color{blue}{{\left(\left(-eh\right) \cdot \tan t\right)}^{3}}}}{ew}\right) - 0\right| \]

      3. *-commutative 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\color{blue}{\left(\tan t \cdot \left(-eh\right)\right)}}^{3}}}{ew}\right) - 0\right| \]

      4. add-sqr-sqrt 44.2%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)}\right)}^{3}}}{ew}\right) - 0\right| \]

      5. sqrt-unprod 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}\right)}^{3}}}{ew}\right) - 0\right| \]

      6. sqr-neg 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \sqrt{\color{blue}{eh \cdot eh}}\right)}^{3}}}{ew}\right) - 0\right| \]

      7. sqrt-unprod 43.6%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)}\right)}^{3}}}{ew}\right) - 0\right| \]

      8. add-sqr-sqrt 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{eh}\right)}^{3}}}{ew}\right) - 0\right| \]

   7. Applied egg-rr 87.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{\sqrt[3]{{\left(\tan t \cdot eh\right)}^{3}}}}{ew}\right) - 0\right| \]

   if 4.2e10 < eh

   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 eh around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

   4. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. associate-*r/ 99.8%

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

      5. cos-atan 99.8%

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

      6. hypot-1-def 99.8%

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

      7. add-sqr-sqrt 55.4%

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

      8. sqrt-unprod 92.4%

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

      9. sqr-neg 92.4%

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

      10. sqrt-unprod 44.4%

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

      11. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 95.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.28 \cdot 10^{+50}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 42000000000:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot eh\right)}^{3}}}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.75 \cdot 10^{+50} \lor \neg \left(eh \leq 126000000000\right):\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot eh\right)}^{3}}}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.75e+50) (not (<= eh 126000000000.0)))
   (fabs
    (-
     (* ew (/ 1.0 (hypot 1.0 (* (tan t) (/ eh ew)))))
     (* eh (* (sin t) (sin (atan (* (tan t) (/ (- eh) ew))))))))
   (fabs
    (* (* ew (cos t)) (cos (atan (/ (cbrt (pow (* (tan t) eh) 3.0)) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.75e+50) || !(eh <= 126000000000.0)) {
		tmp = fabs(((ew * (1.0 / hypot(1.0, (tan(t) * (eh / ew))))) - (eh * (sin(t) * sin(atan((tan(t) * (-eh / ew))))))));
	} else {
		tmp = fabs(((ew * cos(t)) * cos(atan((cbrt(pow((tan(t) * eh), 3.0)) / ew)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.75e+50) || !(eh <= 126000000000.0)) {
		tmp = Math.abs(((ew * (1.0 / Math.hypot(1.0, (Math.tan(t) * (eh / ew))))) - (eh * (Math.sin(t) * Math.sin(Math.atan((Math.tan(t) * (-eh / ew))))))));
	} else {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(Math.atan((Math.cbrt(Math.pow((Math.tan(t) * eh), 3.0)) / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.75e+50) || !(eh <= 126000000000.0))
		tmp = abs(Float64(Float64(ew * Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew))))) - Float64(eh * Float64(sin(t) * sin(atan(Float64(tan(t) * Float64(Float64(-eh) / ew))))))));
	else
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(atan(Float64(cbrt((Float64(tan(t) * eh) ^ 3.0)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.75e+50], N[Not[LessEqual[eh, 126000000000.0]], $MachinePrecision]], N[Abs[N[(N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[Power[N[Power[N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.75e50 or 1.26e11 < eh

   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 eh around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

   4. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. associate-*r/ 99.8%

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

      5. cos-atan 99.8%

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

      6. hypot-1-def 99.8%

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

      7. add-sqr-sqrt 56.3%

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

      8. sqrt-unprod 91.4%

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

      9. sqr-neg 91.4%

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

      10. sqrt-unprod 43.4%

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

      11. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 93.6%

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

   if -3.75e50 < eh < 1.26e11

   1. Initial program 99.8%

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

   3. Applied egg-rr 87.8%

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

      1. +-inverses 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. *-commutative 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

      3. associate-/l* 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

      4. div0 87.8%

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

   5. Simplified 87.8%

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

      1. add-cbrt-cube 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{\sqrt[3]{\left(\left(\left(-eh\right) \cdot \tan t\right) \cdot \left(\left(-eh\right) \cdot \tan t\right)\right) \cdot \left(\left(-eh\right) \cdot \tan t\right)}}}{ew}\right) - 0\right| \]

      2. pow3 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{\color{blue}{{\left(\left(-eh\right) \cdot \tan t\right)}^{3}}}}{ew}\right) - 0\right| \]

      3. *-commutative 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\color{blue}{\left(\tan t \cdot \left(-eh\right)\right)}}^{3}}}{ew}\right) - 0\right| \]

      4. add-sqr-sqrt 44.2%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)}\right)}^{3}}}{ew}\right) - 0\right| \]

      5. sqrt-unprod 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}\right)}^{3}}}{ew}\right) - 0\right| \]

      6. sqr-neg 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \sqrt{\color{blue}{eh \cdot eh}}\right)}^{3}}}{ew}\right) - 0\right| \]

      7. sqrt-unprod 43.6%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)}\right)}^{3}}}{ew}\right) - 0\right| \]

      8. add-sqr-sqrt 87.8%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot \color{blue}{eh}\right)}^{3}}}{ew}\right) - 0\right| \]

   7. Applied egg-rr 87.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{\sqrt[3]{{\left(\tan t \cdot eh\right)}^{3}}}}{ew}\right) - 0\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.75 \cdot 10^{+50} \lor \neg \left(eh \leq 126000000000\right):\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\sqrt[3]{{\left(\tan t \cdot eh\right)}^{3}}}{ew}\right)\right|\\ \end{array} \]

Alternative 7: 62.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) * (cbrt(cos(atan(Float64(tan(t) * Float64(eh / ew))))) ^ 3.0)))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

   1. add-cube-cbrt 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt[3]{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right) \cdot \sqrt[3]{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right)} - 0\right| \]

   2. pow3 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right)}^{3}} - 0\right| \]

   3. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\color{blue}{\tan t \cdot \left(-eh\right)}}{ew}\right)}\right)}^{3} - 0\right| \]

   4. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \color{blue}{\left(\frac{\tan t}{\frac{ew}{-eh}}\right)}}\right)}^{3} - 0\right| \]

   5. add-sqr-sqrt 34.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{-eh} \cdot \sqrt{-eh}}}}\right)}\right)}^{3} - 0\right| \]

   6. sqrt-unprod 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}}}\right)}\right)}^{3} - 0\right| \]

   7. sqr-neg 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\sqrt{\color{blue}{eh \cdot eh}}}}\right)}\right)}^{3} - 0\right| \]

   8. sqrt-unprod 34.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{eh} \cdot \sqrt{eh}}}}\right)}\right)}^{3} - 0\right| \]

   9. add-sqr-sqrt 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{eh}}}\right)}\right)}^{3} - 0\right| \]

   10. un-div-inv 68.8%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \color{blue}{\left(\tan t \cdot \frac{1}{\frac{ew}{eh}}\right)}}\right)}^{3} - 0\right| \]

   11. clear-num 68.8%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot {\left(\sqrt[3]{\cos \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{eh}{ew}}\right)}\right)}^{3} - 0\right| \]

7. Applied egg-rr 68.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)}\right)}^{3}} - 0\right| \]

8. Final simplification 68.8%

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

Alternative 8: 62.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) * expm1(log1p(cos(atan(Float64(tan(t) * Float64(eh / ew))))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

   1. expm1-log1p-u 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)} - 0\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\color{blue}{\tan t \cdot \left(-eh\right)}}{ew}\right)\right)\right) - 0\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\frac{\tan t}{\frac{ew}{-eh}}\right)}\right)\right) - 0\right| \]

   4. add-sqr-sqrt 34.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{-eh} \cdot \sqrt{-eh}}}}\right)\right)\right) - 0\right| \]

   5. sqrt-unprod 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}}}\right)\right)\right) - 0\right| \]

   6. sqr-neg 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\sqrt{\color{blue}{eh \cdot eh}}}}\right)\right)\right) - 0\right| \]

   7. sqrt-unprod 34.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{eh} \cdot \sqrt{eh}}}}\right)\right)\right) - 0\right| \]

   8. add-sqr-sqrt 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{eh}}}\right)\right)\right) - 0\right| \]

   9. un-div-inv 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\tan t \cdot \frac{1}{\frac{ew}{eh}}\right)}\right)\right) - 0\right| \]

   10. clear-num 68.8%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{eh}{ew}}\right)\right)\right) - 0\right| \]

7. Applied egg-rr 68.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)} - 0\right| \]

8. Final simplification 68.8%

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

Alternative 9: 62.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

   1. expm1-log1p-u 61.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-eh\right) \cdot \tan t\right)\right)}}{ew}\right) - 0\right| \]

   2. expm1-udef 61.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-eh\right) \cdot \tan t\right)} - 1}}{ew}\right) - 0\right| \]

   3. *-commutative 61.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\tan t \cdot \left(-eh\right)}\right)} - 1}{ew}\right) - 0\right| \]

   4. add-sqr-sqrt 28.6%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\tan t \cdot \color{blue}{\left(\sqrt{-eh} \cdot \sqrt{-eh}\right)}\right)} - 1}{ew}\right) - 0\right| \]

   5. sqrt-unprod 56.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\tan t \cdot \color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}\right)} - 1}{ew}\right) - 0\right| \]

   6. sqr-neg 56.5%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\tan t \cdot \sqrt{\color{blue}{eh \cdot eh}}\right)} - 1}{ew}\right) - 0\right| \]

   7. sqrt-unprod 31.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\tan t \cdot \color{blue}{\left(\sqrt{eh} \cdot \sqrt{eh}\right)}\right)} - 1}{ew}\right) - 0\right| \]

   8. add-sqr-sqrt 63.3%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{e^{\mathsf{log1p}\left(\tan t \cdot \color{blue}{eh}\right)} - 1}{ew}\right) - 0\right| \]

7. Applied egg-rr 63.3%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\tan t \cdot eh\right)} - 1}}{ew}\right) - 0\right| \]
8. Step-by-step derivation

   1. expm1-def 63.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan t \cdot eh\right)\right)}}{ew}\right) - 0\right| \]

   2. expm1-log1p 68.8%

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

   3. *-commutative 68.8%

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

9. Simplified 68.8%

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

10. Final simplification 68.8%

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

Alternative 10: 52.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

6. Taylor expanded in t around 0 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

   3. associate-*r* 59.5%

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

   4. neg-mul-1 59.5%

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

8. Simplified 59.5%

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

9. Final simplification 59.5%

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

Alternative 11: 61.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

   1. expm1-log1p-u 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)} - 0\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\color{blue}{\tan t \cdot \left(-eh\right)}}{ew}\right)\right)\right) - 0\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\frac{\tan t}{\frac{ew}{-eh}}\right)}\right)\right) - 0\right| \]

   4. add-sqr-sqrt 34.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{-eh} \cdot \sqrt{-eh}}}}\right)\right)\right) - 0\right| \]

   5. sqrt-unprod 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}}}\right)\right)\right) - 0\right| \]

   6. sqr-neg 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\sqrt{\color{blue}{eh \cdot eh}}}}\right)\right)\right) - 0\right| \]

   7. sqrt-unprod 34.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{eh} \cdot \sqrt{eh}}}}\right)\right)\right) - 0\right| \]

   8. add-sqr-sqrt 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{eh}}}\right)\right)\right) - 0\right| \]

   9. un-div-inv 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\tan t \cdot \frac{1}{\frac{ew}{eh}}\right)}\right)\right) - 0\right| \]

   10. clear-num 68.8%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{eh}{ew}}\right)\right)\right) - 0\right| \]

7. Applied egg-rr 68.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)} - 0\right| \]

9. Applied egg-rr 23.9%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} \cdot \cos t\right)} - 1\right)} - 0\right| \]
10. Step-by-step derivation

    1. expm1-def 41.7%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} \cdot \cos t\right)\right)} - 0\right| \]

    2. expm1-log1p 68.5%

       \[\leadsto \left|\color{blue}{\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} \cdot \cos t} - 0\right| \]

    3. associate-/r/ 68.5%

       \[\leadsto \left|\color{blue}{\frac{ew}{\frac{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}{\cos t}}} - 0\right| \]

11. Simplified 68.5%

    \[\leadsto \left|\color{blue}{\frac{ew}{\frac{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}{\cos t}}} - 0\right| \]

12. Final simplification 68.5%

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

Alternative 12: 61.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

   1. expm1-log1p-u 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)} - 0\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\color{blue}{\tan t \cdot \left(-eh\right)}}{ew}\right)\right)\right) - 0\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\frac{\tan t}{\frac{ew}{-eh}}\right)}\right)\right) - 0\right| \]

   4. add-sqr-sqrt 34.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{-eh} \cdot \sqrt{-eh}}}}\right)\right)\right) - 0\right| \]

   5. sqrt-unprod 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{\left(-eh\right) \cdot \left(-eh\right)}}}}\right)\right)\right) - 0\right| \]

   6. sqr-neg 62.4%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\sqrt{\color{blue}{eh \cdot eh}}}}\right)\right)\right) - 0\right| \]

   7. sqrt-unprod 34.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{\sqrt{eh} \cdot \sqrt{eh}}}}\right)\right)\right) - 0\right| \]

   8. add-sqr-sqrt 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{\color{blue}{eh}}}\right)\right)\right) - 0\right| \]

   9. un-div-inv 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \color{blue}{\left(\tan t \cdot \frac{1}{\frac{ew}{eh}}\right)}\right)\right) - 0\right| \]

   10. clear-num 68.8%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{eh}{ew}}\right)\right)\right) - 0\right| \]

7. Applied egg-rr 68.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)} - 0\right| \]
8. Step-by-step derivation

   1. expm1-log1p-u 68.8%

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

   2. cos-atan 68.5%

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

   3. un-div-inv 68.5%

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

   4. hypot-1-def 68.5%

      \[\leadsto \left|\frac{ew \cdot \cos t}{\color{blue}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}} - 0\right| \]

9. Applied egg-rr 68.5%

   \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}} - 0\right| \]

10. Final simplification 68.5%

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

Alternative 13: 42.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|ew\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs ew))

C

double code(double eh, double ew, double t) {
	return fabs(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)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(ew);
}

Python

def code(eh, ew, t):
	return math.fabs(ew)

Julia

function code(eh, ew, t)
	return abs(ew)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(ew);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 68.8%

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

   1. +-inverses 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. *-commutative 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \frac{\color{blue}{0 \cdot eh}}{2}\right| \]

   3. associate-/l* 68.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\frac{0}{\frac{2}{eh}}}\right| \]

   4. div0 68.8%

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

5. Simplified 68.8%

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

6. Taylor expanded in t around 0 47.1%

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

   1. add-cbrt-cube 19.7%

      \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}} - 0\right| \]

   2. pow1/3 10.4%

      \[\leadsto \left|\color{blue}{{\left(\left(\left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}^{0.3333333333333333}} - 0\right| \]

8. Applied egg-rr 10.3%

   \[\leadsto \left|\color{blue}{{\left({\left(\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}^{3}\right)}^{0.3333333333333333}} - 0\right| \]

9. Taylor expanded in ew around inf 47.3%

   \[\leadsto \left|\color{blue}{ew} - 0\right| \]

10. Final simplification 47.3%

    \[\leadsto \left|ew\right| \]

Reproduce

herbie shell --seed 2023293 
(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)))))))