Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 26.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. add-sqr-sqrt 47.6%

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

   2. sqrt-unprod 96.7%

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

   3. sqr-neg 96.7%

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

   4. sqrt-unprod 52.2%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(eh, ew, t)
	return abs(fma(ew, Float64(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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Cos[t], $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. 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. Step-by-step derivation

   1. add-sqr-sqrt 47.6%

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

   2. sqrt-unprod 96.7%

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

   3. sqr-neg 96.7%

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

   4. sqrt-unprod 52.2%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\left({\left(\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)\right)}^{-0.5}\right)} \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)\right)\right)\right| \]
8. Step-by-step derivation

   1. pow-sqr 99.8%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{{\left(\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)\right)}^{\left(2 \cdot -0.5\right)}} \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)\right)\right)\right| \]

   2. metadata-eval 99.8%

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

   3. unpow-1 99.8%

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

   4. associate-*r/ 99.8%

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

   5. associate-*l/ 99.8%

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

   6. *-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-rgt-neg-out 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. associate-*r/ 99.8%

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

   5. expm1-log1p-u 99.8%

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

   6. expm1-udef 99.8%

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

3. Applied egg-rr 99.8%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) * (1.0 / math.hypot(1.0, (math.tan(t) / (ew / eh))))) - ((eh * math.sin(t)) * math.sin(math.atan(((eh * -math.tan(t)) / 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(ew / eh))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew))))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-out 99.8%

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

   5. distribute-lft-neg-out 99.8%

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

   6. associate-*r/ 99.8%

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

   7. add-sqr-sqrt 47.6%

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

   8. sqrt-unprod 96.7%

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

   9. sqr-neg 96.7%

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

   10. sqrt-unprod 52.2%

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

   11. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 99.2%

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

   1. associate-*r/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-*r* 98.8%

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

   4. neg-mul-1 98.8%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 6: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 78.7%

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

   3. associate-*r/ 75.5%

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

   4. add-sqr-sqrt 37.9%

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

   5. sqrt-unprod 74.5%

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

   6. sqr-neg 74.5%

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

   7. sqrt-unprod 37.3%

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

   8. add-sqr-sqrt 75.2%

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

   9. hypot-1-def 77.8%

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

5. Applied egg-rr 77.8%

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

   1. associate-/l* 85.5%

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

   2. associate-/r/ 85.4%

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

   3. associate-*r/ 85.5%

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

   4. associate-/l* 85.4%

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

   5. associate-*r/ 86.1%

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

   6. associate-/l* 86.2%

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

7. Simplified 86.2%

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

   1. expm1-log1p-u 86.2%

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

   2. expm1-udef 86.0%

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

9. Applied egg-rr 86.4%

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

    1. expm1-def 86.5%

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

    2. expm1-log1p 86.6%

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

    3. associate-*r/ 86.6%

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

    4. associate-*l/ 86.6%

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

    5. *-commutative 86.6%

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

11. Simplified 86.6%

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

12. Taylor expanded in eh around inf 99.1%

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

13. Final simplification 99.1%

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

Alternative 7: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -8.2 \cdot 10^{-143} \lor \neg \left(eh \leq 2.05 \cdot 10^{-54}\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -8.2e-143) (not (<= eh 2.05e-54)))
   (fabs (- ew (* (* eh (sin t)) (sin (atan (/ (* eh (- (tan t))) ew))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.2e-143) || !(eh <= 2.05e-54)) {
		tmp = fabs((ew - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-8.2d-143)) .or. (.not. (eh <= 2.05d-54))) then
        tmp = abs((ew - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.2e-143) || !(eh <= 2.05e-54)) {
		tmp = Math.abs((ew - ((eh * Math.sin(t)) * Math.sin(Math.atan(((eh * -Math.tan(t)) / ew))))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -8.2e-143) or not (eh <= 2.05e-54):
		tmp = math.fabs((ew - ((eh * math.sin(t)) * math.sin(math.atan(((eh * -math.tan(t)) / ew))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -8.2e-143) || !(eh <= 2.05e-54))
		tmp = abs(Float64(ew - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -8.2e-143) || ~((eh <= 2.05e-54)))
		tmp = abs((ew - ((eh * sin(t)) * sin(atan(((eh * -tan(t)) / ew))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -8.2e-143], N[Not[LessEqual[eh, 2.05e-54]], $MachinePrecision]], N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.2e-143 or 2.05e-54 < 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. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-out 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. associate-*r/ 99.8%

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

      7. add-sqr-sqrt 41.1%

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

      8. sqrt-unprod 94.9%

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

      9. sqr-neg 94.9%

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

      10. sqrt-unprod 58.7%

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

      11. add-sqr-sqrt 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in t around 0 99.2%

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

   7. Taylor expanded in t around 0 90.3%

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

   if -8.2e-143 < eh < 2.05e-54

   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. Step-by-step derivation

      1. sin-mult 91.8%

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

      2. associate-*r/ 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. +-inverses 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-/l* 91.8%

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

      4. div0 91.8%

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

   7. Simplified 91.8%

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

      1. add-cube-cbrt 89.9%

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

      2. pow3 90.0%

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

   9. Applied egg-rr 89.9%

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

   10. Taylor expanded in eh around 0 91.9%

       \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot \left(ew \cdot \cos t\right)}\right| \]
   11. Step-by-step derivation

       1. pow-base-1 91.9%

          \[\leadsto \left|\color{blue}{1} \cdot \left(ew \cdot \cos t\right)\right| \]

       2. *-lft-identity 91.9%

          \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   12. Simplified 91.9%

       \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.2 \cdot 10^{-143} \lor \neg \left(eh \leq 2.05 \cdot 10^{-54}\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]

Alternative 8: 98.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-out 99.8%

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

   5. distribute-lft-neg-out 99.8%

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

   6. associate-*r/ 99.8%

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

   7. add-sqr-sqrt 47.6%

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

   8. sqrt-unprod 96.7%

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

   9. sqr-neg 96.7%

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

   10. sqrt-unprod 52.2%

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

   11. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-/l* 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 99.1%

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

7. Taylor expanded in t around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-*r* 98.8%

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

   4. neg-mul-1 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 9: 61.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $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. Step-by-step derivation

   1. sin-mult 63.3%

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

   2. associate-*r/ 63.3%

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

5. Applied egg-rr 61.7%

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

   1. +-inverses 61.7%

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

   2. *-commutative 61.7%

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

   3. associate-/l* 61.7%

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

   4. div0 61.7%

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

7. Simplified 61.7%

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

   1. add-cube-cbrt 60.6%

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

   2. pow3 60.6%

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

9. Applied egg-rr 60.3%

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

10. Taylor expanded in eh around 0 61.9%

    \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot \left(ew \cdot \cos t\right)}\right| \]
11. Step-by-step derivation

    1. pow-base-1 61.9%

       \[\leadsto \left|\color{blue}{1} \cdot \left(ew \cdot \cos t\right)\right| \]

    2. *-lft-identity 61.9%

       \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

12. Simplified 61.9%

    \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

13. Final simplification 61.9%

    \[\leadsto \left|ew \cdot \cos t\right| \]

Alternative 10: 42.4% 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%

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

   1. sin-mult 63.3%

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

   2. associate-*r/ 63.3%

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

5. Applied egg-rr 61.7%

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

   1. +-inverses 61.7%

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

   2. *-commutative 61.7%

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

   3. associate-/l* 61.7%

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

   4. div0 61.7%

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

7. Simplified 61.7%

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

   1. add-cube-cbrt 60.6%

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

   2. pow3 60.6%

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

9. Applied egg-rr 60.3%

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

10. Taylor expanded in t around 0 43.2%

    \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew}\right| \]
11. Step-by-step derivation

    1. pow-base-1 43.2%

       \[\leadsto \left|\color{blue}{1} \cdot ew\right| \]

    2. *-lft-identity 43.2%

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

12. Simplified 43.2%

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

13. Final simplification 43.2%

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

Reproduce

herbie shell --seed 2023275 
(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)))))))