Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.0s

Alternatives: 11

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, 0.8× speedup

Math

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

FPCore

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

C

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r/ 99.8%

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

   2. add-cbrt-cube 99.8%

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

   3. pow3 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   ew
   (* (cos t) (/ -1.0 (hypot 1.0 (* (tan t) (/ eh ew)))))
   (* (sin t) (* eh (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))))), (sin(t) * (eh * 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(sin(t) * Float64(eh * 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[(N[Sin[t], $MachinePrecision] * N[(eh * 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-define 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%

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

5. Taylor expanded in eh around 0 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

   4. mul-1-neg 99.8%

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

   5. distribute-frac-neg2 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-/l* 99.8%

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

7. Simplified 99.8%

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

   1. add-sqr-sqrt 47.5%

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

   2. sqrt-unprod 90.0%

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

   3. sqr-neg 90.0%

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

   4. sqrt-unprod 50.1%

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

   5. add-sqr-sqrt 97.6%

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

   6. associate-*r/ 97.6%

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

9. Applied egg-rr 99.8%

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

    1. cos-atan 99.8%

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

    2. hypot-1-def 99.8%

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

    3. *-commutative 99.8%

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

    4. associate-*r/ 99.8%

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

11. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) * Math.cos(Math.atan((eh * (Math.tan(t) / ew))))) - ((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((eh * (math.tan(t) / ew))))) - ((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(eh * Float64(tan(t) / ew))))) - 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((eh * (tan(t) / ew))))) - ((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[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r/ 99.8%

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

   2. add-sqr-sqrt 48.3%

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

   3. sqrt-unprod 91.4%

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

   4. sqr-neg 91.4%

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

   5. sqrt-unprod 51.4%

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

   6. add-sqr-sqrt 99.8%

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

   7. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (/ 1.0 (hypot 1.0 (* (tan t) (/ eh ew)))))
   (* (* 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) * (eh / ew))))) - ((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) * (eh / ew))))) - ((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) * (eh / ew))))) - ((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(eh / ew))))) - 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) * (eh / ew))))) - ((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[(eh / ew), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r/ 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

   4. add-sqr-sqrt 48.3%

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

   5. sqrt-unprod 91.2%

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

   6. sqr-neg 91.2%

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

   7. sqrt-unprod 51.4%

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

   8. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 98.9% accurate, 1.1× speedup

Math

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

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(t * Float64(-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[(t * (-N[(eh / ew), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 97.8%

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

   1. associate-*r/ 97.8%

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

   2. associate-*r* 97.8%

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

   3. associate-*l/ 97.8%

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

   4. associate-*r/ 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. neg-mul-1 97.8%

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

5. Simplified 97.8%

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

6. Final simplification 97.8%

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

Alternative 6: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   1. sin-atan 83.0%

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

   2. associate-*r/ 82.9%

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

   3. add-sqr-sqrt 40.2%

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

   4. sqrt-unprod 65.6%

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

   5. sqr-neg 65.6%

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

   6. sqrt-unprod 41.6%

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

   7. add-sqr-sqrt 81.6%

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

   8. hypot-1-def 90.2%

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

   9. add-sqr-sqrt 43.6%

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

   10. sqrt-unprod 75.0%

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

   11. sqr-neg 75.0%

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

   12. sqrt-unprod 46.4%

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

   13. add-sqr-sqrt 90.2%

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

6. Applied egg-rr 90.2%

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

   1. *-commutative 90.2%

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

   2. associate-*l* 88.6%

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

   3. associate-/l* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. associate-*l/ 86.6%

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

   7. associate-*r/ 85.8%

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

8. Simplified 85.8%

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

9. Taylor expanded in eh around -inf 97.6%

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

    1. mul-1-neg 84.1%

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

11. Simplified 97.6%

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

12. Final simplification 97.6%

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

Alternative 7: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. 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-define 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%

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

   1. sin-atan 83.0%

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

   2. associate-*r/ 82.9%

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

   3. add-sqr-sqrt 40.2%

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

   4. sqrt-unprod 65.6%

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

   5. sqr-neg 65.6%

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

   6. sqrt-unprod 41.6%

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

   7. add-sqr-sqrt 81.6%

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

   8. hypot-1-def 90.2%

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

   9. add-sqr-sqrt 43.6%

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

   10. sqrt-unprod 75.0%

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

   11. sqr-neg 75.0%

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

   12. sqrt-unprod 46.4%

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

   13. add-sqr-sqrt 90.2%

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

6. Applied egg-rr 90.2%

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

   1. *-commutative 90.2%

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

   2. associate-*l* 88.6%

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

   3. associate-/l* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. associate-*l/ 86.6%

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

   7. associate-*r/ 85.8%

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

8. Simplified 85.8%

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

9. Taylor expanded in eh around inf 97.6%

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

    1. add-sqr-sqrt 47.5%

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

    2. sqrt-unprod 90.0%

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

    3. sqr-neg 90.0%

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

    4. sqrt-unprod 50.1%

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

    5. add-sqr-sqrt 97.6%

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

    6. associate-*r/ 97.6%

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

11. Applied egg-rr 97.6%

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

12. Final simplification 97.6%

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

Alternative 8: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(-\cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right), eh \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 1.75e+139)
   (fabs
    (fma
     ew
     (* (- (cos t)) (cos (atan (* (- eh) (/ t ew)))))
     (* eh (- (sin t)))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.75e+139) {
		tmp = fabs(fma(ew, (-cos(t) * cos(atan((-eh * (t / ew))))), (eh * -sin(t))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 1.75e+139)
		tmp = abs(fma(ew, Float64(Float64(-cos(t)) * cos(atan(Float64(Float64(-eh) * Float64(t / ew))))), Float64(eh * Float64(-sin(t)))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 1.75e+139], N[Abs[N[(ew * N[((-N[Cos[t], $MachinePrecision]) * N[Cos[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(eh * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 1.74999999999999989e139

   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-define 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%

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

      1. sin-atan 80.1%

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

      2. associate-*r/ 80.0%

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

      3. add-sqr-sqrt 41.3%

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

      4. sqrt-unprod 62.2%

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

      5. sqr-neg 62.2%

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

      6. sqrt-unprod 37.8%

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

      7. add-sqr-sqrt 78.8%

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

      8. hypot-1-def 88.9%

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

      9. add-sqr-sqrt 45.3%

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

      10. sqrt-unprod 72.6%

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

      11. sqr-neg 72.6%

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

      12. sqrt-unprod 43.5%

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

      13. add-sqr-sqrt 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 87.0%

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

      3. associate-/l* 84.8%

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

      4. *-commutative 84.8%

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

      5. *-commutative 84.8%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in t around 0 70.5%

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

   10. Taylor expanded in eh around -inf 85.4%

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

       1. mul-1-neg 85.4%

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

   12. Simplified 85.4%

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

   if 1.74999999999999989e139 < ew

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      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-define 99.7%

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

   3. Simplified 99.7%

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

      1. sin-mult 96.6%

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

      2. associate-*r/ 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. +-inverses 96.6%

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

      2. mul0-rgt 96.6%

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

      3. metadata-eval 96.6%

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

   8. Simplified 96.6%

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

      1. add-sqr-sqrt 26.1%

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

      2. pow2 26.1%

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

   10. Applied egg-rr 70.1%

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

   11. Taylor expanded in ew around inf 96.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(-\cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right), eh \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 4.35 \cdot 10^{+139}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(-\cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right), eh \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 4.35e+139)
   (fabs
    (fma ew (* (- (cos t)) (cos (atan (* (- eh) (/ t ew))))) (* eh (sin t))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 4.35e+139) {
		tmp = fabs(fma(ew, (-cos(t) * cos(atan((-eh * (t / ew))))), (eh * sin(t))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 4.35e+139)
		tmp = abs(fma(ew, Float64(Float64(-cos(t)) * cos(atan(Float64(Float64(-eh) * Float64(t / ew))))), Float64(eh * sin(t))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 4.35e+139], N[Abs[N[(ew * N[((-N[Cos[t], $MachinePrecision]) * N[Cos[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 4.3499999999999998e139

   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-define 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%

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

      1. sin-atan 80.1%

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

      2. associate-*r/ 80.0%

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

      3. add-sqr-sqrt 41.3%

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

      4. sqrt-unprod 62.2%

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

      5. sqr-neg 62.2%

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

      6. sqrt-unprod 37.8%

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

      7. add-sqr-sqrt 78.8%

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

      8. hypot-1-def 88.9%

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

      9. add-sqr-sqrt 45.3%

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

      10. sqrt-unprod 72.6%

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

      11. sqr-neg 72.6%

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

      12. sqrt-unprod 43.5%

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

      13. add-sqr-sqrt 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 87.0%

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

      3. associate-/l* 84.8%

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

      4. *-commutative 84.8%

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

      5. *-commutative 84.8%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in eh around inf 97.8%

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

   10. Taylor expanded in t around 0 85.3%

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

   if 4.3499999999999998e139 < ew

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      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-define 99.7%

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

   3. Simplified 99.7%

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

      1. sin-mult 96.6%

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

      2. associate-*r/ 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. +-inverses 96.6%

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

      2. mul0-rgt 96.6%

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

      3. metadata-eval 96.6%

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

   8. Simplified 96.6%

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

      1. add-sqr-sqrt 26.1%

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

      2. pow2 26.1%

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

   10. Applied egg-rr 70.1%

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

   11. Taylor expanded in ew around inf 96.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq 4.35 \cdot 10^{+139}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(-\cos t\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right), eh \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% 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-define 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%

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

   1. sin-mult 65.9%

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

   2. associate-*r/ 65.9%

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

6. Applied egg-rr 65.5%

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

   1. +-inverses 65.5%

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

   2. mul0-rgt 65.5%

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

   3. metadata-eval 65.5%

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

8. Simplified 65.5%

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

   1. add-sqr-sqrt 30.2%

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

   2. pow2 30.2%

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

10. Applied egg-rr 35.1%

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

11. Taylor expanded in ew around inf 65.7%

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

12. Final simplification 65.7%

    \[\leadsto \left|ew \cdot \cos t\right| \]
13. Add Preprocessing

Alternative 11: 42.2% 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-define 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%

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

   1. sin-mult 65.9%

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

   2. associate-*r/ 65.9%

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

6. Applied egg-rr 65.5%

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

   1. +-inverses 65.5%

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

   2. mul0-rgt 65.5%

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

   3. metadata-eval 65.5%

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

8. Simplified 65.5%

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

   1. add-sqr-sqrt 30.2%

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

   2. pow2 30.2%

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

10. Applied egg-rr 35.1%

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

11. Taylor expanded in t around 0 41.1%

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

12. Final simplification 41.1%

    \[\leadsto \left|ew\right| \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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)))))))