Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 27.1s

Alternatives: 13

Speedup: 1.1×

• could not determine a ground truth

  1. eh = 1.7826328125617583e+168
  2. ew = -2133315068632668000.0
  3. t = 4.4642945700821746e+45

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. add-sqr-sqrt 54.6%

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

   2. sqrt-unprod 92.7%

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

   3. sqr-neg 92.7%

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

   4. sqrt-unprod 45.2%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * cos(atan((eh / (ew / tan(t)))))) - ((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[(ew / N[Tan[t], $MachinePrecision]), $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 54.6%

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

   3. sqrt-unprod 92.7%

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

   4. sqr-neg 92.7%

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

   5. sqrt-unprod 45.2%

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

   6. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-/r/ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * (1.0 / hypot(1.0, (eh / (ew / tan(t)))))) - ((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[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $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/ 62.5%

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

   2. cos-atan 62.2%

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

   3. hypot-1-def 62.2%

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

   4. add-sqr-sqrt 34.8%

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

   5. sqrt-unprod 55.4%

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

   6. sqr-neg 55.4%

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

   7. sqrt-unprod 27.5%

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

   8. add-sqr-sqrt 62.2%

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

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. associate-*r/ 62.2%

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

   2. associate-*l/ 62.2%

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

   3. associate-/r/ 62.2%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. 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 54.6%

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

   3. sqrt-unprod 92.7%

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

   4. sqr-neg 92.7%

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

   5. sqrt-unprod 45.2%

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

   6. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-/r/ 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 98.9%

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

   1. mul-1-neg 43.1%

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

   2. distribute-rgt-neg-in 43.1%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 5: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. 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. expm1-log1p-u 99.8%

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

   3. expm1-undefine 99.8%

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

   4. associate-*r/ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. add-sqr-sqrt 54.6%

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

   7. sqrt-unprod 92.6%

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

   8. sqr-neg 92.6%

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

   9. sqrt-unprod 45.2%

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

   10. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 45.2%

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

   4. sqrt-unprod 92.6%

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

   5. sqr-neg 92.6%

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

   6. sqrt-unprod 54.6%

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

   7. add-sqr-sqrt 99.8%

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

   8. associate-/l* 99.8%

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

   9. *-commutative 99.8%

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

   10. associate-/l* 99.8%

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

   11. add-sqr-sqrt 54.6%

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

   12. sqrt-unprod 92.7%

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

   13. sqr-neg 92.7%

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

   14. sqrt-unprod 45.2%

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

   15. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in t around 0 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 62.5% accurate, 1.8× speedup

Math

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

FPCore

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

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)))))
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)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $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-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Final simplification 62.5%

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

Alternative 7: 62.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $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-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

   1. associate-*r/ 62.5%

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

   2. cos-atan 62.2%

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

   3. hypot-1-def 62.2%

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

   4. add-sqr-sqrt 34.8%

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

   5. sqrt-unprod 55.4%

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

   6. sqr-neg 55.4%

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

   7. sqrt-unprod 27.5%

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

   8. add-sqr-sqrt 62.2%

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

8. Applied egg-rr 62.2%

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

   1. associate-*r/ 62.2%

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

   2. associate-*l/ 62.2%

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

   3. associate-/r/ 62.2%

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

10. Simplified 62.2%

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

11. Final simplification 62.2%

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

Alternative 8: 52.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in t around 0 52.0%

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

   1. mul-1-neg 43.1%

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

   2. distribute-rgt-neg-in 43.1%

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

9. Simplified 52.0%

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

10. Final simplification 52.0%

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

Alternative 9: 42.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in t around 0 44.2%

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

   1. add-sqr-sqrt 22.8%

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

   2. sqrt-unprod 38.9%

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

   3. sqr-neg 38.9%

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

   4. sqrt-unprod 21.4%

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

   5. add-sqr-sqrt 44.2%

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

   6. pow1 44.2%

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

9. Applied egg-rr 44.2%

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

    1. unpow1 44.2%

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

11. Simplified 44.2%

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

12. Final simplification 44.2%

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

Alternative 10: 41.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in t around 0 44.2%

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

   1. cos-atan 43.9%

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

   2. un-div-inv 43.9%

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

   3. hypot-1-def 43.9%

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

   4. *-commutative 43.9%

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

   5. associate-/l* 43.9%

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

   6. add-sqr-sqrt 22.7%

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

   7. sqrt-unprod 37.9%

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

   8. sqr-neg 37.9%

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

   9. sqrt-unprod 21.2%

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

   10. add-sqr-sqrt 43.9%

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

9. Applied egg-rr 43.9%

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

    1. associate-*r/ 43.9%

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

    2. *-commutative 43.9%

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

    3. associate-/l* 43.9%

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

11. Simplified 43.9%

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

12. Final simplification 43.9%

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

Alternative 11: 40.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in t around 0 44.2%

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

   1. add-sqr-sqrt 22.8%

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

   2. sqrt-unprod 38.9%

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

   3. sqr-neg 38.9%

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

   4. sqrt-unprod 21.4%

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

   5. add-sqr-sqrt 44.2%

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

   6. add-cbrt-cube 41.8%

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

   7. pow3 41.8%

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

9. Applied egg-rr 41.8%

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

10. Taylor expanded in t around 0 43.1%

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

11. Final simplification 43.1%

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

Alternative 12: 39.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in t around 0 44.2%

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

8. Taylor expanded in t around 0 43.1%

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

   1. mul-1-neg 43.1%

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

   2. distribute-rgt-neg-in 43.1%

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

10. Simplified 43.1%

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

    1. cos-atan 42.1%

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

    2. un-div-inv 42.1%

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

    3. hypot-1-def 42.2%

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

    4. associate-/l* 42.2%

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

    5. add-sqr-sqrt 19.7%

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

    6. sqrt-unprod 41.4%

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

    7. sqr-neg 41.4%

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

    8. sqrt-unprod 22.6%

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

    9. add-sqr-sqrt 42.2%

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

12. Applied egg-rr 42.2%

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

13. Final simplification 42.2%

    \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, eh \cdot \frac{t}{ew}\right)}\right| \]
14. Add Preprocessing

Alternative 13: 39.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. associate-*r/ 99.8%

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

   3. sin-mult 64.3%

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

   4. associate-*r/ 64.3%

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

4. Applied egg-rr 62.5%

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

   1. +-inverses 62.5%

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

   2. mul0-rgt 62.5%

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

   3. metadata-eval 62.5%

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

6. Simplified 62.5%

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

7. Taylor expanded in t around 0 44.2%

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

8. Taylor expanded in t around 0 43.1%

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

   1. mul-1-neg 43.1%

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

   2. distribute-rgt-neg-in 43.1%

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

10. Simplified 43.1%

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

    1. cos-atan 42.1%

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

    2. un-div-inv 42.1%

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

    3. hypot-1-def 42.2%

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

    4. associate-/l* 42.2%

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

    5. add-sqr-sqrt 19.7%

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

    6. sqrt-unprod 41.4%

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

    7. sqr-neg 41.4%

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

    8. sqrt-unprod 22.6%

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

    9. add-sqr-sqrt 42.2%

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

12. Applied egg-rr 42.2%

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

    1. *-commutative 42.2%

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

    2. associate-*l/ 42.2%

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

    3. associate-*r/ 42.3%

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

14. Simplified 42.3%

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

15. Final simplification 42.3%

    \[\leadsto \left|\frac{ew}{\mathsf{hypot}\left(1, t \cdot \frac{eh}{ew}\right)}\right| \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))))