Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 28.7s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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. associate-*r/ 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-/l* 99.8%

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

   7. add-sqr-sqrt 49.1%

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

   8. sqrt-unprod 96.3%

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

   9. sqr-neg 96.3%

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

   10. sqrt-unprod 50.7%

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

   11. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 99.3%

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

   1. associate-*r/ 99.3%

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

   2. associate-*r* 99.3%

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

   3. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sin-atan 76.2%

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

   3. associate-*r/ 72.7%

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

   4. associate-*r/ 72.7%

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

   5. *-commutative 72.7%

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

   6. associate-/l* 70.7%

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

   7. add-sqr-sqrt 35.0%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod 35.2%

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

   11. add-sqr-sqrt 69.9%

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

   12. hypot-1-def 75.9%

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

   13. associate-*r/ 75.9%

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

6. Applied egg-rr 75.8%

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

7. Taylor expanded in eh around -inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sin-atan 76.2%

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

   3. associate-*r/ 72.7%

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

   4. associate-*r/ 72.7%

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

   5. *-commutative 72.7%

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

   6. associate-/l* 70.7%

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

   7. add-sqr-sqrt 35.0%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod 35.2%

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

   11. add-sqr-sqrt 69.9%

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

   12. hypot-1-def 75.9%

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

   13. associate-*r/ 75.9%

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

6. Applied egg-rr 75.8%

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

7. Taylor expanded in eh around inf 98.8%

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

8. Final simplification 98.8%

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

Alternative 5: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \left(-t\right)}{ew}\right)\\ t_2 := ew \cdot \cos t\\ \mathbf{if}\;t \leq -0.00155:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+23}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, -\cos t\_1, \sin t\_1 \cdot \left(t \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-0.5, \frac{{\left(eh \cdot \sin t\right)}^{2}}{t\_2}, t\_2\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (- t)) ew))) (t_2 (* ew (cos t))))
   (if (<= t -0.00155)
     (fabs t_2)
     (if (<= t 1.25e+23)
       (fabs (fma ew (- (cos t_1)) (* (sin t_1) (* t eh))))
       (fabs (fma -0.5 (/ (pow (* eh (sin t)) 2.0) t_2) t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * -t) / ew));
	double t_2 = ew * cos(t);
	double tmp;
	if (t <= -0.00155) {
		tmp = fabs(t_2);
	} else if (t <= 1.25e+23) {
		tmp = fabs(fma(ew, -cos(t_1), (sin(t_1) * (t * eh))));
	} else {
		tmp = fabs(fma(-0.5, (pow((eh * sin(t)), 2.0) / t_2), t_2));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * Float64(-t)) / ew))
	t_2 = Float64(ew * cos(t))
	tmp = 0.0
	if (t <= -0.00155)
		tmp = abs(t_2);
	elseif (t <= 1.25e+23)
		tmp = abs(fma(ew, Float64(-cos(t_1)), Float64(sin(t_1) * Float64(t * eh))));
	else
		tmp = abs(fma(-0.5, Float64((Float64(eh * sin(t)) ^ 2.0) / t_2), t_2));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * (-t)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.00155], N[Abs[t$95$2], $MachinePrecision], If[LessEqual[t, 1.25e+23], N[Abs[N[(ew * (-N[Cos[t$95$1], $MachinePrecision]) + N[(N[Sin[t$95$1], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(-0.5 * N[(N[Power[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.00154999999999999995

   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* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      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 50.5%

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

      2. associate-*r/ 50.5%

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

   6. Applied egg-rr 50.5%

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

      1. +-inverses 50.5%

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

      2. *-commutative 50.5%

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

      3. associate-/l* 50.5%

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

      4. mul0-lft 50.5%

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

   8. Simplified 50.5%

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

      1. fma-undefine 50.5%

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

      2. +-rgt-identity 50.5%

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

      3. add-sqr-sqrt 28.0%

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

      4. associate-*l* 28.1%

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

      5. *-commutative 28.1%

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

      6. cos-atan 27.9%

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

      7. un-div-inv 27.9%

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

      8. add-sqr-sqrt 20.6%

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

      9. sqrt-unprod 27.9%

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

      10. sqr-neg 27.9%

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

      11. sqrt-unprod 7.3%

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

      12. add-sqr-sqrt 27.9%

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

      13. hypot-1-def 27.9%

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

   10. Applied egg-rr 27.9%

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

   11. Taylor expanded in ew around inf 50.7%

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

   if -0.00154999999999999995 < t < 1.25e23

   1. Initial program 99.9%

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

      1. fabs-sub 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 100.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 93.3%

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

      1. associate-*r* 93.3%

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

      2. associate-*r/ 93.3%

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

      3. associate-*r* 93.3%

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

      4. *-commutative 93.3%

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

      5. associate-*l/ 93.3%

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

      6. associate-*r/ 93.3%

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

      7. mul-1-neg 93.3%

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

   7. Simplified 93.3%

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

   8. Taylor expanded in t around 0 93.3%

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

      1. mul-1-neg 93.3%

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

      2. *-commutative 93.3%

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

      3. distribute-neg-frac2 93.3%

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

      4. *-commutative 93.3%

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

   10. Simplified 93.3%

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

   11. Taylor expanded in t around 0 93.3%

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

   12. Taylor expanded in t around 0 93.0%

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

       1. mul-1-neg 93.0%

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

       2. associate-*r/ 93.0%

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

       3. neg-mul-1 93.0%

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

       4. distribute-rgt-neg-in 93.0%

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

   14. Simplified 93.0%

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

   if 1.25e23 < t

   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* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      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 48.5%

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

      2. associate-*r/ 48.5%

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

   6. Applied egg-rr 48.5%

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

      1. +-inverses 48.5%

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

      2. *-commutative 48.5%

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

      3. associate-/l* 48.5%

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

      4. mul0-lft 48.5%

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

   8. Simplified 48.5%

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

      1. fma-undefine 48.5%

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

      2. +-rgt-identity 48.5%

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

      3. add-sqr-sqrt 21.2%

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

      4. associate-*l* 21.2%

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

      5. *-commutative 21.2%

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

      6. cos-atan 21.1%

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

      7. un-div-inv 21.1%

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

      8. add-sqr-sqrt 8.0%

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

      9. sqrt-unprod 21.1%

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

      10. sqr-neg 21.1%

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

      11. sqrt-unprod 13.0%

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

      12. add-sqr-sqrt 21.1%

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

      13. hypot-1-def 21.1%

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

   10. Applied egg-rr 21.1%

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

   11. Taylor expanded in eh around 0 49.6%

       \[\leadsto \left|\color{blue}{-0.5 \cdot \frac{{eh}^{2} \cdot {\sin t}^{2}}{ew \cdot \cos t} + ew \cdot \cos t}\right| \]
   12. Step-by-step derivation

       1. fma-define 49.6%

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

       2. unpow2 49.6%

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

       3. unpow2 49.6%

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

       4. swap-sqr 49.6%

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

       5. unpow2 49.6%

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

   13. Simplified 49.6%

       \[\leadsto \left|\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(eh \cdot \sin t\right)}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00155:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+23}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, -\cos \tan^{-1} \left(\frac{eh \cdot \left(-t\right)}{ew}\right), \sin \tan^{-1} \left(\frac{eh \cdot \left(-t\right)}{ew}\right) \cdot \left(t \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-0.5, \frac{{\left(eh \cdot \sin t\right)}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.7% 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 61.9%

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

   2. associate-*r/ 61.9%

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

6. Applied egg-rr 59.1%

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

   1. +-inverses 59.1%

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

   2. *-commutative 59.1%

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

   3. associate-/l* 59.1%

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

   4. mul0-lft 59.1%

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

8. Simplified 59.1%

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

   1. fma-undefine 59.1%

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

   2. +-rgt-identity 59.1%

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

   3. add-sqr-sqrt 28.3%

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

   4. associate-*l* 28.3%

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

   5. *-commutative 28.3%

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

   6. cos-atan 28.2%

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

   7. un-div-inv 28.2%

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

   8. add-sqr-sqrt 6.5%

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

   9. sqrt-unprod 28.2%

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

   10. sqr-neg 28.2%

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

   11. sqrt-unprod 21.7%

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

   12. add-sqr-sqrt 28.2%

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

   13. hypot-1-def 28.2%

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

10. Applied egg-rr 28.2%

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

11. Taylor expanded in ew around inf 59.3%

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

12. Final simplification 59.3%

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

Alternative 7: 41.8% 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 61.9%

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

   2. associate-*r/ 61.9%

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

6. Applied egg-rr 59.1%

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

   1. +-inverses 59.1%

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

   2. *-commutative 59.1%

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

   3. associate-/l* 59.1%

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

   4. mul0-lft 59.1%

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

8. Simplified 59.1%

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

   1. fma-undefine 59.1%

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

   2. +-rgt-identity 59.1%

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

   3. add-sqr-sqrt 28.3%

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

   4. associate-*l* 28.3%

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

   5. *-commutative 28.3%

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

   6. cos-atan 28.2%

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

   7. un-div-inv 28.2%

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

   8. add-sqr-sqrt 6.5%

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

   9. sqrt-unprod 28.2%

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

   10. sqr-neg 28.2%

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

   11. sqrt-unprod 21.7%

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

   12. add-sqr-sqrt 28.2%

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

   13. hypot-1-def 28.2%

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

10. Applied egg-rr 28.2%

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

11. Taylor expanded in t around 0 42.8%

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

12. Final simplification 42.8%

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

Reproduce

herbie shell --seed 2024055 
(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)))))))