Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (cos t_1) (* ew (cos t))) (* (* 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(((cos(t_1) * (ew * cos(t))) - ((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(((cos(t_1) * (ew * cos(t))) - ((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(((Math.cos(t_1) * (ew * Math.cos(t))) - ((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(((math.cos(t_1) * (ew * math.cos(t))) - ((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(cos(t_1) * Float64(ew * cos(t))) - 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(((cos(t_1) * (ew * cos(t))) - ((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[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $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. Final simplification 99.8%

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

Alternative 3: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.4 \cdot 10^{-67} \lor \neg \left(ew \leq 1.15 \cdot 10^{-126}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \left(\sin t \cdot \frac{\sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(eh \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.4e-67) (not (<= ew 1.15e-126)))
   (fabs
    (*
     ew
     (+
      (cos t)
      (* eh (* (sin t) (/ (sin (atan (* (tan t) (/ eh (- ew))))) ew))))))
   (fabs (+ ew (* (sin (atan (* eh (/ (tan t) (- ew))))) (* eh (sin t)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.4e-67) || !(ew <= 1.15e-126)) {
		tmp = fabs((ew * (cos(t) + (eh * (sin(t) * (sin(atan((tan(t) * (eh / -ew)))) / ew))))));
	} else {
		tmp = fabs((ew + (sin(atan((eh * (tan(t) / -ew)))) * (eh * sin(t)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.4d-67)) .or. (.not. (ew <= 1.15d-126))) then
        tmp = abs((ew * (cos(t) + (eh * (sin(t) * (sin(atan((tan(t) * (eh / -ew)))) / ew))))))
    else
        tmp = abs((ew + (sin(atan((eh * (tan(t) / -ew)))) * (eh * sin(t)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.4e-67) || !(ew <= 1.15e-126)) {
		tmp = Math.abs((ew * (Math.cos(t) + (eh * (Math.sin(t) * (Math.sin(Math.atan((Math.tan(t) * (eh / -ew)))) / ew))))));
	} else {
		tmp = Math.abs((ew + (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) * (eh * Math.sin(t)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.4e-67) or not (ew <= 1.15e-126):
		tmp = math.fabs((ew * (math.cos(t) + (eh * (math.sin(t) * (math.sin(math.atan((math.tan(t) * (eh / -ew)))) / ew))))))
	else:
		tmp = math.fabs((ew + (math.sin(math.atan((eh * (math.tan(t) / -ew)))) * (eh * math.sin(t)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.4e-67) || !(ew <= 1.15e-126))
		tmp = abs(Float64(ew * Float64(cos(t) + Float64(eh * Float64(sin(t) * Float64(sin(atan(Float64(tan(t) * Float64(eh / Float64(-ew))))) / ew))))));
	else
		tmp = abs(Float64(ew + Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) * Float64(eh * sin(t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.4e-67) || ~((ew <= 1.15e-126)))
		tmp = abs((ew * (cos(t) + (eh * (sin(t) * (sin(atan((tan(t) * (eh / -ew)))) / ew))))));
	else
		tmp = abs((ew + (sin(atan((eh * (tan(t) / -ew)))) * (eh * sin(t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.4e-67], N[Not[LessEqual[ew, 1.15e-126]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.40000000000000005e-67 or 1.15000000000000005e-126 < ew

   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-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ew around inf 98.6%

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

      1. associate-/l* 98.6%

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

      2. associate-/l* 98.6%

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

      3. mul-1-neg 98.6%

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

      4. associate-*l/ 98.6%

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

      5. *-commutative 98.6%

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

      6. distribute-rgt-neg-in 98.6%

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

      7. mul-1-neg 98.6%

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

      8. associate-*r/ 98.6%

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

      9. neg-mul-1 98.6%

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

   9. Simplified 98.6%

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

   if -1.40000000000000005e-67 < ew < 1.15000000000000005e-126

   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 99.9%

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

   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. Step-by-step derivation

      1. add-cbrt-cube 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in t around 0 95.4%

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

   8. Taylor expanded in ew around 0 95.4%

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

      1. associate-*r* 95.4%

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

      2. mul-1-neg 95.4%

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

      3. associate-*r/ 95.4%

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

   10. Simplified 95.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.4 \cdot 10^{-67} \lor \neg \left(ew \leq 1.15 \cdot 10^{-126}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \left(\sin t \cdot \frac{\sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(eh \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.5× speedup

Math

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

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(Float64(-eh) * tan(t)) / 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. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in ew around -inf 99.1%

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

   1. associate-*r* 99.1%

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

   2. neg-mul-1 99.1%

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

   3. rem-cube-cbrt 99.1%

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

   4. neg-mul-1 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.1 \cdot 10^{-92} \lor \neg \left(eh \leq 4 \cdot 10^{-255}\right):\\ \;\;\;\;\left|ew + \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(eh \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -5.1e-92) (not (<= eh 4e-255)))
   (fabs (+ ew (* (sin (atan (* eh (/ (tan t) (- ew))))) (* eh (sin t)))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -5.1e-92) || !(eh <= 4e-255)) {
		tmp = fabs((ew + (sin(atan((eh * (tan(t) / -ew)))) * (eh * sin(t)))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-5.1d-92)) .or. (.not. (eh <= 4d-255))) then
        tmp = abs((ew + (sin(atan((eh * (tan(t) / -ew)))) * (eh * sin(t)))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -5.1e-92) || !(eh <= 4e-255)) {
		tmp = Math.abs((ew + (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) * (eh * Math.sin(t)))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -5.1e-92) or not (eh <= 4e-255):
		tmp = math.fabs((ew + (math.sin(math.atan((eh * (math.tan(t) / -ew)))) * (eh * math.sin(t)))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -5.1e-92) || !(eh <= 4e-255))
		tmp = abs(Float64(ew + Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) * Float64(eh * sin(t)))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -5.1e-92) || ~((eh <= 4e-255)))
		tmp = abs((ew + (sin(atan((eh * (tan(t) / -ew)))) * (eh * sin(t)))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -5.1e-92], N[Not[LessEqual[eh, 4e-255]], $MachinePrecision]], N[Abs[N[(ew + N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.09999999999999972e-92 or 4e-255 < eh

   1. Initial program 99.8%

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

      1. 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 99.9%

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

   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. Step-by-step derivation

      1. add-cbrt-cube 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in t around 0 89.8%

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

   8. Taylor expanded in ew around 0 89.8%

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

      1. associate-*r* 89.8%

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

      2. mul-1-neg 89.8%

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

      3. associate-*r/ 89.8%

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

   10. Simplified 89.8%

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

   if -5.09999999999999972e-92 < eh < 4e-255

   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-cbrt-cube 99.5%

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

      2. pow3 99.5%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in ew around inf 95.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.1 \cdot 10^{-92} \lor \neg \left(eh \leq 4 \cdot 10^{-255}\right):\\ \;\;\;\;\left|ew + \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(eh \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3.8 \cdot 10^{-15} \lor \neg \left(ew \leq 9.5 \cdot 10^{-127}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3.8e-15) (not (<= ew 9.5e-127)))
   (fabs (* ew (cos t)))
   (fabs (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.8e-15) || !(ew <= 9.5e-127)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-3.8d-15)) .or. (.not. (ew <= 9.5d-127))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs(((eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.8e-15) || !(ew <= 9.5e-127)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -3.8e-15) or not (ew <= 9.5e-127):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -3.8e-15) || !(ew <= 9.5e-127))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -3.8e-15) || ~((ew <= 9.5e-127)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -3.8e-15], N[Not[LessEqual[ew, 9.5e-127]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.8000000000000002e-15 or 9.4999999999999997e-127 < ew

   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-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ew around inf 80.7%

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

   if -3.8000000000000002e-15 < ew < 9.4999999999999997e-127

   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 99.9%

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

   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. Step-by-step derivation

      1. add-cbrt-cube 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in t around 0 94.4%

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

      1. associate-*r* 94.4%

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

      2. associate-*r/ 94.4%

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

      3. add-sqr-sqrt 60.9%

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

      4. pow2 60.9%

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

   9. Applied egg-rr 45.3%

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

   10. Taylor expanded in ew around 0 77.0%

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

       1. associate-*r* 77.0%

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

       2. *-commutative 77.0%

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

       3. associate-*r/ 77.0%

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

       4. *-commutative 77.0%

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

   12. Simplified 77.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.8 \cdot 10^{-15} \lor \neg \left(ew \leq 9.5 \cdot 10^{-127}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{-62} \lor \neg \left(ew \leq 4.1 \cdot 10^{-131}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.35e-62) (not (<= ew 4.1e-131)))
   (fabs (* ew (cos t)))
   (- ew (* (* eh (sin t)) (sin (atan (* eh (/ (tan t) ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.35e-62) || !(ew <= 4.1e-131)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = ew - ((eh * sin(t)) * sin(atan((eh * (tan(t) / ew)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.35d-62)) .or. (.not. (ew <= 4.1d-131))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = ew - ((eh * sin(t)) * sin(atan((eh * (tan(t) / ew)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.35e-62) || !(ew <= 4.1e-131)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = ew - ((eh * Math.sin(t)) * Math.sin(Math.atan((eh * (Math.tan(t) / ew)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.35e-62) or not (ew <= 4.1e-131):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = ew - ((eh * math.sin(t)) * math.sin(math.atan((eh * (math.tan(t) / ew)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.35e-62) || !(ew <= 4.1e-131))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = Float64(ew - Float64(Float64(eh * sin(t)) * sin(atan(Float64(eh * Float64(tan(t) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.35e-62) || ~((ew <= 4.1e-131)))
		tmp = abs((ew * cos(t)));
	else
		tmp = ew - ((eh * sin(t)) * sin(atan((eh * (tan(t) / ew)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.35e-62], N[Not[LessEqual[ew, 4.1e-131]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.3500000000000001e-62 or 4.1000000000000002e-131 < ew

   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-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ew around inf 78.8%

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

   if -1.3500000000000001e-62 < ew < 4.1000000000000002e-131

   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 99.9%

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

   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. Step-by-step derivation

      1. add-cbrt-cube 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in t around 0 95.2%

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

      1. *-un-lft-identity 95.2%

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

      2. associate-*r* 95.2%

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

      3. distribute-lft-neg-out 95.2%

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

      4. atan-neg 95.2%

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

   9. Applied egg-rr 95.2%

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

       1. *-lft-identity 95.2%

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

       2. rem-square-sqrt 52.2%

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

       3. fabs-sqr 52.2%

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

       4. rem-square-sqrt 54.1%

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

       5. sin-neg 54.1%

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

       6. associate-*r/ 54.1%

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

       7. *-commutative 54.1%

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

       8. associate-*r/ 54.1%

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

       9. distribute-rgt-neg-out 54.1%

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

   11. Simplified 54.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{-62} \lor \neg \left(ew \leq 4.1 \cdot 10^{-131}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3.2 \cdot 10^{-192} \lor \neg \left(ew \leq 1.45 \cdot 10^{-142}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(eh \cdot t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3.2e-192) (not (<= ew 1.45e-142)))
   (fabs (* ew (cos t)))
   (fabs (* (sin (atan (* (tan t) (/ eh (- ew))))) (* eh t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.2e-192) || !(ew <= 1.45e-142)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((sin(atan((tan(t) * (eh / -ew)))) * (eh * t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-3.2d-192)) .or. (.not. (ew <= 1.45d-142))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((sin(atan((tan(t) * (eh / -ew)))) * (eh * t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.2e-192) || !(ew <= 1.45e-142)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((Math.sin(Math.atan((Math.tan(t) * (eh / -ew)))) * (eh * t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -3.2e-192) or not (ew <= 1.45e-142):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((math.sin(math.atan((math.tan(t) * (eh / -ew)))) * (eh * t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -3.2e-192) || !(ew <= 1.45e-142))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(sin(atan(Float64(tan(t) * Float64(eh / Float64(-ew))))) * Float64(eh * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -3.2e-192) || ~((ew <= 1.45e-142)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((sin(atan((tan(t) * (eh / -ew)))) * (eh * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -3.2e-192], N[Not[LessEqual[ew, 1.45e-142]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.2000000000000002e-192 or 1.44999999999999995e-142 < ew

   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-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ew around inf 74.1%

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

   if -3.2000000000000002e-192 < ew < 1.44999999999999995e-142

   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 99.9%

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

   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 48.4%

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

   6. Taylor expanded in ew around 0 36.8%

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

      1. associate-*r* 36.8%

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

      2. mul-1-neg 36.8%

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

      3. associate-*l/ 36.8%

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

      4. *-commutative 36.8%

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

      5. distribute-rgt-neg-in 36.8%

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

      6. mul-1-neg 36.8%

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

      7. associate-*r/ 36.8%

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

      8. neg-mul-1 36.8%

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

   8. Simplified 36.8%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.2 \cdot 10^{-192} \lor \neg \left(ew \leq 1.45 \cdot 10^{-142}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(eh \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6.2 \cdot 10^{-179} \lor \neg \left(ew \leq 1.3 \cdot 10^{-142}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -6.2e-179) (not (<= ew 1.3e-142)))
   (fabs (* ew (cos t)))
   (fabs (* (* eh t) (sin (atan (/ (* eh t) (- ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6.2e-179) || !(ew <= 1.3e-142)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((eh * t) * sin(atan(((eh * t) / -ew)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-6.2d-179)) .or. (.not. (ew <= 1.3d-142))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs(((eh * t) * sin(atan(((eh * t) / -ew)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6.2e-179) || !(ew <= 1.3e-142)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs(((eh * t) * Math.sin(Math.atan(((eh * t) / -ew)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -6.2e-179) or not (ew <= 1.3e-142):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((eh * t) * math.sin(math.atan(((eh * t) / -ew)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -6.2e-179) || !(ew <= 1.3e-142))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(eh * t) * sin(atan(Float64(Float64(eh * t) / Float64(-ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -6.2e-179) || ~((ew <= 1.3e-142)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((eh * t) * sin(atan(((eh * t) / -ew)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -6.2e-179], N[Not[LessEqual[ew, 1.3e-142]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * t), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.2000000000000005e-179 or 1.3e-142 < ew

   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-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ew around inf 74.1%

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

   if -6.2000000000000005e-179 < ew < 1.3e-142

   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 99.9%

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

   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 48.4%

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

   6. Taylor expanded in ew around 0 36.8%

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

      1. associate-*r* 36.8%

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

      2. mul-1-neg 36.8%

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

      3. associate-*l/ 36.8%

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

      4. *-commutative 36.8%

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

      5. distribute-rgt-neg-in 36.8%

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

      6. mul-1-neg 36.8%

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

      7. associate-*r/ 36.8%

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

      8. neg-mul-1 36.8%

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

   8. Simplified 36.8%

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

   9. Taylor expanded in t around 0 36.8%

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

       1. mul-1-neg 36.8%

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

       2. *-commutative 36.8%

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

       3. associate-/l* 36.8%

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

       4. distribute-rgt-neg-out 36.8%

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

       5. distribute-neg-frac 36.8%

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

       6. associate-*r/ 36.8%

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

   11. Simplified 36.8%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.2 \cdot 10^{-179} \lor \neg \left(ew \leq 1.3 \cdot 10^{-142}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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. add-cbrt-cube 99.7%

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

   2. pow3 99.8%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in ew around inf 60.5%

   \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
8. Add Preprocessing

Alternative 11: 41.4% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-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-cbrt-cube 99.7%

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

   2. pow3 99.8%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in t around 0 46.6%

   \[\leadsto \left|\color{blue}{ew}\right| \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))))