Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (ew * tan(t))))
    code = abs(((eh * (cos(t) * sin(t_1))) + (ew * (cos(t_1) * sin(t)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

Alternative 2: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_3 := \sin t\_2\\ \mathbf{if}\;\left(eh \cdot \cos t\right) \cdot t\_3 + t\_1 \cdot \cos t\_2 \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \cos t \cdot \left(eh \cdot t\_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2 (atan (/ (/ eh ew) (tan t))))
        (t_3 (sin t_2)))
   (if (<= (+ (* (* eh (cos t)) t_3) (* t_1 (cos t_2))) -1e-295)
     (fabs (* (cos t) (* eh (sin (atan (/ eh (* ew (tan t))))))))
     (+ t_1 (* (cos t) (* eh t_3))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	double t_3 = sin(t_2);
	double tmp;
	if ((((eh * cos(t)) * t_3) + (t_1 * cos(t_2))) <= -1e-295) {
		tmp = fabs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = t_1 + (cos(t) * (eh * t_3));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = atan(((eh / ew) / tan(t)))
    t_3 = sin(t_2)
    if ((((eh * cos(t)) * t_3) + (t_1 * cos(t_2))) <= (-1d-295)) then
        tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))))
    else
        tmp = t_1 + (cos(t) * (eh * t_3))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.atan(((eh / ew) / Math.tan(t)));
	double t_3 = Math.sin(t_2);
	double tmp;
	if ((((eh * Math.cos(t)) * t_3) + (t_1 * Math.cos(t_2))) <= -1e-295) {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	} else {
		tmp = t_1 + (Math.cos(t) * (eh * t_3));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.atan(((eh / ew) / math.tan(t)))
	t_3 = math.sin(t_2)
	tmp = 0
	if (((eh * math.cos(t)) * t_3) + (t_1 * math.cos(t_2))) <= -1e-295:
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	else:
		tmp = t_1 + (math.cos(t) * (eh * t_3))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	t_3 = sin(t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(eh * cos(t)) * t_3) + Float64(t_1 * cos(t_2))) <= -1e-295)
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = Float64(t_1 + Float64(cos(t) * Float64(eh * t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = atan(((eh / ew) / tan(t)));
	t_3 = sin(t_2);
	tmp = 0.0;
	if ((((eh * cos(t)) * t_3) + (t_1 * cos(t_2))) <= -1e-295)
		tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	else
		tmp = t_1 + (cos(t) * (eh * t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, If[LessEqual[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-295], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 + N[(N[Cos[t], $MachinePrecision] * N[(eh * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -1.00000000000000006e-295

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*r* 66.6%

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

      3. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   if -1.00000000000000006e-295 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 99.1%

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

      2. fabs-sqr 99.1%

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

      3. add-sqr-sqrt 99.8%

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

      4. fma-undefine 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in ew around inf 98.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin t + \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (fabs
    (+
     (* eh (* (cos t) (sin (atan t_1))))
     (/ (* ew (sin t)) (hypot 1.0 t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	return fabs(((eh * (cos(t) * sin(atan(t_1)))) + ((ew * sin(t)) / hypot(1.0, t_1))));
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	return Math.abs(((eh * (Math.cos(t) * Math.sin(Math.atan(t_1)))) + ((ew * Math.sin(t)) / Math.hypot(1.0, t_1))));
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	return math.fabs(((eh * (math.cos(t) * math.sin(math.atan(t_1)))) + ((ew * math.sin(t)) / math.hypot(1.0, t_1))))

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	return abs(Float64(Float64(eh * Float64(cos(t) * sin(atan(t_1)))) + Float64(Float64(ew * sin(t)) / hypot(1.0, t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	tmp = abs(((eh * (cos(t) * sin(atan(t_1)))) + ((ew * sin(t)) / hypot(1.0, t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l/ 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 98.6%

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

4. Final simplification 98.6%

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

Alternative 6: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in ew around inf 98.4%

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

Alternative 7: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6 \cdot 10^{+72} \lor \neg \left(ew \leq 6 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -6e+72) (not (<= ew 6e+69)))
   (fabs (* (sin t) (* ew (cos (atan (/ (/ eh ew) (tan t)))))))
   (fabs (* (cos t) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6e+72) || !(ew <= 6e+69)) {
		tmp = fabs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs((cos(t) * (eh * sin(atan((eh / (ew * tan(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 <= (-6d+72)) .or. (.not. (ew <= 6d+69))) then
        tmp = abs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))))
    else
        tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6e+72) || !(ew <= 6e+69)) {
		tmp = Math.abs((Math.sin(t) * (ew * Math.cos(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -6e+72) or not (ew <= 6e+69):
		tmp = math.fabs((math.sin(t) * (ew * math.cos(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -6e+72) || !(ew <= 6e+69))
		tmp = abs(Float64(sin(t) * Float64(ew * cos(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -6e+72) || ~((ew <= 6e+69)))
		tmp = abs((sin(t) * (ew * cos(atan(((eh / ew) / tan(t)))))));
	else
		tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -6e+72], N[Not[LessEqual[ew, 6e+69]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(ew * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.00000000000000006e72 or 5.99999999999999967e69 < ew

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/l* 99.7%

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

      3. fma-define 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/l* 99.7%

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

      6. associate-/r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in ew around inf 71.8%

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

      1. associate-*r* 71.8%

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

      2. associate-/r* 71.8%

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

      3. *-commutative 71.8%

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

   10. Simplified 71.8%

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

   if -6.00000000000000006e72 < ew < 5.99999999999999967e69

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*r* 82.8%

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

      3. *-commutative 82.8%

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

   7. Simplified 82.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6 \cdot 10^{+72} \lor \neg \left(ew \leq 6 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\sin t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in ew around 0 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 65.1%

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

   3. *-commutative 65.1%

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

7. Simplified 65.1%

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

Alternative 9: 43.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\left|ew \cdot \left(eh \cdot \left(\cos t \cdot \frac{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -4.5e-7)
   (fabs (* ew (* eh (* (cos t) (/ (sin (atan (/ eh (* t ew)))) ew)))))
   (fabs eh)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -4.5e-7) {
		tmp = fabs((ew * (eh * (cos(t) * (sin(atan((eh / (t * ew)))) / ew)))));
	} else {
		tmp = fabs(eh);
	}
	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 (t <= (-4.5d-7)) then
        tmp = abs((ew * (eh * (cos(t) * (sin(atan((eh / (t * ew)))) / ew)))))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -4.5e-7) {
		tmp = Math.abs((ew * (eh * (Math.cos(t) * (Math.sin(Math.atan((eh / (t * ew)))) / ew)))));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= -4.5e-7:
		tmp = math.fabs((ew * (eh * (math.cos(t) * (math.sin(math.atan((eh / (t * ew)))) / ew)))))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -4.5e-7)
		tmp = abs(Float64(ew * Float64(eh * Float64(cos(t) * Float64(sin(atan(Float64(eh / Float64(t * ew)))) / ew)))));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= -4.5e-7)
		tmp = abs((ew * (eh * (cos(t) * (sin(atan((eh / (t * ew)))) / ew)))));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -4.5e-7], N[Abs[N[(ew * N[(eh * N[(N[Cos[t], $MachinePrecision] * N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.4999999999999998e-7

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ew around inf 86.1%

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

      1. +-commutative 86.1%

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

      2. associate-/l* 86.0%

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

      3. fma-define 86.0%

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

      4. *-commutative 86.0%

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

      5. associate-/l* 86.0%

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

      6. associate-/r* 86.0%

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

      7. *-commutative 86.0%

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

      8. associate-/r* 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in eh around inf 49.9%

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

      1. associate-/l* 49.6%

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

      2. associate-/l* 49.6%

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

      3. associate-/r* 49.6%

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

   10. Simplified 49.6%

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

   11. Taylor expanded in t around 0 25.4%

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

   if -4.4999999999999998e-7 < t

   1. Initial program 99.9%

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

      1. fma-define 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 49.9%

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

   6. Taylor expanded in t around 0 48.3%

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

      1. sin-atan 15.4%

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

      2. hypot-1-def 26.9%

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

   8. Applied egg-rr 26.9%

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

      1. associate-/l/ 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in eh around inf 50.3%

       \[\leadsto \left|\color{blue}{eh}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\left|ew \cdot \left(eh \cdot \left(\cos t \cdot \frac{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.9% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

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(eh)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 40.5%

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

6. Taylor expanded in t around 0 38.1%

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

   1. sin-atan 13.1%

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

   2. hypot-1-def 22.1%

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

8. Applied egg-rr 22.1%

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

   1. associate-/l/ 22.2%

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

10. Simplified 22.2%

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

11. Taylor expanded in eh around inf 40.9%

    \[\leadsto \left|\color{blue}{eh}\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))