Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.2s

Alternatives: 12

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

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

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.sin(t)) * (1.0 / Math.hypot(1.0, ((eh / Math.tan(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)) * (1.0 / math.hypot(1.0, ((eh / math.tan(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)) * Float64(1.0 / hypot(1.0, Float64(Float64(eh / tan(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)) * (1.0 / hypot(1.0, ((eh / tan(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[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

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 t) ew)))))))

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 / t) / ew))))));
}

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 / t) / ew))))));
}

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 / t) / ew))))))

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(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] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 85.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_2 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -8.5 \cdot 10^{-43} \lor \neg \left(ew \leq 1.65 \cdot 10^{-85}\right):\\ \;\;\;\;\left|t\_2 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t\_1 + t\_2 \cdot \frac{ew \cdot t}{eh}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (/ eh ew) (tan t))))) (t_2 (* ew (sin t))))
   (if (or (<= ew -8.5e-43) (not (<= ew 1.65e-85)))
     (fabs (+ (* t_2 (/ 1.0 (hypot 1.0 (/ (/ eh t) ew)))) (* eh t_1)))
     (fabs (+ (* (* eh (cos t)) t_1) (* t_2 (/ (* ew t) eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((eh / ew) / tan(t))));
	double t_2 = ew * sin(t);
	double tmp;
	if ((ew <= -8.5e-43) || !(ew <= 1.65e-85)) {
		tmp = fabs(((t_2 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * t_1)));
	} else {
		tmp = fabs((((eh * cos(t)) * t_1) + (t_2 * ((ew * t) / eh))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan(((eh / ew) / Math.tan(t))));
	double t_2 = ew * Math.sin(t);
	double tmp;
	if ((ew <= -8.5e-43) || !(ew <= 1.65e-85)) {
		tmp = Math.abs(((t_2 * (1.0 / Math.hypot(1.0, ((eh / t) / ew)))) + (eh * t_1)));
	} else {
		tmp = Math.abs((((eh * Math.cos(t)) * t_1) + (t_2 * ((ew * t) / eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((eh / ew) / math.tan(t))))
	t_2 = ew * math.sin(t)
	tmp = 0
	if (ew <= -8.5e-43) or not (ew <= 1.65e-85):
		tmp = math.fabs(((t_2 * (1.0 / math.hypot(1.0, ((eh / t) / ew)))) + (eh * t_1)))
	else:
		tmp = math.fabs((((eh * math.cos(t)) * t_1) + (t_2 * ((ew * t) / eh))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(eh / ew) / tan(t))))
	t_2 = Float64(ew * sin(t))
	tmp = 0.0
	if ((ew <= -8.5e-43) || !(ew <= 1.65e-85))
		tmp = abs(Float64(Float64(t_2 * Float64(1.0 / hypot(1.0, Float64(Float64(eh / t) / ew)))) + Float64(eh * t_1)));
	else
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * t_1) + Float64(t_2 * Float64(Float64(ew * t) / eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((eh / ew) / tan(t))));
	t_2 = ew * sin(t);
	tmp = 0.0;
	if ((ew <= -8.5e-43) || ~((ew <= 1.65e-85)))
		tmp = abs(((t_2 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * t_1)));
	else
		tmp = abs((((eh * cos(t)) * t_1) + (t_2 * ((ew * t) / eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -8.5e-43], N[Not[LessEqual[ew, 1.65e-85]], $MachinePrecision]], N[Abs[N[(N[(t$95$2 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -8.50000000000000056e-43 or 1.64999999999999986e-85 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 99.2%

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

   6. Taylor expanded in t around 0 90.2%

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

   if -8.50000000000000056e-43 < ew < 1.64999999999999986e-85

   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. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 98.9%

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

   6. Taylor expanded in eh around inf 85.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -8.5 \cdot 10^{-43} \lor \neg \left(ew \leq 1.65 \cdot 10^{-85}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \frac{ew \cdot t}{eh}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -3.2 \cdot 10^{-43} \lor \neg \left(ew \leq 7.2 \cdot 10^{-86}\right):\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + t\_1 \cdot \left(ew \cdot \frac{t}{eh}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (or (<= ew -3.2e-43) (not (<= ew 7.2e-86)))
     (fabs
      (+
       (* t_1 (/ 1.0 (hypot 1.0 (/ (/ eh t) ew))))
       (* eh (sin (atan (/ eh (* ew t)))))))
     (fabs
      (+
       (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))
       (* t_1 (* ew (/ t eh))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if ((ew <= -3.2e-43) || !(ew <= 7.2e-86)) {
		tmp = fabs(((t_1 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * sin(atan((eh / (ew * t)))))));
	} else {
		tmp = fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (t_1 * (ew * (t / eh)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double tmp;
	if ((ew <= -3.2e-43) || !(ew <= 7.2e-86)) {
		tmp = Math.abs(((t_1 * (1.0 / Math.hypot(1.0, ((eh / t) / ew)))) + (eh * Math.sin(Math.atan((eh / (ew * t)))))));
	} else {
		tmp = Math.abs((((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))))) + (t_1 * (ew * (t / eh)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	tmp = 0
	if (ew <= -3.2e-43) or not (ew <= 7.2e-86):
		tmp = math.fabs(((t_1 * (1.0 / math.hypot(1.0, ((eh / t) / ew)))) + (eh * math.sin(math.atan((eh / (ew * t)))))))
	else:
		tmp = math.fabs((((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))) + (t_1 * (ew * (t / eh)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if ((ew <= -3.2e-43) || !(ew <= 7.2e-86))
		tmp = abs(Float64(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(Float64(eh / t) / ew)))) + Float64(eh * sin(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(t_1 * Float64(ew * Float64(t / eh)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if ((ew <= -3.2e-43) || ~((ew <= 7.2e-86)))
		tmp = abs(((t_1 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * sin(atan((eh / (ew * t)))))));
	else
		tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (t_1 * (ew * (t / eh)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -3.2e-43], N[Not[LessEqual[ew, 7.2e-86]], $MachinePrecision]], N[Abs[N[(N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(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] + N[(t$95$1 * N[(ew * N[(t / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.19999999999999985e-43 or 7.19999999999999932e-86 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 99.2%

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

   6. Taylor expanded in t around 0 93.5%

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

   7. Taylor expanded in t around 0 89.3%

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

      1. *-commutative 89.3%

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

   9. Simplified 89.3%

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

   if -3.19999999999999985e-43 < ew < 7.19999999999999932e-86

   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. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 98.9%

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

   6. Taylor expanded in eh around inf 85.9%

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

      1. associate-*r/ 77.0%

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

   8. Simplified 84.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.2 \cdot 10^{-43} \lor \neg \left(ew \leq 7.2 \cdot 10^{-86}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \left(ew \cdot \frac{t}{eh}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -1.4 \cdot 10^{-43} \lor \neg \left(ew \leq 1.65 \cdot 10^{-87}\right):\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + t\_1 \cdot \frac{ew \cdot t}{eh}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (or (<= ew -1.4e-43) (not (<= ew 1.65e-87)))
     (fabs
      (+
       (* t_1 (/ 1.0 (hypot 1.0 (/ (/ eh t) ew))))
       (* eh (sin (atan (/ eh (* ew t)))))))
     (fabs
      (+
       (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))
       (* t_1 (/ (* ew t) eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if ((ew <= -1.4e-43) || !(ew <= 1.65e-87)) {
		tmp = fabs(((t_1 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * sin(atan((eh / (ew * t)))))));
	} else {
		tmp = fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (t_1 * ((ew * t) / eh))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double tmp;
	if ((ew <= -1.4e-43) || !(ew <= 1.65e-87)) {
		tmp = Math.abs(((t_1 * (1.0 / Math.hypot(1.0, ((eh / t) / ew)))) + (eh * Math.sin(Math.atan((eh / (ew * t)))))));
	} else {
		tmp = Math.abs((((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))))) + (t_1 * ((ew * t) / eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	tmp = 0
	if (ew <= -1.4e-43) or not (ew <= 1.65e-87):
		tmp = math.fabs(((t_1 * (1.0 / math.hypot(1.0, ((eh / t) / ew)))) + (eh * math.sin(math.atan((eh / (ew * t)))))))
	else:
		tmp = math.fabs((((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))) + (t_1 * ((ew * t) / eh))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if ((ew <= -1.4e-43) || !(ew <= 1.65e-87))
		tmp = abs(Float64(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(Float64(eh / t) / ew)))) + Float64(eh * sin(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(t_1 * Float64(Float64(ew * t) / eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if ((ew <= -1.4e-43) || ~((ew <= 1.65e-87)))
		tmp = abs(((t_1 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * sin(atan((eh / (ew * t)))))));
	else
		tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (t_1 * ((ew * t) / eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -1.4e-43], N[Not[LessEqual[ew, 1.65e-87]], $MachinePrecision]], N[Abs[N[(N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(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] + N[(t$95$1 * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.3999999999999999e-43 or 1.65e-87 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 99.2%

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

   6. Taylor expanded in t around 0 93.5%

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

   7. Taylor expanded in t around 0 89.3%

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

      1. *-commutative 89.3%

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

   9. Simplified 89.3%

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

   if -1.3999999999999999e-43 < ew < 1.65e-87

   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. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 98.9%

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

   6. Taylor expanded in eh around inf 85.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.4 \cdot 10^{-43} \lor \neg \left(ew \leq 1.65 \cdot 10^{-87}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \frac{ew \cdot t}{eh}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0 99.0%

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

6. Taylor expanded in t around 0 90.1%

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

7. Final simplification 90.1%

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

Alternative 7: 81.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ t_3 := \left(eh \cdot \cos t\right) \cdot t\_2\\ \mathbf{if}\;eh \leq -2.4 \cdot 10^{+237}:\\ \;\;\;\;\left|t\_3 + t\_1 \cdot \frac{ew \cdot t}{eh}\right|\\ \mathbf{elif}\;eh \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3 + t\_1 \cdot \left(ew \cdot \frac{t}{eh}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2 (sin (atan (/ eh (* ew t)))))
        (t_3 (* (* eh (cos t)) t_2)))
   (if (<= eh -2.4e+237)
     (fabs (+ t_3 (* t_1 (/ (* ew t) eh))))
     (if (<= eh 5e+119)
       (fabs (+ (* t_1 (/ 1.0 (hypot 1.0 (/ (/ eh t) ew)))) (* eh t_2)))
       (fabs (+ t_3 (* t_1 (* ew (/ t eh)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = sin(atan((eh / (ew * t))));
	double t_3 = (eh * cos(t)) * t_2;
	double tmp;
	if (eh <= -2.4e+237) {
		tmp = fabs((t_3 + (t_1 * ((ew * t) / eh))));
	} else if (eh <= 5e+119) {
		tmp = fabs(((t_1 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * t_2)));
	} else {
		tmp = fabs((t_3 + (t_1 * (ew * (t / eh)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.sin(Math.atan((eh / (ew * t))));
	double t_3 = (eh * Math.cos(t)) * t_2;
	double tmp;
	if (eh <= -2.4e+237) {
		tmp = Math.abs((t_3 + (t_1 * ((ew * t) / eh))));
	} else if (eh <= 5e+119) {
		tmp = Math.abs(((t_1 * (1.0 / Math.hypot(1.0, ((eh / t) / ew)))) + (eh * t_2)));
	} else {
		tmp = Math.abs((t_3 + (t_1 * (ew * (t / eh)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.sin(math.atan((eh / (ew * t))))
	t_3 = (eh * math.cos(t)) * t_2
	tmp = 0
	if eh <= -2.4e+237:
		tmp = math.fabs((t_3 + (t_1 * ((ew * t) / eh))))
	elif eh <= 5e+119:
		tmp = math.fabs(((t_1 * (1.0 / math.hypot(1.0, ((eh / t) / ew)))) + (eh * t_2)))
	else:
		tmp = math.fabs((t_3 + (t_1 * (ew * (t / eh)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = sin(atan(Float64(eh / Float64(ew * t))))
	t_3 = Float64(Float64(eh * cos(t)) * t_2)
	tmp = 0.0
	if (eh <= -2.4e+237)
		tmp = abs(Float64(t_3 + Float64(t_1 * Float64(Float64(ew * t) / eh))));
	elseif (eh <= 5e+119)
		tmp = abs(Float64(Float64(t_1 * Float64(1.0 / hypot(1.0, Float64(Float64(eh / t) / ew)))) + Float64(eh * t_2)));
	else
		tmp = abs(Float64(t_3 + Float64(t_1 * Float64(ew * Float64(t / eh)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = sin(atan((eh / (ew * t))));
	t_3 = (eh * cos(t)) * t_2;
	tmp = 0.0;
	if (eh <= -2.4e+237)
		tmp = abs((t_3 + (t_1 * ((ew * t) / eh))));
	elseif (eh <= 5e+119)
		tmp = abs(((t_1 * (1.0 / hypot(1.0, ((eh / t) / ew)))) + (eh * t_2)));
	else
		tmp = abs((t_3 + (t_1 * (ew * (t / eh)))));
	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[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[eh, -2.4e+237], N[Abs[N[(t$95$3 + N[(t$95$1 * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 5e+119], N[Abs[N[(N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$3 + N[(t$95$1 * N[(ew * N[(t / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -2.3999999999999999e237

   1. Initial program 99.6%

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

      1. cos-atan 99.6%

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

      2. hypot-1-def 99.6%

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

      3. associate-/l/ 99.6%

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

      4. associate-/r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0 99.6%

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

   6. Taylor expanded in t around 0 85.3%

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

   7. Taylor expanded in eh around inf 85.7%

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

   if -2.3999999999999999e237 < eh < 4.9999999999999999e119

   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. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. associate-/r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 99.3%

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

   6. Taylor expanded in t around 0 91.4%

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

   7. Taylor expanded in t around 0 85.5%

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

      1. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   if 4.9999999999999999e119 < eh

   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. Add Preprocessing
   3. Step-by-step derivation

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/l/ 99.9%

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

      4. associate-/r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 97.5%

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

   6. Taylor expanded in t around 0 85.6%

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

   7. Taylor expanded in eh around inf 85.4%

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

      1. associate-*r/ 85.4%

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

   9. Simplified 85.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.4 \cdot 10^{+237}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \frac{ew \cdot t}{eh}\right|\\ \mathbf{elif}\;eh \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\frac{eh}{t}}{ew}\right)} + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \left(ew \cdot \frac{t}{eh}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ t_2 := \sin t\_1\\ \mathbf{if}\;t \leq -70 \lor \neg \left(t \leq 31000\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t\_2 + \left(ew \cdot \sin t\right) \cdot \left(ew \cdot \frac{t}{eh}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \cos t\_1\right) + eh \cdot t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew t)))) (t_2 (sin t_1)))
   (if (or (<= t -70.0) (not (<= t 31000.0)))
     (fabs (+ (* (* eh (cos t)) t_2) (* (* ew (sin t)) (* ew (/ t eh)))))
     (fabs (+ (* ew (* t (cos t_1))) (* eh t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * t)));
	double t_2 = sin(t_1);
	double tmp;
	if ((t <= -70.0) || !(t <= 31000.0)) {
		tmp = fabs((((eh * cos(t)) * t_2) + ((ew * sin(t)) * (ew * (t / eh)))));
	} else {
		tmp = fabs(((ew * (t * cos(t_1))) + (eh * t_2)));
	}
	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) :: tmp
    t_1 = atan((eh / (ew * t)))
    t_2 = sin(t_1)
    if ((t <= (-70.0d0)) .or. (.not. (t <= 31000.0d0))) then
        tmp = abs((((eh * cos(t)) * t_2) + ((ew * sin(t)) * (ew * (t / eh)))))
    else
        tmp = abs(((ew * (t * cos(t_1))) + (eh * t_2)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh / (ew * t)));
	double t_2 = Math.sin(t_1);
	double tmp;
	if ((t <= -70.0) || !(t <= 31000.0)) {
		tmp = Math.abs((((eh * Math.cos(t)) * t_2) + ((ew * Math.sin(t)) * (ew * (t / eh)))));
	} else {
		tmp = Math.abs(((ew * (t * Math.cos(t_1))) + (eh * t_2)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * t)))
	t_2 = math.sin(t_1)
	tmp = 0
	if (t <= -70.0) or not (t <= 31000.0):
		tmp = math.fabs((((eh * math.cos(t)) * t_2) + ((ew * math.sin(t)) * (ew * (t / eh)))))
	else:
		tmp = math.fabs(((ew * (t * math.cos(t_1))) + (eh * t_2)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * t)))
	t_2 = sin(t_1)
	tmp = 0.0
	if ((t <= -70.0) || !(t <= 31000.0))
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * t_2) + Float64(Float64(ew * sin(t)) * Float64(ew * Float64(t / eh)))));
	else
		tmp = abs(Float64(Float64(ew * Float64(t * cos(t_1))) + Float64(eh * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh / (ew * t)));
	t_2 = sin(t_1);
	tmp = 0.0;
	if ((t <= -70.0) || ~((t <= 31000.0)))
		tmp = abs((((eh * cos(t)) * t_2) + ((ew * sin(t)) * (ew * (t / eh)))));
	else
		tmp = abs(((ew * (t * cos(t_1))) + (eh * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[Or[LessEqual[t, -70.0], N[Not[LessEqual[t, 31000.0]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(ew * N[(t / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[(t * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -70 or 31000 < t

   1. Initial program 99.6%

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

      1. cos-atan 99.6%

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

      2. hypot-1-def 99.6%

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

      3. associate-/l/ 99.6%

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

      4. associate-/r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0 97.9%

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

   6. Taylor expanded in t around 0 78.6%

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

   7. Taylor expanded in eh around inf 31.1%

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

      1. associate-*r/ 30.9%

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

   9. Simplified 30.9%

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

   if -70 < t < 31000

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   4. Taylor expanded in t around 0 96.9%

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

   5. Taylor expanded in t around 0 96.9%

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

   6. Taylor expanded in t around 0 96.9%

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

      1. *-commutative 96.9%

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

   8. Simplified 96.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -70 \lor \neg \left(t \leq 31000\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \left(ew \cdot \frac{t}{eh}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right) + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ t_2 := \sin t\_1\\ \mathbf{if}\;t \leq -70 \lor \neg \left(t \leq 31000\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t\_2 + \left(ew \cdot \sin t\right) \cdot \frac{ew \cdot t}{eh}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \cos t\_1\right) + eh \cdot t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew t)))) (t_2 (sin t_1)))
   (if (or (<= t -70.0) (not (<= t 31000.0)))
     (fabs (+ (* (* eh (cos t)) t_2) (* (* ew (sin t)) (/ (* ew t) eh))))
     (fabs (+ (* ew (* t (cos t_1))) (* eh t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * t)));
	double t_2 = sin(t_1);
	double tmp;
	if ((t <= -70.0) || !(t <= 31000.0)) {
		tmp = fabs((((eh * cos(t)) * t_2) + ((ew * sin(t)) * ((ew * t) / eh))));
	} else {
		tmp = fabs(((ew * (t * cos(t_1))) + (eh * t_2)));
	}
	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) :: tmp
    t_1 = atan((eh / (ew * t)))
    t_2 = sin(t_1)
    if ((t <= (-70.0d0)) .or. (.not. (t <= 31000.0d0))) then
        tmp = abs((((eh * cos(t)) * t_2) + ((ew * sin(t)) * ((ew * t) / eh))))
    else
        tmp = abs(((ew * (t * cos(t_1))) + (eh * t_2)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh / (ew * t)));
	double t_2 = Math.sin(t_1);
	double tmp;
	if ((t <= -70.0) || !(t <= 31000.0)) {
		tmp = Math.abs((((eh * Math.cos(t)) * t_2) + ((ew * Math.sin(t)) * ((ew * t) / eh))));
	} else {
		tmp = Math.abs(((ew * (t * Math.cos(t_1))) + (eh * t_2)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * t)))
	t_2 = math.sin(t_1)
	tmp = 0
	if (t <= -70.0) or not (t <= 31000.0):
		tmp = math.fabs((((eh * math.cos(t)) * t_2) + ((ew * math.sin(t)) * ((ew * t) / eh))))
	else:
		tmp = math.fabs(((ew * (t * math.cos(t_1))) + (eh * t_2)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * t)))
	t_2 = sin(t_1)
	tmp = 0.0
	if ((t <= -70.0) || !(t <= 31000.0))
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * t_2) + Float64(Float64(ew * sin(t)) * Float64(Float64(ew * t) / eh))));
	else
		tmp = abs(Float64(Float64(ew * Float64(t * cos(t_1))) + Float64(eh * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh / (ew * t)));
	t_2 = sin(t_1);
	tmp = 0.0;
	if ((t <= -70.0) || ~((t <= 31000.0)))
		tmp = abs((((eh * cos(t)) * t_2) + ((ew * sin(t)) * ((ew * t) / eh))));
	else
		tmp = abs(((ew * (t * cos(t_1))) + (eh * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[Or[LessEqual[t, -70.0], N[Not[LessEqual[t, 31000.0]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[(t * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -70 or 31000 < t

   1. Initial program 99.6%

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

      1. cos-atan 99.6%

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

      2. hypot-1-def 99.6%

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

      3. associate-/l/ 99.6%

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

      4. associate-/r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0 97.9%

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

   6. Taylor expanded in t around 0 78.6%

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

   7. Taylor expanded in eh around inf 31.1%

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

   if -70 < t < 31000

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   4. Taylor expanded in t around 0 96.9%

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

   5. Taylor expanded in t around 0 96.9%

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

   6. Taylor expanded in t around 0 96.9%

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

      1. *-commutative 96.9%

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

   8. Simplified 96.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -70 \lor \neg \left(t \leq 31000\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \frac{ew \cdot t}{eh}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right) + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t * N[(ew / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $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 81.6%

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

4. Taylor expanded in t around 0 57.3%

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

5. Taylor expanded in t around 0 57.2%

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

   1. associate-*r* 57.2%

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

   2. associate-/l/ 57.2%

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

   3. cos-atan 57.2%

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

   4. hypot-1-def 57.2%

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

   5. un-div-inv 57.2%

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

   6. *-commutative 57.2%

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

   7. associate-/l/ 57.2%

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

   8. associate-/r* 57.2%

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

7. Applied egg-rr 57.2%

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

   1. associate-/l* 57.2%

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

   2. associate-/l/ 57.2%

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

   3. *-commutative 57.2%

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

9. Simplified 57.2%

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

10. Final simplification 57.2%

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

Alternative 11: 54.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in t around 0 81.6%

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

4. Taylor expanded in t around 0 57.3%

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

5. Taylor expanded in t around 0 57.2%

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

   1. associate-*r* 57.2%

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

   2. associate-/l/ 57.2%

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

   3. cos-atan 57.2%

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

   4. hypot-1-def 57.2%

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

   5. un-div-inv 57.2%

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

   6. *-commutative 57.2%

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

   7. associate-/l/ 57.2%

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

   8. associate-/r* 57.2%

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

7. Applied egg-rr 57.2%

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

8. Final simplification 57.2%

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

Alternative 12: 54.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(ew * N[(t * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[t$95$1], $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 81.6%

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

4. Taylor expanded in t around 0 57.3%

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

5. Taylor expanded in t around 0 57.2%

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

6. Taylor expanded in t around 0 57.1%

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

   1. *-commutative 57.1%

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

8. Simplified 57.1%

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

9. Final simplification 57.1%

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

Reproduce

herbie shell --seed 2024077 
(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))))))))