Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.7s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $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. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((ew * sin(t)) * cos(atan((eh / (ew * 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)) * cos(atan((eh / (ew * 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)) * Math.cos(Math.atan((eh / (ew * 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)) * math.cos(math.atan((eh / (ew * t)))))))

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)) * cos(atan(Float64(eh / Float64(ew * t)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((ew * sin(t)) * cos(atan((eh / (ew * t)))))));
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[Cos[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. Taylor expanded in t around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $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. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 99.0%

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

Alternative 4: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;ew \leq -230000000 \lor \neg \left(ew \leq 2.8 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \cos t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew (tan t))))))
   (if (or (<= ew -230000000.0) (not (<= ew 2.8e+18)))
     (fabs (* (* ew (sin t)) (cos t_1)))
     (fabs (* (cos t) (* eh (sin t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	double tmp;
	if ((ew <= -230000000.0) || !(ew <= 2.8e+18)) {
		tmp = fabs(((ew * sin(t)) * cos(t_1)));
	} else {
		tmp = fabs((cos(t) * (eh * sin(t_1))));
	}
	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) :: tmp
    t_1 = atan((eh / (ew * tan(t))))
    if ((ew <= (-230000000.0d0)) .or. (.not. (ew <= 2.8d+18))) then
        tmp = abs(((ew * sin(t)) * cos(t_1)))
    else
        tmp = abs((cos(t) * (eh * sin(t_1))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh / (ew * Math.tan(t))));
	double tmp;
	if ((ew <= -230000000.0) || !(ew <= 2.8e+18)) {
		tmp = Math.abs(((ew * Math.sin(t)) * Math.cos(t_1)));
	} else {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * math.tan(t))))
	tmp = 0
	if (ew <= -230000000.0) or not (ew <= 2.8e+18):
		tmp = math.fabs(((ew * math.sin(t)) * math.cos(t_1)))
	else:
		tmp = math.fabs((math.cos(t) * (eh * math.sin(t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	tmp = 0.0
	if ((ew <= -230000000.0) || !(ew <= 2.8e+18))
		tmp = abs(Float64(Float64(ew * sin(t)) * cos(t_1)));
	else
		tmp = abs(Float64(cos(t) * Float64(eh * sin(t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh / (ew * tan(t))));
	tmp = 0.0;
	if ((ew <= -230000000.0) || ~((ew <= 2.8e+18)))
		tmp = abs(((ew * sin(t)) * cos(t_1)));
	else
		tmp = abs((cos(t) * (eh * sin(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -230000000.0], N[Not[LessEqual[ew, 2.8e+18]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.3e8 or 2.8e18 < 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 eh around inf 78.3%

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

      1. +-commutative 78.3%

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

      2. associate-/l* 78.1%

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

      3. fma-define 78.1%

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

      4. *-commutative 78.1%

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

      5. associate-/r* 78.1%

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

      6. associate-/l* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in eh around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l* 71.3%

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

   10. Simplified 71.3%

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

   if -2.3e8 < ew < 2.8e18

   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 ew around 0 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*r* 85.4%

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

      3. *-commutative 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -230000000 \lor \neg \left(ew \leq 2.8 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ \mathbf{if}\;ew \leq -290000000 \lor \neg \left(ew \leq 5.4 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (if (or (<= ew -290000000.0) (not (<= ew 5.4e+18)))
     (fabs (* (* ew (sin t)) (/ 1.0 (hypot 1.0 t_1))))
     (fabs (* (cos t) (* eh (sin (atan t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	double tmp;
	if ((ew <= -290000000.0) || !(ew <= 5.4e+18)) {
		tmp = fabs(((ew * sin(t)) * (1.0 / hypot(1.0, t_1))));
	} else {
		tmp = fabs((cos(t) * (eh * sin(atan(t_1)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	double tmp;
	if ((ew <= -290000000.0) || !(ew <= 5.4e+18)) {
		tmp = Math.abs(((ew * Math.sin(t)) * (1.0 / Math.hypot(1.0, t_1))));
	} else {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan(t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	tmp = 0
	if (ew <= -290000000.0) or not (ew <= 5.4e+18):
		tmp = math.fabs(((ew * math.sin(t)) * (1.0 / math.hypot(1.0, t_1))))
	else:
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan(t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	tmp = 0.0
	if ((ew <= -290000000.0) || !(ew <= 5.4e+18))
		tmp = abs(Float64(Float64(ew * sin(t)) * Float64(1.0 / hypot(1.0, t_1))));
	else
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	tmp = 0.0;
	if ((ew <= -290000000.0) || ~((ew <= 5.4e+18)))
		tmp = abs(((ew * sin(t)) * (1.0 / hypot(1.0, t_1))));
	else
		tmp = abs((cos(t) * (eh * sin(atan(t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -290000000.0], N[Not[LessEqual[ew, 5.4e+18]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.9e8 or 5.4e18 < 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 eh around inf 78.3%

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

      1. +-commutative 78.3%

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

      2. associate-/l* 78.1%

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

      3. fma-define 78.1%

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

      4. *-commutative 78.1%

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

      5. associate-/r* 78.1%

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

      6. associate-/l* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in eh around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l* 71.3%

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

   10. Simplified 71.3%

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

       1. associate-/r* 71.3%

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

       2. cos-atan 70.9%

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

       3. hypot-1-def 70.9%

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

       4. associate-/r* 70.9%

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

   12. Applied egg-rr 70.9%

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

   if -2.9e8 < ew < 5.4e18

   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 ew around 0 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*r* 85.4%

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

      3. *-commutative 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -290000000 \lor \neg \left(ew \leq 5.4 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-113} \lor \neg \left(t \leq 1.1 \cdot 10^{-17}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (if (or (<= t -5.8e-113) (not (<= t 1.1e-17)))
     (fabs (* (* ew (sin t)) (/ 1.0 (hypot 1.0 t_1))))
     (fabs (* eh (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	double tmp;
	if ((t <= -5.8e-113) || !(t <= 1.1e-17)) {
		tmp = fabs(((ew * sin(t)) * (1.0 / hypot(1.0, t_1))));
	} else {
		tmp = fabs((eh * sin(atan(t_1))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	double tmp;
	if ((t <= -5.8e-113) || !(t <= 1.1e-17)) {
		tmp = Math.abs(((ew * Math.sin(t)) * (1.0 / Math.hypot(1.0, t_1))));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan(t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	tmp = 0
	if (t <= -5.8e-113) or not (t <= 1.1e-17):
		tmp = math.fabs(((ew * math.sin(t)) * (1.0 / math.hypot(1.0, t_1))))
	else:
		tmp = math.fabs((eh * math.sin(math.atan(t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	tmp = 0.0
	if ((t <= -5.8e-113) || !(t <= 1.1e-17))
		tmp = abs(Float64(Float64(ew * sin(t)) * Float64(1.0 / hypot(1.0, t_1))));
	else
		tmp = abs(Float64(eh * sin(atan(t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	tmp = 0.0;
	if ((t <= -5.8e-113) || ~((t <= 1.1e-17)))
		tmp = abs(((ew * sin(t)) * (1.0 / hypot(1.0, t_1))));
	else
		tmp = abs((eh * sin(atan(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -5.8e-113], N[Not[LessEqual[t, 1.1e-17]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.80000000000000008e-113 or 1.1e-17 < t

   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 99.8%

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

      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/ 99.7%

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

   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 eh around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. associate-/l* 82.1%

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

      3. fma-define 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-/r* 82.1%

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

      6. associate-/l* 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in eh around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-*l* 60.9%

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

   10. Simplified 60.9%

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

       1. associate-/r* 60.9%

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

       2. cos-atan 60.6%

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

       3. hypot-1-def 60.6%

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

       4. associate-/r* 60.6%

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

   12. Applied egg-rr 60.6%

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

   if -5.80000000000000008e-113 < t < 1.1e-17

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 86.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-113} \lor \neg \left(t \leq 1.1 \cdot 10^{-17}\right):\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-113} \lor \neg \left(t \leq 1.9 \cdot 10^{-22}\right):\\ \;\;\;\;ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -5.8e-113) (not (<= t 1.9e-22)))
   (+ (* ew (sin t)) (* eh (sin (atan (/ (/ eh ew) (tan t))))))
   (fabs (* eh (sin (atan (/ eh (* ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -5.8e-113) || !(t <= 1.9e-22)) {
		tmp = (ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))));
	} else {
		tmp = fabs((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 ((t <= (-5.8d-113)) .or. (.not. (t <= 1.9d-22))) then
        tmp = (ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))
    else
        tmp = abs((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 ((t <= -5.8e-113) || !(t <= 1.9e-22)) {
		tmp = (ew * Math.sin(t)) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -5.8e-113) or not (t <= 1.9e-22):
		tmp = (ew * math.sin(t)) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))
	else:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -5.8e-113) || !(t <= 1.9e-22))
		tmp = Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t))))));
	else
		tmp = abs(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 ((t <= -5.8e-113) || ~((t <= 1.9e-22)))
		tmp = (ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))));
	else
		tmp = abs((eh * sin(atan((eh / (ew * tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -5.8e-113], N[Not[LessEqual[t, 1.9e-22]], $MachinePrecision]], N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.80000000000000008e-113 or 1.90000000000000012e-22 < t

   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 99.8%

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

      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/ 99.7%

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

   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. Step-by-step derivation

      1. add-sqr-sqrt 55.4%

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

      2. fabs-sqr 55.4%

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

      3. add-sqr-sqrt 56.2%

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

      4. fma-undefine 56.2%

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

      5. associate-/l/ 56.2%

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

      6. associate-*l* 56.2%

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

   6. Applied egg-rr 56.2%

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

   7. Taylor expanded in eh around 0 55.8%

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

   8. Taylor expanded in t around 0 42.3%

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

      1. associate-/r* 42.3%

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

   10. Simplified 42.3%

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

   if -5.80000000000000008e-113 < t < 1.90000000000000012e-22

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 86.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-113} \lor \neg \left(t \leq 1.9 \cdot 10^{-22}\right):\\ \;\;\;\;ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ \mathbf{if}\;t \leq -1.08 \cdot 10^{-63} \lor \neg \left(t \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (if (or (<= t -1.08e-63) (not (<= t 3.4e-16)))
     (/ (* ew (sin t)) (hypot 1.0 t_1))
     (fabs (* eh (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	double tmp;
	if ((t <= -1.08e-63) || !(t <= 3.4e-16)) {
		tmp = (ew * sin(t)) / hypot(1.0, t_1);
	} else {
		tmp = fabs((eh * sin(atan(t_1))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	double tmp;
	if ((t <= -1.08e-63) || !(t <= 3.4e-16)) {
		tmp = (ew * Math.sin(t)) / Math.hypot(1.0, t_1);
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan(t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	tmp = 0
	if (t <= -1.08e-63) or not (t <= 3.4e-16):
		tmp = (ew * math.sin(t)) / math.hypot(1.0, t_1)
	else:
		tmp = math.fabs((eh * math.sin(math.atan(t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	tmp = 0.0
	if ((t <= -1.08e-63) || !(t <= 3.4e-16))
		tmp = Float64(Float64(ew * sin(t)) / hypot(1.0, t_1));
	else
		tmp = abs(Float64(eh * sin(atan(t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	tmp = 0.0;
	if ((t <= -1.08e-63) || ~((t <= 3.4e-16)))
		tmp = (ew * sin(t)) / hypot(1.0, t_1);
	else
		tmp = abs((eh * sin(atan(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -1.08e-63], N[Not[LessEqual[t, 3.4e-16]], $MachinePrecision]], N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.07999999999999994e-63 or 3.4e-16 < t

   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 eh around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. associate-/l* 82.1%

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

      3. fma-define 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-/r* 82.1%

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

      6. associate-/l* 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in eh around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

   10. Simplified 61.1%

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

       1. log1p-expm1-u 61.0%

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

       2. associate-/r* 61.0%

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

       3. cos-atan 60.7%

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

       4. hypot-1-def 60.7%

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

       5. div-inv 60.7%

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

       6. add-sqr-sqrt 36.8%

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

       7. fabs-sqr 36.8%

          \[\leadsto \color{blue}{\sqrt{\frac{ew \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin t\right)\right)}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}} \cdot \sqrt{\frac{ew \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin t\right)\right)}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}}} \]

       8. add-sqr-sqrt 37.3%

          \[\leadsto \color{blue}{\frac{ew \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin t\right)\right)}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}} \]

       9. log1p-expm1-u 37.3%

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

       10. associate-/r* 37.3%

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

   12. Applied egg-rr 37.3%

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

   if -1.07999999999999994e-63 < t < 3.4e-16

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 83.7%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{-63} \lor \neg \left(t \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{-63} \lor \neg \left(t \leq 1.46 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.08e-63) (not (<= t 1.46e-16)))
   (/ (* ew (sin t)) (hypot 1.0 (/ eh (* ew (tan t)))))
   (fabs (* eh (sin (atan (/ (/ eh ew) t)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.08e-63) || !(t <= 1.46e-16)) {
		tmp = (ew * sin(t)) / hypot(1.0, (eh / (ew * tan(t))));
	} else {
		tmp = fabs((eh * sin(atan(((eh / ew) / t)))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.08e-63) || !(t <= 1.46e-16)) {
		tmp = (ew * Math.sin(t)) / Math.hypot(1.0, (eh / (ew * Math.tan(t))));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan(((eh / ew) / t)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -1.08e-63) or not (t <= 1.46e-16):
		tmp = (ew * math.sin(t)) / math.hypot(1.0, (eh / (ew * math.tan(t))))
	else:
		tmp = math.fabs((eh * math.sin(math.atan(((eh / ew) / t)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.08e-63) || !(t <= 1.46e-16))
		tmp = Float64(Float64(ew * sin(t)) / hypot(1.0, Float64(eh / Float64(ew * tan(t)))));
	else
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(eh / ew) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -1.08e-63) || ~((t <= 1.46e-16)))
		tmp = (ew * sin(t)) / hypot(1.0, (eh / (ew * tan(t))));
	else
		tmp = abs((eh * sin(atan(((eh / ew) / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.08e-63], N[Not[LessEqual[t, 1.46e-16]], $MachinePrecision]], N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.07999999999999994e-63 or 1.4600000000000001e-16 < t

   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 eh around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. associate-/l* 82.1%

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

      3. fma-define 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-/r* 82.1%

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

      6. associate-/l* 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in eh around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

   10. Simplified 61.1%

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

       1. log1p-expm1-u 61.0%

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

       2. associate-/r* 61.0%

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

       3. cos-atan 60.7%

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

       4. hypot-1-def 60.7%

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

       5. div-inv 60.7%

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

       6. add-sqr-sqrt 36.8%

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

       7. fabs-sqr 36.8%

          \[\leadsto \color{blue}{\sqrt{\frac{ew \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin t\right)\right)}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}} \cdot \sqrt{\frac{ew \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin t\right)\right)}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}}} \]

       8. add-sqr-sqrt 37.3%

          \[\leadsto \color{blue}{\frac{ew \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin t\right)\right)}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}} \]

       9. log1p-expm1-u 37.3%

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

       10. associate-/r* 37.3%

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

   12. Applied egg-rr 37.3%

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

   if -1.07999999999999994e-63 < t < 1.4600000000000001e-16

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 83.7%

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

   6. Taylor expanded in t around 0 82.8%

      \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

   7. Taylor expanded in t around 0 83.7%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{-63} \lor \neg \left(t \leq 1.46 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / 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 99.9%

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

   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/ 99.8%

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

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 44.1%

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

6. Taylor expanded in t around 0 42.6%

   \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

7. Taylor expanded in t around 0 42.5%

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

Alternative 11: 26.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \mathbf{if}\;ew \leq -1.22 \cdot 10^{-145} \lor \neg \left(ew \leq 2.6 \cdot 10^{-121}\right):\\ \;\;\;\;\left|eh \cdot \frac{eh}{\left(ew \cdot t\right) \cdot \mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))))
   (if (or (<= ew -1.22e-145) (not (<= ew 2.6e-121)))
     (fabs (* eh (/ eh (* (* ew t) (hypot 1.0 t_1)))))
     (* eh (sin (atan t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double tmp;
	if ((ew <= -1.22e-145) || !(ew <= 2.6e-121)) {
		tmp = fabs((eh * (eh / ((ew * t) * hypot(1.0, t_1)))));
	} else {
		tmp = eh * sin(atan(t_1));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double tmp;
	if ((ew <= -1.22e-145) || !(ew <= 2.6e-121)) {
		tmp = Math.abs((eh * (eh / ((ew * t) * Math.hypot(1.0, t_1)))));
	} else {
		tmp = eh * Math.sin(Math.atan(t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	tmp = 0
	if (ew <= -1.22e-145) or not (ew <= 2.6e-121):
		tmp = math.fabs((eh * (eh / ((ew * t) * math.hypot(1.0, t_1)))))
	else:
		tmp = eh * math.sin(math.atan(t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	tmp = 0.0
	if ((ew <= -1.22e-145) || !(ew <= 2.6e-121))
		tmp = abs(Float64(eh * Float64(eh / Float64(Float64(ew * t) * hypot(1.0, t_1)))));
	else
		tmp = Float64(eh * sin(atan(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	tmp = 0.0;
	if ((ew <= -1.22e-145) || ~((ew <= 2.6e-121)))
		tmp = abs((eh * (eh / ((ew * t) * hypot(1.0, t_1)))));
	else
		tmp = eh * sin(atan(t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -1.22e-145], N[Not[LessEqual[ew, 2.6e-121]], $MachinePrecision]], N[Abs[N[(eh * N[(eh / N[(N[(ew * t), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(eh * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.2199999999999999e-145 or 2.59999999999999986e-121 < 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 t around 0 36.7%

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

   6. Taylor expanded in t around 0 36.8%

      \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

   7. Taylor expanded in t around 0 35.3%

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

      1. sin-atan 16.1%

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

      2. associate-/l/ 15.4%

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

      3. hypot-1-def 25.9%

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

      4. associate-/l/ 26.3%

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

   9. Applied egg-rr 26.3%

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

       1. associate-/l/ 26.4%

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

       2. *-commutative 26.4%

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

       3. *-commutative 26.4%

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

   11. Simplified 26.4%

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

   if -1.2199999999999999e-145 < ew < 2.59999999999999986e-121

   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 60.8%

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

   6. Taylor expanded in t around 0 55.9%

      \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

   7. Taylor expanded in t around 0 58.9%

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

      1. add-sqr-sqrt 26.9%

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

      2. fabs-sqr 26.9%

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

      3. add-sqr-sqrt 28.1%

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

      4. *-commutative 28.1%

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

      5. associate-/l/ 28.1%

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

   9. Applied egg-rr 28.1%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.22 \cdot 10^{-145} \lor \neg \left(ew \leq 2.6 \cdot 10^{-121}\right):\\ \;\;\;\;\left|eh \cdot \frac{eh}{\left(ew \cdot t\right) \cdot \mathsf{hypot}\left(1, \frac{eh}{ew \cdot t}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 21.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * 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 99.9%

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

   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/ 99.8%

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

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 44.1%

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

6. Taylor expanded in t around 0 42.6%

   \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-0.3333333333333333 \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

7. Taylor expanded in t around 0 42.5%

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

   1. add-sqr-sqrt 18.3%

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

   2. fabs-sqr 18.3%

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

   3. add-sqr-sqrt 19.0%

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

   4. *-commutative 19.0%

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

   5. associate-/l/ 19.0%

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

9. Applied egg-rr 19.0%

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

10. Final simplification 19.0%

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

Reproduce

herbie shell --seed 2024188 
(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))))))))