Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 12.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \sin t \cdot eh\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19} \lor \neg \left(ew \leq 1.02 \cdot 10^{-91}\right):\\ \;\;\;\;\left|\frac{t\_2 \cdot t\_1 + \cos t \cdot ew}{{\cos \tan^{-1} t\_1}^{-1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh)) (t_2 (* (sin t) eh)))
   (if (or (<= ew -1.9e+19) (not (<= ew 1.02e-91)))
     (fabs (/ (+ (* t_2 t_1) (* (cos t) ew)) (pow (cos (atan t_1)) -1.0)))
     (fabs (* (sin (atan (* (/ eh (cos t)) (/ (sin t) ew)))) t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = sin(t) * eh;
	double tmp;
	if ((ew <= -1.9e+19) || !(ew <= 1.02e-91)) {
		tmp = fabs((((t_2 * t_1) + (cos(t) * ew)) / pow(cos(atan(t_1)), -1.0)));
	} else {
		tmp = fabs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * t_2));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (tan(t) / ew) * eh
    t_2 = sin(t) * eh
    if ((ew <= (-1.9d+19)) .or. (.not. (ew <= 1.02d-91))) then
        tmp = abs((((t_2 * t_1) + (cos(t) * ew)) / (cos(atan(t_1)) ** (-1.0d0))))
    else
        tmp = abs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (Math.tan(t) / ew) * eh;
	double t_2 = Math.sin(t) * eh;
	double tmp;
	if ((ew <= -1.9e+19) || !(ew <= 1.02e-91)) {
		tmp = Math.abs((((t_2 * t_1) + (Math.cos(t) * ew)) / Math.pow(Math.cos(Math.atan(t_1)), -1.0)));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / Math.cos(t)) * (Math.sin(t) / ew)))) * t_2));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (math.tan(t) / ew) * eh
	t_2 = math.sin(t) * eh
	tmp = 0
	if (ew <= -1.9e+19) or not (ew <= 1.02e-91):
		tmp = math.fabs((((t_2 * t_1) + (math.cos(t) * ew)) / math.pow(math.cos(math.atan(t_1)), -1.0)))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / math.cos(t)) * (math.sin(t) / ew)))) * t_2))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = Float64(sin(t) * eh)
	tmp = 0.0
	if ((ew <= -1.9e+19) || !(ew <= 1.02e-91))
		tmp = abs(Float64(Float64(Float64(t_2 * t_1) + Float64(cos(t) * ew)) / (cos(atan(t_1)) ^ -1.0)));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / cos(t)) * Float64(sin(t) / ew)))) * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (tan(t) / ew) * eh;
	t_2 = sin(t) * eh;
	tmp = 0.0;
	if ((ew <= -1.9e+19) || ~((ew <= 1.02e-91)))
		tmp = abs((((t_2 * t_1) + (cos(t) * ew)) / (cos(atan(t_1)) ^ -1.0)));
	else
		tmp = abs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[Or[LessEqual[ew, -1.9e+19], N[Not[LessEqual[ew, 1.02e-91]], $MachinePrecision]], N[Abs[N[(N[(N[(t$95$2 * t$95$1), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.9e19 or 1.01999999999999994e-91 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 87.4%

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

   if -1.9e19 < ew < 1.01999999999999994e-91

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 77.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19} \lor \neg \left(ew \leq 1.02 \cdot 10^{-91}\right):\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \frac{\tan t}{ew} \cdot eh\\ t_3 := \tan^{-1} t\_2\\ \mathbf{if}\;ew \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \cos t, ew, t\_1 \cdot \sin t\_3\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_1 \cdot t\_2 + \cos t \cdot ew}{{\cos t\_3}^{-1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (* (/ (tan t) ew) eh)) (t_3 (atan t_2)))
   (if (<= ew 7.5e+48)
     (fabs (fma (* (cos (atan (* (/ t ew) eh))) (cos t)) ew (* t_1 (sin t_3))))
     (fabs (/ (+ (* t_1 t_2) (* (cos t) ew)) (pow (cos t_3) -1.0))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = (tan(t) / ew) * eh;
	double t_3 = atan(t_2);
	double tmp;
	if (ew <= 7.5e+48) {
		tmp = fabs(fma((cos(atan(((t / ew) * eh))) * cos(t)), ew, (t_1 * sin(t_3))));
	} else {
		tmp = fabs((((t_1 * t_2) + (cos(t) * ew)) / pow(cos(t_3), -1.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(Float64(tan(t) / ew) * eh)
	t_3 = atan(t_2)
	tmp = 0.0
	if (ew <= 7.5e+48)
		tmp = abs(fma(Float64(cos(atan(Float64(Float64(t / ew) * eh))) * cos(t)), ew, Float64(t_1 * sin(t_3))));
	else
		tmp = abs(Float64(Float64(Float64(t_1 * t_2) + Float64(cos(t) * ew)) / (cos(t_3) ^ -1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$2], $MachinePrecision]}, If[LessEqual[ew, 7.5e+48], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew + N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[t$95$3], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < 7.5000000000000006e48

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 7.5000000000000006e48 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 95.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \cos t, ew, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \sin t \cdot eh\\ t_3 := t\_1 \cdot eh\\ t_4 := \cos \tan^{-1} t\_3\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{t\_2 \cdot t\_3 + \cos t \cdot ew}{{t\_4}^{-1}}\right|\\ \mathbf{elif}\;ew \leq 1.02 \cdot 10^{-91}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_1 \cdot \left(\left(-eh\right) \cdot \sin t\right), eh, \left(-\cos t\right) \cdot ew\right)}{\frac{-1}{t\_4}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* (sin t) eh))
        (t_3 (* t_1 eh))
        (t_4 (cos (atan t_3))))
   (if (<= ew -1.9e+19)
     (fabs (/ (+ (* t_2 t_3) (* (cos t) ew)) (pow t_4 -1.0)))
     (if (<= ew 1.02e-91)
       (fabs (* (sin (atan (* (/ eh (cos t)) (/ (sin t) ew)))) t_2))
       (fabs
        (/
         (fma (* t_1 (* (- eh) (sin t))) eh (* (- (cos t)) ew))
         (/ -1.0 t_4)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = sin(t) * eh;
	double t_3 = t_1 * eh;
	double t_4 = cos(atan(t_3));
	double tmp;
	if (ew <= -1.9e+19) {
		tmp = fabs((((t_2 * t_3) + (cos(t) * ew)) / pow(t_4, -1.0)));
	} else if (ew <= 1.02e-91) {
		tmp = fabs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * t_2));
	} else {
		tmp = fabs((fma((t_1 * (-eh * sin(t))), eh, (-cos(t) * ew)) / (-1.0 / t_4)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(sin(t) * eh)
	t_3 = Float64(t_1 * eh)
	t_4 = cos(atan(t_3))
	tmp = 0.0
	if (ew <= -1.9e+19)
		tmp = abs(Float64(Float64(Float64(t_2 * t_3) + Float64(cos(t) * ew)) / (t_4 ^ -1.0)));
	elseif (ew <= 1.02e-91)
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / cos(t)) * Float64(sin(t) / ew)))) * t_2));
	else
		tmp = abs(Float64(fma(Float64(t_1 * Float64(Float64(-eh) * sin(t))), eh, Float64(Float64(-cos(t)) * ew)) / Float64(-1.0 / t_4)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * eh), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.9e+19], N[Abs[N[(N[(N[(t$95$2 * t$95$3), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$4, -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.02e-91], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$1 * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh + N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.9e19

   1. Initial program 99.9%

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

   4. Applied rewrites 92.9%

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

   if -1.9e19 < ew < 1.01999999999999994e-91

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 77.8%

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

   if 1.01999999999999994e-91 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 85.2

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

   6. Applied rewrites 85.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \mathbf{elif}\;ew \leq 1.02 \cdot 10^{-91}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\tan t}{ew} \cdot \left(\left(-eh\right) \cdot \sin t\right), eh, \left(-\cos t\right) \cdot ew\right)}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19} \lor \neg \left(ew \leq 1.02 \cdot 10^{-91}\right):\\ \;\;\;\;\left|\left(t\_1 \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right| \cdot \cos \tan^{-1} t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh)))
   (if (or (<= ew -1.9e+19) (not (<= ew 1.02e-91)))
     (* (fabs (+ (* (* t_1 eh) (sin t)) (* (cos t) ew))) (cos (atan t_1)))
     (fabs
      (* (sin (atan (* (/ eh (cos t)) (/ (sin t) ew)))) (* (sin t) eh))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double tmp;
	if ((ew <= -1.9e+19) || !(ew <= 1.02e-91)) {
		tmp = fabs((((t_1 * eh) * sin(t)) + (cos(t) * ew))) * cos(atan(t_1));
	} else {
		tmp = fabs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * (sin(t) * eh)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (tan(t) / ew) * eh
    if ((ew <= (-1.9d+19)) .or. (.not. (ew <= 1.02d-91))) then
        tmp = abs((((t_1 * eh) * sin(t)) + (cos(t) * ew))) * cos(atan(t_1))
    else
        tmp = abs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * (sin(t) * eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (Math.tan(t) / ew) * eh;
	double tmp;
	if ((ew <= -1.9e+19) || !(ew <= 1.02e-91)) {
		tmp = Math.abs((((t_1 * eh) * Math.sin(t)) + (Math.cos(t) * ew))) * Math.cos(Math.atan(t_1));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / Math.cos(t)) * (Math.sin(t) / ew)))) * (Math.sin(t) * eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (math.tan(t) / ew) * eh
	tmp = 0
	if (ew <= -1.9e+19) or not (ew <= 1.02e-91):
		tmp = math.fabs((((t_1 * eh) * math.sin(t)) + (math.cos(t) * ew))) * math.cos(math.atan(t_1))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / math.cos(t)) * (math.sin(t) / ew)))) * (math.sin(t) * eh)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	tmp = 0.0
	if ((ew <= -1.9e+19) || !(ew <= 1.02e-91))
		tmp = Float64(abs(Float64(Float64(Float64(t_1 * eh) * sin(t)) + Float64(cos(t) * ew))) * cos(atan(t_1)));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / cos(t)) * Float64(sin(t) / ew)))) * Float64(sin(t) * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (tan(t) / ew) * eh;
	tmp = 0.0;
	if ((ew <= -1.9e+19) || ~((ew <= 1.02e-91)))
		tmp = abs((((t_1 * eh) * sin(t)) + (cos(t) * ew))) * cos(atan(t_1));
	else
		tmp = abs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * (sin(t) * eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, If[Or[LessEqual[ew, -1.9e+19], N[Not[LessEqual[ew, 1.02e-91]], $MachinePrecision]], N[(N[Abs[N[(N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.9e19 or 1.01999999999999994e-91 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 87.4%

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

   if -1.9e19 < ew < 1.01999999999999994e-91

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 77.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19} \lor \neg \left(ew \leq 1.02 \cdot 10^{-91}\right):\\ \;\;\;\;\left|\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19} \lor \neg \left(ew \leq 2.45 \cdot 10^{-83}\right):\\ \;\;\;\;\left|\left(-\cos t\right) \cdot ew\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.9e+19) (not (<= ew 2.45e-83)))
   (* (fabs (* (- (cos t)) ew)) (cos (atan (* (/ (tan t) ew) eh))))
   (fabs (* (sin (atan (* (/ eh (cos t)) (/ (sin t) ew)))) (* (sin t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.9e+19) || !(ew <= 2.45e-83)) {
		tmp = fabs((-cos(t) * ew)) * cos(atan(((tan(t) / ew) * eh)));
	} else {
		tmp = fabs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * (sin(t) * eh)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.9d+19)) .or. (.not. (ew <= 2.45d-83))) then
        tmp = abs((-cos(t) * ew)) * cos(atan(((tan(t) / ew) * eh)))
    else
        tmp = abs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * (sin(t) * eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.9e+19) || !(ew <= 2.45e-83)) {
		tmp = Math.abs((-Math.cos(t) * ew)) * Math.cos(Math.atan(((Math.tan(t) / ew) * eh)));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / Math.cos(t)) * (Math.sin(t) / ew)))) * (Math.sin(t) * eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.9e+19) or not (ew <= 2.45e-83):
		tmp = math.fabs((-math.cos(t) * ew)) * math.cos(math.atan(((math.tan(t) / ew) * eh)))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / math.cos(t)) * (math.sin(t) / ew)))) * (math.sin(t) * eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.9e+19) || !(ew <= 2.45e-83))
		tmp = Float64(abs(Float64(Float64(-cos(t)) * ew)) * cos(atan(Float64(Float64(tan(t) / ew) * eh))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / cos(t)) * Float64(sin(t) / ew)))) * Float64(sin(t) * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.9e+19) || ~((ew <= 2.45e-83)))
		tmp = abs((-cos(t) * ew)) * cos(atan(((tan(t) / ew) * eh)));
	else
		tmp = abs((sin(atan(((eh / cos(t)) * (sin(t) / ew)))) * (sin(t) * eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.9e+19], N[Not[LessEqual[ew, 2.45e-83]], $MachinePrecision]], N[(N[Abs[N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.9e19 or 2.45e-83 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 86.2

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

   7. Applied rewrites 86.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. fabs-mul N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

   9. Applied rewrites 86.2%

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

   if -1.9e19 < ew < 2.45e-83

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 84.3%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 77.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+19} \lor \neg \left(ew \leq 2.45 \cdot 10^{-83}\right):\\ \;\;\;\;\left|\left(-\cos t\right) \cdot ew\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 59.8

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

7. Applied rewrites 59.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. fabs-mul N/A

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

   5. fabs-div N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

9. Applied rewrites 59.8%

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

Alternative 8: 52.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (/ (* (- ew) (cos t)) (/ -1.0 (cos (atan (* (/ eh ew) t)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((-ew * cos(t)) / (-1.0 / 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(((-ew * cos(t)) / ((-1.0d0) / cos(atan(((eh / ew) * t))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 59.8

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

7. Applied rewrites 59.8%

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

8. Taylor expanded in t around 0

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

   1. *-commutative N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \color{blue}{\left(\left(\frac{1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}\right) \cdot t\right)}}}\right| \]

   2. associate-*r/ N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\color{blue}{\frac{\frac{1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{ew}} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   3. associate-*r* N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\frac{\color{blue}{\left(\frac{1}{3} \cdot eh\right) \cdot {t}^{2}}}{ew} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   4. associate-*l/ N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\color{blue}{\frac{\frac{1}{3} \cdot eh}{ew} \cdot {t}^{2}} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   5. associate-*r/ N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{eh}{ew}\right)} \cdot {t}^{2} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\color{blue}{\left(\frac{eh}{ew} \cdot \frac{1}{3}\right)} \cdot {t}^{2} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   7. metadata-eval N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\left(\frac{eh}{ew} \cdot \color{blue}{\left(\frac{-1}{6} - \frac{-1}{2}\right)}\right) \cdot {t}^{2} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\color{blue}{\left(\frac{-1}{6} \cdot \frac{eh}{ew} - \frac{-1}{2} \cdot \frac{eh}{ew}\right)} \cdot {t}^{2} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   9. *-commutative N/A

      \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\left(\color{blue}{{t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{eh}{ew} - \frac{-1}{2} \cdot \frac{eh}{ew}\right)} + \frac{eh}{ew}\right) \cdot t\right)}}\right| \]

   10. lower-*.f64 N/A

       \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \color{blue}{\left(\left({t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{eh}{ew} - \frac{-1}{2} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}\right) \cdot t\right)}}}\right| \]

10. Applied rewrites 41.9%

    \[\leadsto \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{eh}{ew}\right) \cdot t\right)}}}\right| \]

11. Taylor expanded in t around 0

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

13. Applied rewrites 50.6%

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

14. Final simplification 50.6%

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

Alternative 9: 43.3% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{ew}{1}\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (/ ew 1.0)))

C

double code(double eh, double ew, double t) {
	return fabs((ew / 1.0));
}

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 / 1.0d0))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew / 1.0));
}

Python

def code(eh, ew, t):
	return math.fabs((ew / 1.0))

Julia

function code(eh, ew, t)
	return abs(Float64(ew / 1.0))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew / 1.0));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 40.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 38.8%

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

10. Applied rewrites 37.9%

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

11. Taylor expanded in eh around 0

    \[\leadsto \left|\frac{ew}{1}\right| \]
12. Step-by-step derivation

13. Applied rewrites 40.1%

    \[\leadsto \left|\frac{ew}{1}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))