Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.1s

Alternatives: 16

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) + ew \cdot \sqrt[3]{{\cos t}^{3}}\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* eh (* (sin t) (sin (atan (* (tan t) (/ eh ew))))))
   (* ew (cbrt (pow (cos t) 3.0))))))

C

double code(double eh, double ew, double t) {
	return fabs(((eh * (sin(t) * sin(atan((tan(t) * (eh / ew)))))) + (ew * cbrt(pow(cos(t), 3.0)))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((eh * (Math.sin(t) * Math.sin(Math.atan((Math.tan(t) * (eh / ew)))))) + (ew * Math.cbrt(Math.pow(Math.cos(t), 3.0)))));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(eh * Float64(sin(t) * sin(atan(Float64(tan(t) * Float64(eh / ew)))))) + Float64(ew * cbrt((cos(t) ^ 3.0)))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ew * N[Power[N[Power[N[Cos[t], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-cbrt-cube 99.6%

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

   2. pow3 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in t around 0 98.6%

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

8. Final simplification 98.6%

   \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) + ew \cdot \sqrt[3]{{\cos t}^{3}}\right| \]
9. Add Preprocessing

Alternative 3: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.5 \cdot 10^{-108} \lor \neg \left(ew \leq 6000\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(ew \cdot \frac{\cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.5e-108) (not (<= ew 6000.0)))
   (fabs
    (*
     ew
     (+ (cos t) (* eh (/ (* (sin t) (sin (atan (* eh (/ (tan t) ew))))) ew)))))
   (fabs
    (*
     eh
     (-
      (* ew (/ (cos t) eh))
      (* (sin t) (sin (atan (* (tan t) (/ eh (- ew)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.5e-108) || !(ew <= 6000.0)) {
		tmp = fabs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / ew))))) / ew)))));
	} else {
		tmp = fabs((eh * ((ew * (cos(t) / eh)) - (sin(t) * sin(atan((tan(t) * (eh / -ew))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.5d-108)) .or. (.not. (ew <= 6000.0d0))) then
        tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / ew))))) / ew)))))
    else
        tmp = abs((eh * ((ew * (cos(t) / eh)) - (sin(t) * sin(atan((tan(t) * (eh / -ew))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.5e-108) || !(ew <= 6000.0)) {
		tmp = Math.abs((ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(Math.atan((eh * (Math.tan(t) / ew))))) / ew)))));
	} else {
		tmp = Math.abs((eh * ((ew * (Math.cos(t) / eh)) - (Math.sin(t) * Math.sin(Math.atan((Math.tan(t) * (eh / -ew))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.5e-108) or not (ew <= 6000.0):
		tmp = math.fabs((ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(math.atan((eh * (math.tan(t) / ew))))) / ew)))))
	else:
		tmp = math.fabs((eh * ((ew * (math.cos(t) / eh)) - (math.sin(t) * math.sin(math.atan((math.tan(t) * (eh / -ew))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.5e-108) || !(ew <= 6000.0))
		tmp = abs(Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / ew))))) / ew)))));
	else
		tmp = abs(Float64(eh * Float64(Float64(ew * Float64(cos(t) / eh)) - Float64(sin(t) * sin(atan(Float64(tan(t) * Float64(eh / Float64(-ew)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.5e-108) || ~((ew <= 6000.0)))
		tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / ew))))) / ew)))));
	else
		tmp = abs((eh * ((ew * (cos(t) / eh)) - (sin(t) * sin(atan((tan(t) * (eh / -ew))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.5e-108], N[Not[LessEqual[ew, 6000.0]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[(ew * N[(N[Cos[t], $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.49999999999999996e-108 or 6e3 < 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
   3. Step-by-step derivation

      1. sub-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. add-exp-log 77.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in ew around inf 98.9%

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

      1. associate-/l* 99.0%

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

      2. associate-/l* 99.0%

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

   9. Simplified 99.0%

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

   if -1.49999999999999996e-108 < ew < 6e3

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 53.1%

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

      2. pow2 53.1%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in ew around inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. unsub-neg 80.2%

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

      3. associate-/l* 80.1%

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

      4. mul-1-neg 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*r/ 80.1%

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

      7. distribute-rgt-neg-in 80.1%

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

      8. distribute-neg-frac2 80.1%

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

   7. Simplified 80.1%

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

   8. Taylor expanded in eh around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. mul-1-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. *-commutative 98.4%

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

      5. distribute-frac-neg2 98.4%

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

      6. associate-*r/ 98.4%

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

      7. unsub-neg 98.4%

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

      8. associate-/l* 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.5 \cdot 10^{-108} \lor \neg \left(ew \leq 6000\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(ew \cdot \frac{\cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.18 \cdot 10^{+122}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.18e+122)
   (fabs (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh (- ew)))))))
   (fabs
    (*
     ew
     (+
      (cos t)
      (* eh (/ (* (sin t) (sin (atan (* eh (/ (tan t) ew))))) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.18e+122) {
		tmp = fabs(((eh * sin(t)) * sin(atan((tan(t) * (eh / -ew))))));
	} else {
		tmp = fabs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / ew))))) / ew)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.18d+122)) then
        tmp = abs(((eh * sin(t)) * sin(atan((tan(t) * (eh / -ew))))))
    else
        tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / ew))))) / ew)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.18e+122) {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / -ew))))));
	} else {
		tmp = Math.abs((ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(Math.atan((eh * (Math.tan(t) / ew))))) / ew)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.18e+122:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / -ew))))))
	else:
		tmp = math.fabs((ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(math.atan((eh * (math.tan(t) / ew))))) / ew)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.18e+122)
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / Float64(-ew)))))));
	else
		tmp = abs(Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / ew))))) / ew)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.18e+122)
		tmp = abs(((eh * sin(t)) * sin(atan((tan(t) * (eh / -ew))))));
	else
		tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / ew))))) / ew)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.18e+122], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.18000000000000003e122

   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 ew around 0 75.7%

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

      1. mul-1-neg 75.7%

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

      2. associate-*r* 75.8%

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

      3. distribute-rgt-neg-in 75.8%

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

      4. mul-1-neg 75.8%

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

      5. associate-*r/ 75.7%

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

      6. *-commutative 75.7%

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

      7. associate-*l/ 75.8%

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

      8. associate-*r/ 75.8%

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

      9. distribute-rgt-neg-in 75.8%

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

      10. distribute-neg-frac2 75.8%

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

   5. Simplified 75.8%

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

   if -1.18000000000000003e122 < eh

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. add-exp-log 71.1%

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

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in ew around inf 95.9%

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

      1. associate-/l* 95.9%

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

      2. associate-/l* 95.9%

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

   9. Simplified 95.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.18 \cdot 10^{+122}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\\ \mathbf{if}\;ew \leq -1.1 \cdot 10^{-102} \lor \neg \left(ew \leq 1.55 \cdot 10^{-87}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (tan t) (/ eh (- ew))))))
   (if (or (<= ew -1.1e-102) (not (<= ew 1.55e-87)))
     (fabs (* (* ew (cos t)) (cos t_1)))
     (fabs (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((tan(t) * (eh / -ew)));
	double tmp;
	if ((ew <= -1.1e-102) || !(ew <= 1.55e-87)) {
		tmp = fabs(((ew * cos(t)) * cos(t_1)));
	} else {
		tmp = fabs(((eh * sin(t)) * 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((tan(t) * (eh / -ew)))
    if ((ew <= (-1.1d-102)) .or. (.not. (ew <= 1.55d-87))) then
        tmp = abs(((ew * cos(t)) * cos(t_1)))
    else
        tmp = abs(((eh * sin(t)) * 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((Math.tan(t) * (eh / -ew)));
	double tmp;
	if ((ew <= -1.1e-102) || !(ew <= 1.55e-87)) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(t_1)));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((math.tan(t) * (eh / -ew)))
	tmp = 0
	if (ew <= -1.1e-102) or not (ew <= 1.55e-87):
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(t_1)))
	else:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(tan(t) * Float64(eh / Float64(-ew))))
	tmp = 0.0
	if ((ew <= -1.1e-102) || !(ew <= 1.55e-87))
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(t_1)));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((tan(t) * (eh / -ew)));
	tmp = 0.0;
	if ((ew <= -1.1e-102) || ~((ew <= 1.55e-87)))
		tmp = abs(((ew * cos(t)) * cos(t_1)));
	else
		tmp = abs(((eh * sin(t)) * sin(t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -1.1e-102], N[Not[LessEqual[ew, 1.55e-87]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.10000000000000006e-102 or 1.54999999999999999e-87 < 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

   3. Taylor expanded in ew around inf 86.7%

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

      1. associate-*r* 86.7%

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

      2. associate-*r/ 86.7%

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

      3. neg-mul-1 86.7%

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

      4. *-commutative 86.7%

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

      5. distribute-rgt-neg-in 86.7%

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

      6. mul-1-neg 86.7%

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

      7. associate-*r/ 86.7%

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

      8. mul-1-neg 86.7%

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

   5. Simplified 86.7%

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

   if -1.10000000000000006e-102 < ew < 1.54999999999999999e-87

   1. Initial program 99.7%

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

   3. Taylor expanded in ew around 0 78.5%

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

      1. mul-1-neg 78.5%

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

      2. associate-*r* 78.5%

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

      3. distribute-rgt-neg-in 78.5%

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

      4. mul-1-neg 78.5%

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

      5. associate-*r/ 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*l/ 78.5%

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

      8. associate-*r/ 78.5%

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

      9. distribute-rgt-neg-in 78.5%

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

      10. distribute-neg-frac2 78.5%

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

   5. Simplified 78.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.1 \cdot 10^{-102} \lor \neg \left(ew \leq 1.55 \cdot 10^{-87}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\\ \mathbf{if}\;t \leq -240000 \lor \neg \left(t \leq 0.078\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (tan t) (/ eh (- ew))))))
   (if (or (<= t -240000.0) (not (<= t 0.078)))
     (fabs (* (* ew (cos t)) (cos t_1)))
     (fabs (+ ew (* t (- (* -0.5 (* ew t)) (* eh (sin t_1)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((tan(t) * (eh / -ew)));
	double tmp;
	if ((t <= -240000.0) || !(t <= 0.078)) {
		tmp = fabs(((ew * cos(t)) * cos(t_1)));
	} else {
		tmp = fabs((ew + (t * ((-0.5 * (ew * 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((tan(t) * (eh / -ew)))
    if ((t <= (-240000.0d0)) .or. (.not. (t <= 0.078d0))) then
        tmp = abs(((ew * cos(t)) * cos(t_1)))
    else
        tmp = abs((ew + (t * (((-0.5d0) * (ew * 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((Math.tan(t) * (eh / -ew)));
	double tmp;
	if ((t <= -240000.0) || !(t <= 0.078)) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(t_1)));
	} else {
		tmp = Math.abs((ew + (t * ((-0.5 * (ew * t)) - (eh * Math.sin(t_1))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((math.tan(t) * (eh / -ew)))
	tmp = 0
	if (t <= -240000.0) or not (t <= 0.078):
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(t_1)))
	else:
		tmp = math.fabs((ew + (t * ((-0.5 * (ew * t)) - (eh * math.sin(t_1))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(tan(t) * Float64(eh / Float64(-ew))))
	tmp = 0.0
	if ((t <= -240000.0) || !(t <= 0.078))
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(t_1)));
	else
		tmp = abs(Float64(ew + Float64(t * Float64(Float64(-0.5 * Float64(ew * t)) - Float64(eh * sin(t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((tan(t) * (eh / -ew)));
	tmp = 0.0;
	if ((t <= -240000.0) || ~((t <= 0.078)))
		tmp = abs(((ew * cos(t)) * cos(t_1)));
	else
		tmp = abs((ew + (t * ((-0.5 * (ew * t)) - (eh * sin(t_1))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t, -240000.0], N[Not[LessEqual[t, 0.078]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(t * N[(N[(-0.5 * N[(ew * t), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4e5 or 0.0779999999999999999 < t

   1. Initial program 99.5%

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

   3. Taylor expanded in ew around inf 54.1%

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

      1. associate-*r* 54.1%

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

      2. associate-*r/ 54.1%

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

      3. neg-mul-1 54.1%

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

      4. *-commutative 54.1%

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

      5. distribute-rgt-neg-in 54.1%

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

      6. mul-1-neg 54.1%

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

      7. associate-*r/ 54.1%

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

      8. mul-1-neg 54.1%

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

   5. Simplified 54.1%

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

   if -2.4e5 < t < 0.0779999999999999999

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 52.2%

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

      2. pow2 52.2%

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

   4. Applied egg-rr 60.0%

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

   5. Taylor expanded in ew around inf 94.3%

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

      1. mul-1-neg 94.3%

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

      2. unsub-neg 94.3%

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

      3. associate-/l* 94.2%

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

      4. mul-1-neg 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*r/ 94.2%

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

      7. distribute-rgt-neg-in 94.2%

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

      8. distribute-neg-frac2 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in t around 0 97.5%

      \[\leadsto \left|\color{blue}{ew + t \cdot \left(-1 \cdot \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + -0.5 \cdot \left(ew \cdot t\right)\right)}\right| \]
   9. Step-by-step derivation

      1. +-commutative 97.5%

         \[\leadsto \left|ew + t \cdot \color{blue}{\left(-0.5 \cdot \left(ew \cdot t\right) + -1 \cdot \left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      2. mul-1-neg 97.5%

         \[\leadsto \left|ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) + \color{blue}{\left(-eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right)\right| \]

      3. unsub-neg 97.5%

         \[\leadsto \left|ew + t \cdot \color{blue}{\left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

      4. mul-1-neg 97.5%

         \[\leadsto \left|ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(-\frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

      5. associate-*l/ 97.5%

         \[\leadsto \left|ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \left(-\color{blue}{\frac{eh}{ew} \cdot \tan t}\right)\right)\right| \]

      6. distribute-rgt-neg-in 97.5%

         \[\leadsto \left|ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)}\right)\right| \]

   10. Simplified 97.5%

       \[\leadsto \left|\color{blue}{ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -240000 \lor \neg \left(t \leq 0.078\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + t \cdot \left(-0.5 \cdot \left(ew \cdot t\right) - eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 53.2%

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

   2. pow2 53.2%

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

4. Applied egg-rr 62.1%

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

5. Taylor expanded in t around 0 59.6%

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

   1. mul-1-neg 59.6%

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

   2. unsub-neg 59.6%

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

   3. mul-1-neg 59.6%

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

   4. *-commutative 59.6%

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

   5. associate-*r/ 59.6%

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

   6. distribute-rgt-neg-in 59.6%

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

   7. distribute-neg-frac2 59.6%

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

7. Simplified 59.6%

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

8. Final simplification 59.6%

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

Alternative 8: 55.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-exp-log 67.6%

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

6. Applied egg-rr 67.6%

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

7. Taylor expanded in t around 0 59.6%

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

   1. associate-*r* 59.4%

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

   2. *-commutative 59.4%

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

   3. associate-/l* 59.4%

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

9. Simplified 59.4%

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

10. Final simplification 59.4%

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

Alternative 9: 42.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Final simplification 46.2%

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

Alternative 10: 42.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

   1. associate-*r/ 46.2%

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

   2. *-commutative 46.2%

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

   3. cos-atan 46.0%

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

   4. hypot-1-def 46.0%

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

   5. *-commutative 46.0%

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

   6. associate-*r/ 46.0%

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

7. Applied egg-rr 46.0%

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

   1. hypot-undefine 45.9%

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

   2. swap-sqr 43.8%

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

   3. distribute-frac-neg 43.8%

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

   4. distribute-frac-neg 43.8%

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

   5. sqr-neg 43.8%

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

   6. swap-sqr 45.9%

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

   7. hypot-undefine 46.0%

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

   8. associate-*r/ 46.0%

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

   9. *-commutative 46.0%

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

   10. associate-/l* 46.0%

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

9. Simplified 46.0%

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

Alternative 11: 41.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(N[Cos[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Abs[ew], $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 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in t around 0 45.3%

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

   1. mul-1-neg 45.3%

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

   2. associate-/l* 45.3%

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

   3. distribute-rgt-neg-in 45.3%

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

8. Simplified 45.3%

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

   1. *-un-lft-identity 45.3%

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

   2. distribute-rgt-neg-out 45.3%

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

   3. atan-neg 45.3%

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

10. Applied egg-rr 45.3%

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

    1. *-lft-identity 45.3%

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

    2. fabs-mul 45.3%

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

    3. *-commutative 45.3%

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

    4. rem-square-sqrt 45.3%

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

    5. fabs-sqr 45.3%

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

    6. rem-square-sqrt 45.3%

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

    7. cos-neg 45.3%

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

    8. associate-*r/ 45.3%

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

    9. *-commutative 45.3%

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

12. Simplified 45.3%

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

Alternative 12: 41.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \left(-0.5 \cdot {\left(t \cdot \frac{eh}{ew}\right)}^{2} + 1\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* ew (+ (* -0.5 (pow (* t (/ eh ew)) 2.0)) 1.0))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * ((-0.5 * pow((t * (eh / ew)), 2.0)) + 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 * (((-0.5d0) * ((t * (eh / ew)) ** 2.0d0)) + 1.0d0)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * ((-0.5 * Math.pow((t * (eh / ew)), 2.0)) + 1.0)));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * ((-0.5 * math.pow((t * (eh / ew)), 2.0)) + 1.0)))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * Float64(Float64(-0.5 * (Float64(t * Float64(eh / ew)) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * ((-0.5 * ((t * (eh / ew)) ^ 2.0)) + 1.0)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[(-0.5 * N[Power[N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in t around 0 45.3%

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

   1. mul-1-neg 45.3%

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

   2. associate-/l* 45.3%

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

   3. distribute-rgt-neg-in 45.3%

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

8. Simplified 45.3%

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

   1. cos-atan 44.6%

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

   2. hypot-1-def 44.7%

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

   3. distribute-neg-frac2 44.7%

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

10. Applied egg-rr 44.7%

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

    1. distribute-frac-neg2 44.7%

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

    2. distribute-neg-frac 44.7%

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

12. Simplified 44.7%

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

13. Taylor expanded in ew around inf 32.0%

    \[\leadsto \left|\color{blue}{ew \cdot \left(1 + -0.5 \cdot \frac{{eh}^{2} \cdot {t}^{2}}{{ew}^{2}}\right)}\right| \]
14. Step-by-step derivation

    1. associate-/l* 31.8%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \color{blue}{\left({eh}^{2} \cdot \frac{{t}^{2}}{{ew}^{2}}\right)}\right)\right| \]

    2. unpow2 31.8%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(eh \cdot eh\right)} \cdot \frac{{t}^{2}}{{ew}^{2}}\right)\right)\right| \]

    3. unpow2 31.8%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \left(\left(eh \cdot eh\right) \cdot \frac{\color{blue}{t \cdot t}}{{ew}^{2}}\right)\right)\right| \]

    4. unpow2 31.8%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{\color{blue}{ew \cdot ew}}\right)\right)\right| \]

    5. times-frac 36.9%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \left(\left(eh \cdot eh\right) \cdot \color{blue}{\left(\frac{t}{ew} \cdot \frac{t}{ew}\right)}\right)\right)\right| \]

    6. swap-sqr 45.2%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(eh \cdot \frac{t}{ew}\right) \cdot \left(eh \cdot \frac{t}{ew}\right)\right)}\right)\right| \]

    7. unpow1 45.2%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \left(\color{blue}{{\left(eh \cdot \frac{t}{ew}\right)}^{1}} \cdot \left(eh \cdot \frac{t}{ew}\right)\right)\right)\right| \]

    8. pow-plus 45.2%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot \color{blue}{{\left(eh \cdot \frac{t}{ew}\right)}^{\left(1 + 1\right)}}\right)\right| \]

    9. *-commutative 45.2%

       \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(\frac{t}{ew} \cdot eh\right)}}^{\left(1 + 1\right)}\right)\right| \]

    10. associate-*l/ 45.2%

        \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(\frac{t \cdot eh}{ew}\right)}}^{\left(1 + 1\right)}\right)\right| \]

    11. associate-/l* 45.2%

        \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(t \cdot \frac{eh}{ew}\right)}}^{\left(1 + 1\right)}\right)\right| \]

    12. metadata-eval 45.2%

        \[\leadsto \left|ew \cdot \left(1 + -0.5 \cdot {\left(t \cdot \frac{eh}{ew}\right)}^{\color{blue}{2}}\right)\right| \]

15. Simplified 45.2%

    \[\leadsto \left|\color{blue}{ew \cdot \left(1 + -0.5 \cdot {\left(t \cdot \frac{eh}{ew}\right)}^{2}\right)}\right| \]

16. Final simplification 45.2%

    \[\leadsto \left|ew \cdot \left(-0.5 \cdot {\left(t \cdot \frac{eh}{ew}\right)}^{2} + 1\right)\right| \]
17. Add Preprocessing

Alternative 13: 40.7% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in t around 0 45.3%

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

   1. mul-1-neg 45.3%

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

   2. associate-/l* 45.3%

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

   3. distribute-rgt-neg-in 45.3%

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

8. Simplified 45.3%

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

   1. cos-atan 44.6%

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

   2. hypot-1-def 44.7%

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

   3. distribute-neg-frac2 44.7%

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

10. Applied egg-rr 44.7%

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

    1. distribute-frac-neg2 44.7%

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

    2. distribute-neg-frac 44.7%

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

12. Simplified 44.7%

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

    1. un-div-inv 44.7%

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

    2. distribute-frac-neg 44.7%

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

14. Applied egg-rr 44.7%

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

15. Final simplification 44.7%

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

Alternative 14: 40.7% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(N[Abs[ew], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $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 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in t around 0 45.3%

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

   1. mul-1-neg 45.3%

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

   2. associate-/l* 45.3%

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

   3. distribute-rgt-neg-in 45.3%

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

8. Simplified 45.3%

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

   1. cos-atan 44.6%

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

   2. hypot-1-def 44.7%

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

   3. distribute-neg-frac2 44.7%

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

10. Applied egg-rr 44.7%

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

    1. distribute-frac-neg2 44.7%

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

    2. distribute-neg-frac 44.7%

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

12. Simplified 44.7%

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

    1. *-un-lft-identity 44.7%

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

    2. un-div-inv 44.7%

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

    3. distribute-frac-neg 44.7%

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

14. Applied egg-rr 44.7%

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

    1. *-lft-identity 44.7%

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

    2. fabs-div 44.7%

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

    3. rem-square-sqrt 44.7%

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

    4. fabs-sqr 44.7%

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

    5. rem-square-sqrt 44.7%

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

    6. hypot-undefine 44.6%

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

    7. distribute-rgt-neg-out 44.6%

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

    8. distribute-rgt-neg-out 44.6%

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

    9. sqr-neg 44.6%

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

    10. hypot-undefine 44.7%

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

    11. *-commutative 44.7%

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

    12. associate-*l/ 44.5%

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

    13. associate-/l* 44.6%

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

16. Simplified 44.6%

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

Alternative 15: 5.1% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * (1.0d0 / (t * (eh / ew)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in t around 0 45.3%

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

   1. mul-1-neg 45.3%

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

   2. associate-/l* 45.3%

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

   3. distribute-rgt-neg-in 45.3%

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

8. Simplified 45.3%

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

   1. cos-atan 44.6%

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

   2. hypot-1-def 44.7%

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

   3. distribute-neg-frac2 44.7%

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

10. Applied egg-rr 44.7%

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

    1. distribute-frac-neg2 44.7%

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

    2. distribute-neg-frac 44.7%

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

12. Simplified 44.7%

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

13. Taylor expanded in eh around inf 5.1%

    \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\frac{eh \cdot t}{ew}}}\right| \]
14. Step-by-step derivation

    1. *-commutative 5.1%

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

    2. associate-/l* 5.2%

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

15. Simplified 5.2%

    \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{t \cdot \frac{eh}{ew}}}\right| \]
16. Add Preprocessing

Alternative 16: 5.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \frac{ew}{t \cdot eh}\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(ew / N[(t * 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

3. Taylor expanded in t around 0 46.2%

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

   1. associate-*r/ 46.2%

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

   2. neg-mul-1 46.2%

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

   3. *-commutative 46.2%

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

   4. distribute-rgt-neg-in 46.2%

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

   5. mul-1-neg 46.2%

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

   6. associate-*r/ 46.2%

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

   7. mul-1-neg 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in t around 0 45.3%

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

   1. mul-1-neg 45.3%

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

   2. associate-/l* 45.3%

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

   3. distribute-rgt-neg-in 45.3%

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

8. Simplified 45.3%

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

   1. cos-atan 44.6%

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

   2. hypot-1-def 44.7%

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

   3. distribute-neg-frac2 44.7%

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

10. Applied egg-rr 44.7%

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

    1. distribute-frac-neg2 44.7%

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

    2. distribute-neg-frac 44.7%

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

12. Simplified 44.7%

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

13. Taylor expanded in eh around inf 5.1%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{ew}{eh \cdot t}}\right| \]
14. Step-by-step derivation

    1. *-commutative 5.1%

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

15. Simplified 5.1%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{ew}{t \cdot eh}}\right| \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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)))))))