Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.6s

Alternatives: 8

Speedup: N/A×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/l* 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 59.3%

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

   2. pow3 59.3%

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

6. Applied egg-rr 59.6%

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

   1. rem-cbrt-cube 99.8%

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

   2. *-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/l* 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 98.1%

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

   1. associate-/l* 98.4%

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

   2. count-2 98.4%

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

   3. associate-/r* 98.4%

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

   4. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in t around 0 98.4%

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

10. Final simplification 98.4%

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

Alternative 3: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/l* 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 59.3%

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

   2. pow3 59.3%

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

6. Applied egg-rr 59.6%

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

7. Taylor expanded in ew around inf 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 83.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;\left|ew - t\_1 \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 6.6 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{ew}{\frac{1}{\cos \left(t + \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)}} - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - t\_1 \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= eh -2.65e-17)
     (fabs (- ew (* t_1 (sin (atan (/ (* t (- eh)) ew))))))
     (if (<= eh 6.6e-111)
       (fabs
        (-
         (/ ew (/ 1.0 (cos (+ t (atan (/ (* t eh) ew))))))
         (* (* t eh) (sin (atan (* t (/ (- eh) ew)))))))
       (fabs (- ew (* t_1 (sin (atan (* eh (/ (tan t) ew)))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (eh <= -2.65e-17) {
		tmp = fabs((ew - (t_1 * sin(atan(((t * -eh) / ew))))));
	} else if (eh <= 6.6e-111) {
		tmp = fabs(((ew / (1.0 / cos((t + atan(((t * eh) / ew)))))) - ((t * eh) * sin(atan((t * (-eh / ew)))))));
	} else {
		tmp = fabs((ew - (t_1 * sin(atan((eh * (tan(t) / 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) :: t_1
    real(8) :: tmp
    t_1 = eh * sin(t)
    if (eh <= (-2.65d-17)) then
        tmp = abs((ew - (t_1 * sin(atan(((t * -eh) / ew))))))
    else if (eh <= 6.6d-111) then
        tmp = abs(((ew / (1.0d0 / cos((t + atan(((t * eh) / ew)))))) - ((t * eh) * sin(atan((t * (-eh / ew)))))))
    else
        tmp = abs((ew - (t_1 * sin(atan((eh * (tan(t) / ew)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if (eh <= -2.65e-17) {
		tmp = Math.abs((ew - (t_1 * Math.sin(Math.atan(((t * -eh) / ew))))));
	} else if (eh <= 6.6e-111) {
		tmp = Math.abs(((ew / (1.0 / Math.cos((t + Math.atan(((t * eh) / ew)))))) - ((t * eh) * Math.sin(Math.atan((t * (-eh / ew)))))));
	} else {
		tmp = Math.abs((ew - (t_1 * Math.sin(Math.atan((eh * (Math.tan(t) / ew)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if eh <= -2.65e-17:
		tmp = math.fabs((ew - (t_1 * math.sin(math.atan(((t * -eh) / ew))))))
	elif eh <= 6.6e-111:
		tmp = math.fabs(((ew / (1.0 / math.cos((t + math.atan(((t * eh) / ew)))))) - ((t * eh) * math.sin(math.atan((t * (-eh / ew)))))))
	else:
		tmp = math.fabs((ew - (t_1 * math.sin(math.atan((eh * (math.tan(t) / ew)))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (eh <= -2.65e-17)
		tmp = abs(Float64(ew - Float64(t_1 * sin(atan(Float64(Float64(t * Float64(-eh)) / ew))))));
	elseif (eh <= 6.6e-111)
		tmp = abs(Float64(Float64(ew / Float64(1.0 / cos(Float64(t + atan(Float64(Float64(t * eh) / ew)))))) - Float64(Float64(t * eh) * sin(atan(Float64(t * Float64(Float64(-eh) / ew)))))));
	else
		tmp = abs(Float64(ew - Float64(t_1 * sin(atan(Float64(eh * Float64(tan(t) / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (eh <= -2.65e-17)
		tmp = abs((ew - (t_1 * sin(atan(((t * -eh) / ew))))));
	elseif (eh <= 6.6e-111)
		tmp = abs(((ew / (1.0 / cos((t + atan(((t * eh) / ew)))))) - ((t * eh) * sin(atan((t * (-eh / ew)))))));
	else
		tmp = abs((ew - (t_1 * sin(atan((eh * (tan(t) / ew)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.65e-17], N[Abs[N[(ew - N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 6.6e-111], N[Abs[N[(N[(ew / N[(1.0 / N[Cos[N[(t + N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(t$95$1 * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -2.6499999999999999e-17

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. cancel-sign-sub 99.9%

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

      4. associate-*l* 99.9%

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

      5. associate-/l* 99.9%

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

      6. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

      1. add-cbrt-cube 70.4%

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

      2. pow3 70.4%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in t around 0 90.6%

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

   8. Taylor expanded in t around 0 90.6%

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

      1. associate-*r/ 90.6%

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

      2. mul-1-neg 90.6%

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

   10. Simplified 90.6%

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

   if -2.6499999999999999e-17 < eh < 6.6e-111

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. cancel-sign-sub 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/l* 99.8%

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

      6. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 98.7%

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

      1. associate-/l* 98.7%

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

      2. count-2 98.7%

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

      3. associate-/r* 98.7%

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

      4. metadata-eval 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in t around 0 98.7%

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

   10. Taylor expanded in t around 0 98.0%

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

       1. associate-*r/ 59.8%

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

       2. mul-1-neg 59.8%

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

   12. Simplified 98.0%

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

   13. Taylor expanded in t around 0 85.1%

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

       1. associate-*r* 53.8%

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

       2. *-commutative 53.8%

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

       3. mul-1-neg 53.8%

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

       4. associate-*l/ 53.8%

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

       5. distribute-rgt-neg-in 53.8%

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

   15. Simplified 85.1%

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

   if 6.6e-111 < 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. cancel-sign-sub 99.8%

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

      4. associate-*l* 99.9%

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

      5. associate-/l* 99.9%

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

      6. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

      1. add-cbrt-cube 67.1%

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

      2. pow3 67.0%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in t around 0 84.4%

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

      1. expm1-log1p-u 65.9%

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

      2. expm1-udef 57.6%

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

      3. associate-/r/ 57.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 47.1%

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

      6. sqr-neg 47.1%

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

      7. sqrt-unprod 48.2%

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

      8. add-sqr-sqrt 48.2%

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

      9. *-commutative 48.2%

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

   9. Applied egg-rr 48.2%

      \[\leadsto \left|ew - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)} - 1\right)}\right| \]
   10. Step-by-step derivation

       1. expm1-def 56.6%

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

       2. expm1-log1p 84.4%

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

       3. associate-*r/ 84.4%

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

       4. associate-*l/ 84.4%

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

   11. Simplified 84.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 6.6 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{ew}{\frac{1}{\cos \left(t + \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)}} - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\\ \mathbf{if}\;t \leq -0.058 \lor \neg \left(t \leq 1.5 \cdot 10^{+64}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(t \cdot eh\right) \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (* t (/ (- eh) ew))))))
   (if (or (<= t -0.058) (not (<= t 1.5e+64)))
     (fabs (* (* eh (sin t)) t_1))
     (fabs (- ew (* (* t eh) t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan((t * (-eh / ew))));
	double tmp;
	if ((t <= -0.058) || !(t <= 1.5e+64)) {
		tmp = fabs(((eh * sin(t)) * t_1));
	} else {
		tmp = fabs((ew - ((t * eh) * 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 = sin(atan((t * (-eh / ew))))
    if ((t <= (-0.058d0)) .or. (.not. (t <= 1.5d+64))) then
        tmp = abs(((eh * sin(t)) * t_1))
    else
        tmp = abs((ew - ((t * eh) * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan((t * (-eh / ew))));
	double tmp;
	if ((t <= -0.058) || !(t <= 1.5e+64)) {
		tmp = Math.abs(((eh * Math.sin(t)) * t_1));
	} else {
		tmp = Math.abs((ew - ((t * eh) * t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan((t * (-eh / ew))))
	tmp = 0
	if (t <= -0.058) or not (t <= 1.5e+64):
		tmp = math.fabs(((eh * math.sin(t)) * t_1))
	else:
		tmp = math.fabs((ew - ((t * eh) * t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(t * Float64(Float64(-eh) / ew))))
	tmp = 0.0
	if ((t <= -0.058) || !(t <= 1.5e+64))
		tmp = abs(Float64(Float64(eh * sin(t)) * t_1));
	else
		tmp = abs(Float64(ew - Float64(Float64(t * eh) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan((t * (-eh / ew))));
	tmp = 0.0;
	if ((t <= -0.058) || ~((t <= 1.5e+64)))
		tmp = abs(((eh * sin(t)) * t_1));
	else
		tmp = abs((ew - ((t * eh) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t, -0.058], N[Not[LessEqual[t, 1.5e+64]], $MachinePrecision]], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[(t * eh), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0580000000000000029 or 1.5000000000000001e64 < t

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. cancel-sign-sub 99.6%

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

      4. associate-*l* 99.7%

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

      5. associate-/l* 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. add-cbrt-cube 66.5%

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

      2. pow3 66.5%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in t around 0 57.3%

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

   8. Taylor expanded in t around 0 57.3%

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

      1. associate-*r/ 57.3%

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

      2. mul-1-neg 57.3%

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

   10. Simplified 57.3%

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

   11. Taylor expanded in ew around 0 49.5%

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

       1. mul-1-neg 49.5%

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

       2. associate-*r* 49.5%

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

       3. distribute-rgt-neg-in 49.5%

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

       4. mul-1-neg 49.5%

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

       5. associate-*l/ 49.5%

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

       6. *-commutative 49.5%

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

       7. distribute-rgt-neg-in 49.5%

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

       8. distribute-neg-frac 49.5%

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

   13. Simplified 49.5%

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

   if -0.0580000000000000029 < t < 1.5000000000000001e64

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. associate-*l* 100.0%

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

      5. associate-/l* 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 52.9%

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

      2. pow3 52.9%

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

   6. Applied egg-rr 52.9%

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

   7. Taylor expanded in t around 0 92.7%

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

   8. Taylor expanded in t around 0 92.7%

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

      1. associate-*r/ 92.7%

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

      2. mul-1-neg 92.7%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in t around 0 91.7%

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

       1. associate-*r* 91.7%

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

       2. *-commutative 91.7%

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

       3. mul-1-neg 91.7%

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

       4. associate-*l/ 91.7%

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

       5. distribute-rgt-neg-in 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.058 \lor \neg \left(t \leq 1.5 \cdot 10^{+64}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/l* 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 59.3%

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

   2. pow3 59.3%

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

6. Applied egg-rr 59.6%

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

7. Taylor expanded in t around 0 76.3%

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

8. Taylor expanded in t around 0 76.3%

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

   1. associate-*r/ 76.3%

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

   2. mul-1-neg 76.3%

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

10. Simplified 76.3%

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

11. Final simplification 76.3%

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

Alternative 7: 79.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/l* 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 59.3%

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

   2. pow3 59.3%

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

6. Applied egg-rr 59.6%

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

7. Taylor expanded in t around 0 76.3%

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

8. Taylor expanded in t around 0 76.3%

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

   1. associate-*r/ 76.3%

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

   2. mul-1-neg 76.3%

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

10. Simplified 76.3%

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

    1. add-sqr-sqrt 47.9%

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

    2. sqrt-unprod 74.8%

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

    3. sqr-neg 74.8%

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

    4. sqrt-unprod 39.7%

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

    5. add-sqr-sqrt 76.3%

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

    6. associate-/l* 76.3%

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

    7. associate-/r/ 76.3%

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

12. Applied egg-rr 76.3%

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

13. Final simplification 76.3%

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

Alternative 8: 54.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/l* 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 59.3%

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

   2. pow3 59.3%

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

6. Applied egg-rr 59.6%

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

7. Taylor expanded in t around 0 76.3%

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

8. Taylor expanded in t around 0 76.3%

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

   1. associate-*r/ 76.3%

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

   2. mul-1-neg 76.3%

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

10. Simplified 76.3%

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

11. Taylor expanded in t around 0 54.2%

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

    1. associate-*r* 54.2%

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

    2. *-commutative 54.2%

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

    3. mul-1-neg 54.2%

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

    4. associate-*l/ 54.2%

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

    5. distribute-rgt-neg-in 54.2%

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

13. Simplified 54.2%

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

14. Final simplification 54.2%

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

Reproduce

herbie shell --seed 2024031 
(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)))))))