Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.4s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(1.0 / N[(ew / N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-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. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. clear-num 99.9%

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

   3. add-sqr-sqrt 51.5%

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

   4. sqrt-unprod 93.5%

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

   5. sqr-neg 93.5%

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

   6. sqrt-unprod 48.4%

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

   7. add-sqr-sqrt 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

   1. cos-atan 99.0%

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

   2. hypot-1-def 99.0%

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

   3. div-inv 99.0%

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

   4. div-inv 99.0%

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

   5. add-sqr-sqrt 51.5%

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

   6. sqrt-unprod 93.1%

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

   7. sqr-neg 93.1%

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

   8. sqrt-unprod 47.6%

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

   9. add-sqr-sqrt 99.0%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. associate-/l* 99.0%

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

   3. distribute-neg-frac 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 5: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. associate-/l* 99.0%

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

   3. distribute-neg-frac 99.0%

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

7. Simplified 99.0%

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

   1. cos-atan 99.0%

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

   2. hypot-1-def 99.0%

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

   3. div-inv 99.0%

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

   4. div-inv 99.0%

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

   5. add-sqr-sqrt 51.5%

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

   6. sqrt-unprod 93.1%

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

   7. sqr-neg 93.1%

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

   8. sqrt-unprod 47.6%

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

   9. add-sqr-sqrt 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

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

Alternative 6: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

   1. cos-atan 99.0%

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

   2. hypot-1-def 99.0%

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

   3. div-inv 99.0%

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

   4. div-inv 99.0%

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

   5. add-sqr-sqrt 51.5%

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

   6. sqrt-unprod 93.1%

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

   7. sqr-neg 93.1%

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

   8. sqrt-unprod 47.6%

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

   9. add-sqr-sqrt 99.0%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in t around 0 98.3%

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

8. Taylor expanded in ew around 0 98.3%

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

9. Final simplification 98.3%

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

Alternative 7: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6.5 \cdot 10^{+120} \lor \neg \left(ew \leq 8.5 \cdot 10^{+81}\right):\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left|\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right| - eh \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -6.5e+120) (not (<= ew 8.5e+81)))
   (fabs (- (* ew (cos t)) (* eh (* t (sin (atan (* t (/ (- eh) ew))))))))
   (fabs (- (fabs (/ ew (hypot 1.0 (* (tan t) (/ eh ew))))) (* eh (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6.5e+120) || !(ew <= 8.5e+81)) {
		tmp = fabs(((ew * cos(t)) - (eh * (t * sin(atan((t * (-eh / ew))))))));
	} else {
		tmp = fabs((fabs((ew / hypot(1.0, (tan(t) * (eh / ew))))) - (eh * sin(t))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -6.5e+120) || !(ew <= 8.5e+81)) {
		tmp = Math.abs(((ew * Math.cos(t)) - (eh * (t * Math.sin(Math.atan((t * (-eh / ew))))))));
	} else {
		tmp = Math.abs((Math.abs((ew / Math.hypot(1.0, (Math.tan(t) * (eh / ew))))) - (eh * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -6.5e+120) or not (ew <= 8.5e+81):
		tmp = math.fabs(((ew * math.cos(t)) - (eh * (t * math.sin(math.atan((t * (-eh / ew))))))))
	else:
		tmp = math.fabs((math.fabs((ew / math.hypot(1.0, (math.tan(t) * (eh / ew))))) - (eh * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -6.5e+120) || !(ew <= 8.5e+81))
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(t * sin(atan(Float64(t * Float64(Float64(-eh) / ew))))))));
	else
		tmp = abs(Float64(abs(Float64(ew / hypot(1.0, Float64(tan(t) * Float64(eh / ew))))) - Float64(eh * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -6.5e+120) || ~((ew <= 8.5e+81)))
		tmp = abs(((ew * cos(t)) - (eh * (t * sin(atan((t * (-eh / ew))))))));
	else
		tmp = abs((abs((ew / hypot(1.0, (tan(t) * (eh / ew))))) - (eh * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -6.5e+120], N[Not[LessEqual[ew, 8.5e+81]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(t * N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Abs[N[(ew / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.4999999999999997e120 or 8.49999999999999986e81 < 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. Step-by-step derivation

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   7. Simplified 99.8%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. div-inv 99.9%

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

      4. div-inv 99.9%

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

      5. add-sqr-sqrt 50.6%

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

      6. sqrt-unprod 99.4%

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

      7. sqr-neg 99.4%

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

      8. sqrt-unprod 49.3%

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

      9. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in t around 0 91.7%

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

       1. mul-1-neg 91.7%

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

       2. associate-*l/ 91.7%

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

       3. distribute-rgt-neg-in 91.7%

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

   12. Simplified 91.7%

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

   13. Taylor expanded in t around 0 91.4%

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

   if -6.4999999999999997e120 < ew < 8.49999999999999986e81

   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. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. sin-atan 72.1%

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

      3. associate-*l/ 70.8%

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

      4. div-inv 70.7%

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

      5. div-inv 70.8%

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

      6. add-sqr-sqrt 37.4%

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

      7. sqrt-unprod 52.8%

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

      8. sqr-neg 52.8%

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

      9. sqrt-unprod 32.0%

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

      10. add-sqr-sqrt 69.2%

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

      11. hypot-1-def 75.5%

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

      12. div-inv 75.4%

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

      13. div-inv 75.5%

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

      14. add-sqr-sqrt 40.6%

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

      15. sqrt-unprod 63.4%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in eh around inf 97.8%

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

   8. Taylor expanded in t around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. *-commutative 83.7%

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

      3. neg-mul-1 83.7%

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

      4. distribute-lft-neg-in 83.7%

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

      5. associate-*r/ 83.7%

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

   10. Simplified 83.7%

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

       1. add-sqr-sqrt 40.8%

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

       2. sqrt-unprod 77.7%

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

       3. pow2 77.7%

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

   12. Applied egg-rr 77.7%

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

       1. unpow2 77.7%

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

       2. rem-sqrt-square 83.7%

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

       3. associate-/l* 83.7%

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

       4. *-commutative 83.7%

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

       5. associate-*l/ 83.7%

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

       6. *-commutative 83.7%

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

   14. Simplified 83.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.5 \cdot 10^{+120} \lor \neg \left(ew \leq 8.5 \cdot 10^{+81}\right):\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left|\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right| - eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 80.7%

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

   3. associate-*l/ 79.4%

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

   4. div-inv 79.3%

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

   5. div-inv 79.4%

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

   6. add-sqr-sqrt 41.4%

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

   7. sqrt-unprod 65.8%

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

   8. sqr-neg 65.8%

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

   9. sqrt-unprod 36.9%

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

   10. add-sqr-sqrt 78.3%

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

   11. hypot-1-def 82.6%

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

   12. div-inv 82.5%

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

   13. div-inv 82.6%

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

   14. add-sqr-sqrt 43.7%

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

   15. sqrt-unprod 73.2%

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

6. Applied egg-rr 82.6%

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

7. Taylor expanded in eh around inf 98.1%

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

8. Taylor expanded in t around 0 87.3%

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

   1. mul-1-neg 99.0%

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

   2. associate-/l* 99.0%

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

   3. distribute-neg-frac 99.0%

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

10. Simplified 87.3%

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

11. Final simplification 87.3%

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

Alternative 9: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

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))));
}

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))))

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 80.7%

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

   3. associate-*l/ 79.4%

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

   4. div-inv 79.3%

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

   5. div-inv 79.4%

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

   6. add-sqr-sqrt 41.4%

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

   7. sqrt-unprod 65.8%

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

   8. sqr-neg 65.8%

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

   9. sqrt-unprod 36.9%

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

   10. add-sqr-sqrt 78.3%

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

   11. hypot-1-def 82.6%

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

   12. div-inv 82.5%

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

   13. div-inv 82.6%

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

   14. add-sqr-sqrt 43.7%

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

   15. sqrt-unprod 73.2%

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

6. Applied egg-rr 82.6%

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

7. Taylor expanded in eh around inf 98.1%

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

   1. expm1-log1p-u 69.8%

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

   2. expm1-udef 49.7%

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

9. Applied egg-rr 50.5%

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

    1. expm1-def 70.6%

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

    2. expm1-log1p 98.1%

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

    3. associate-*r/ 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 10: 84.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.15 \cdot 10^{+121} \lor \neg \left(ew \leq 1.55 \cdot 10^{+82}\right):\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew}{\mathsf{hypot}\left(1, \frac{\tan t}{\frac{ew}{eh}}\right)} - eh \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.15e+121) (not (<= ew 1.55e+82)))
   (fabs (- (* ew (cos t)) (* eh (* t (sin (atan (* t (/ (- eh) ew))))))))
   (fabs (- (/ ew (hypot 1.0 (/ (tan t) (/ ew eh)))) (* eh (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.15e+121) || !(ew <= 1.55e+82)) {
		tmp = fabs(((ew * cos(t)) - (eh * (t * sin(atan((t * (-eh / ew))))))));
	} else {
		tmp = fabs(((ew / hypot(1.0, (tan(t) / (ew / eh)))) - (eh * sin(t))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.15e+121) || !(ew <= 1.55e+82)) {
		tmp = Math.abs(((ew * Math.cos(t)) - (eh * (t * Math.sin(Math.atan((t * (-eh / ew))))))));
	} else {
		tmp = Math.abs(((ew / Math.hypot(1.0, (Math.tan(t) / (ew / eh)))) - (eh * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.15e+121) or not (ew <= 1.55e+82):
		tmp = math.fabs(((ew * math.cos(t)) - (eh * (t * math.sin(math.atan((t * (-eh / ew))))))))
	else:
		tmp = math.fabs(((ew / math.hypot(1.0, (math.tan(t) / (ew / eh)))) - (eh * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.15e+121) || !(ew <= 1.55e+82))
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(t * sin(atan(Float64(t * Float64(Float64(-eh) / ew))))))));
	else
		tmp = abs(Float64(Float64(ew / hypot(1.0, Float64(tan(t) / Float64(ew / eh)))) - Float64(eh * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.15e+121) || ~((ew <= 1.55e+82)))
		tmp = abs(((ew * cos(t)) - (eh * (t * sin(atan((t * (-eh / ew))))))));
	else
		tmp = abs(((ew / hypot(1.0, (tan(t) / (ew / eh)))) - (eh * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.15e+121], N[Not[LessEqual[ew, 1.55e+82]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(t * N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] / N[(ew / eh), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.1499999999999999e121 or 1.55000000000000016e82 < 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. Step-by-step derivation

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   7. Simplified 99.8%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. div-inv 99.9%

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

      4. div-inv 99.9%

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

      5. add-sqr-sqrt 50.6%

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

      6. sqrt-unprod 99.4%

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

      7. sqr-neg 99.4%

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

      8. sqrt-unprod 49.3%

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

      9. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in t around 0 91.7%

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

       1. mul-1-neg 91.7%

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

       2. associate-*l/ 91.7%

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

       3. distribute-rgt-neg-in 91.7%

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

   12. Simplified 91.7%

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

   13. Taylor expanded in t around 0 91.4%

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

   if -1.1499999999999999e121 < ew < 1.55000000000000016e82

   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. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. sin-atan 72.1%

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

      3. associate-*l/ 70.8%

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

      4. div-inv 70.7%

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

      5. div-inv 70.8%

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

      6. add-sqr-sqrt 37.4%

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

      7. sqrt-unprod 52.8%

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

      8. sqr-neg 52.8%

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

      9. sqrt-unprod 32.0%

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

      10. add-sqr-sqrt 69.2%

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

      11. hypot-1-def 75.5%

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

      12. div-inv 75.4%

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

      13. div-inv 75.5%

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

      14. add-sqr-sqrt 40.6%

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

      15. sqrt-unprod 63.4%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in eh around inf 97.8%

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

   8. Taylor expanded in t around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. *-commutative 83.7%

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

      3. neg-mul-1 83.7%

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

      4. distribute-lft-neg-in 83.7%

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

      5. associate-*r/ 83.7%

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

   10. Simplified 83.7%

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

       1. cos-atan 83.7%

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

       2. un-div-inv 83.7%

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

       3. hypot-1-def 83.7%

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

       4. clear-num 83.7%

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

       5. add-sqr-sqrt 45.7%

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

       6. sqrt-unprod 79.6%

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

       7. sqr-neg 79.6%

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

       8. sqrt-unprod 38.0%

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

       9. add-sqr-sqrt 83.7%

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

       10. un-div-inv 83.7%

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

   12. Applied egg-rr 83.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.15 \cdot 10^{+121} \lor \neg \left(ew \leq 1.55 \cdot 10^{+82}\right):\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew}{\mathsf{hypot}\left(1, \frac{\tan t}{\frac{ew}{eh}}\right)} - eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-8} \lor \neg \left(t \leq 0.00052\right):\\ \;\;\;\;\left|\frac{0}{\frac{2}{ew}} - eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right) - t \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -2.4e-8) (not (<= t 0.00052)))
   (fabs (- (/ 0.0 (/ 2.0 ew)) (* eh (sin t))))
   (fabs (- (* ew (cos (atan (* (tan t) (/ (- eh) ew))))) (* t eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2.4e-8) || !(t <= 0.00052)) {
		tmp = fabs(((0.0 / (2.0 / ew)) - (eh * sin(t))));
	} else {
		tmp = fabs(((ew * cos(atan((tan(t) * (-eh / ew))))) - (t * eh)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.4d-8)) .or. (.not. (t <= 0.00052d0))) then
        tmp = abs(((0.0d0 / (2.0d0 / ew)) - (eh * sin(t))))
    else
        tmp = abs(((ew * cos(atan((tan(t) * (-eh / ew))))) - (t * eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2.4e-8) || !(t <= 0.00052)) {
		tmp = Math.abs(((0.0 / (2.0 / ew)) - (eh * Math.sin(t))));
	} else {
		tmp = Math.abs(((ew * Math.cos(Math.atan((Math.tan(t) * (-eh / ew))))) - (t * eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -2.4e-8) or not (t <= 0.00052):
		tmp = math.fabs(((0.0 / (2.0 / ew)) - (eh * math.sin(t))))
	else:
		tmp = math.fabs(((ew * math.cos(math.atan((math.tan(t) * (-eh / ew))))) - (t * eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -2.4e-8) || !(t <= 0.00052))
		tmp = abs(Float64(Float64(0.0 / Float64(2.0 / ew)) - Float64(eh * sin(t))));
	else
		tmp = abs(Float64(Float64(ew * cos(atan(Float64(tan(t) * Float64(Float64(-eh) / ew))))) - Float64(t * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -2.4e-8) || ~((t <= 0.00052)))
		tmp = abs(((0.0 / (2.0 / ew)) - (eh * sin(t))));
	else
		tmp = abs(((ew * cos(atan((tan(t) * (-eh / ew))))) - (t * eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -2.4e-8], N[Not[LessEqual[t, 0.00052]], $MachinePrecision]], N[Abs[N[(N[(0.0 / N[(2.0 / ew), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.39999999999999998e-8 or 5.19999999999999954e-4 < t

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. sin-atan 68.7%

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

      3. associate-*l/ 67.7%

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

      4. div-inv 67.6%

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

      5. div-inv 67.7%

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

      6. add-sqr-sqrt 38.3%

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

      7. sqrt-unprod 55.2%

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

      8. sqr-neg 55.2%

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

      9. sqrt-unprod 27.9%

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

      10. add-sqr-sqrt 66.2%

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

      11. hypot-1-def 75.2%

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

      12. div-inv 75.1%

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

      13. div-inv 75.2%

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

      14. add-sqr-sqrt 43.2%

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

      15. sqrt-unprod 61.6%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in eh around inf 98.0%

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

      1. cos-mult 97.6%

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

      2. associate-*r/ 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. *-commutative 97.6%

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

       2. associate-/l* 97.5%

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

   11. Simplified 97.5%

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

   13. Applied egg-rr 0.0%

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

       1. +-inverses 52.7%

          \[\leadsto \left|\frac{\color{blue}{0}}{\frac{2}{ew}} - \sin t \cdot eh\right| \]

   15. Simplified 52.7%

       \[\leadsto \left|\frac{\color{blue}{0}}{\frac{2}{ew}} - \sin t \cdot eh\right| \]

   if -2.39999999999999998e-8 < t < 5.19999999999999954e-4

   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. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. sin-atan 91.7%

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

      3. associate-*l/ 90.2%

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

      4. div-inv 90.2%

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

      5. div-inv 90.2%

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

      6. add-sqr-sqrt 44.3%

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

      7. sqrt-unprod 75.7%

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

      8. sqr-neg 75.7%

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

      9. sqrt-unprod 45.3%

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

      10. add-sqr-sqrt 89.4%

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

      11. hypot-1-def 89.4%

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

      12. div-inv 89.3%

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

      13. div-inv 89.4%

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

      14. add-sqr-sqrt 44.1%

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

      15. sqrt-unprod 83.9%

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

   6. Applied egg-rr 89.4%

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

   7. Taylor expanded in eh around inf 98.2%

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

   8. Taylor expanded in t around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. *-commutative 98.2%

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

      3. neg-mul-1 98.2%

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

      4. distribute-lft-neg-in 98.2%

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

      5. associate-*r/ 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in t around 0 98.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-8} \lor \neg \left(t \leq 0.00052\right):\\ \;\;\;\;\left|\frac{0}{\frac{2}{ew}} - eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right) - t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 80.7%

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

   3. associate-*l/ 79.4%

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

   4. div-inv 79.3%

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

   5. div-inv 79.4%

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

   6. add-sqr-sqrt 41.4%

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

   7. sqrt-unprod 65.8%

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

   8. sqr-neg 65.8%

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

   9. sqrt-unprod 36.9%

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

   10. add-sqr-sqrt 78.3%

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

   11. hypot-1-def 82.6%

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

   12. div-inv 82.5%

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

   13. div-inv 82.6%

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

   14. add-sqr-sqrt 43.7%

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

   15. sqrt-unprod 73.2%

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

6. Applied egg-rr 82.6%

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

7. Taylor expanded in eh around inf 98.1%

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

8. Taylor expanded in t around 0 80.1%

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

   1. associate-*r/ 80.1%

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

   2. *-commutative 80.1%

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

   3. neg-mul-1 80.1%

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

   4. distribute-lft-neg-in 80.1%

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

   5. associate-*r/ 80.1%

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

10. Simplified 80.1%

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

    1. cos-atan 80.1%

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

    2. un-div-inv 80.1%

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

    3. hypot-1-def 80.1%

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

    4. clear-num 80.1%

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

    5. add-sqr-sqrt 43.6%

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

    6. sqrt-unprod 77.2%

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

    7. sqr-neg 77.2%

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

    8. sqrt-unprod 36.5%

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

    9. add-sqr-sqrt 80.1%

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

    10. un-div-inv 80.1%

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

12. Applied egg-rr 80.1%

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

13. Final simplification 80.1%

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

Alternative 13: 42.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \left|\frac{0}{\frac{2}{ew}} - eh \cdot \sin t\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (- (/ 0.0 (/ 2.0 ew)) (* eh (sin t)))))

C

double code(double eh, double ew, double t) {
	return fabs(((0.0 / (2.0 / ew)) - (eh * sin(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(((0.0d0 / (2.0d0 / ew)) - (eh * sin(t))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((0.0 / (2.0 / ew)) - (eh * Math.sin(t))));
}

Python

def code(eh, ew, t):
	return math.fabs(((0.0 / (2.0 / ew)) - (eh * math.sin(t))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(0.0 / Float64(2.0 / ew)) - Float64(eh * sin(t))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((0.0 / (2.0 / ew)) - (eh * sin(t))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(0.0 / N[(2.0 / ew), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[t], $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. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 80.7%

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

   3. associate-*l/ 79.4%

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

   4. div-inv 79.3%

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

   5. div-inv 79.4%

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

   6. add-sqr-sqrt 41.4%

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

   7. sqrt-unprod 65.8%

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

   8. sqr-neg 65.8%

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

   9. sqrt-unprod 36.9%

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

   10. add-sqr-sqrt 78.3%

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

   11. hypot-1-def 82.6%

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

   12. div-inv 82.5%

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

   13. div-inv 82.6%

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

   14. add-sqr-sqrt 43.7%

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

   15. sqrt-unprod 73.2%

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

6. Applied egg-rr 82.6%

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

7. Taylor expanded in eh around inf 98.1%

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

   1. cos-mult 97.9%

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

   2. associate-*r/ 97.9%

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

9. Applied egg-rr 97.9%

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

    1. *-commutative 97.9%

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

    2. associate-/l* 97.7%

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

11. Simplified 97.7%

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

13. Applied egg-rr 0.0%

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

    1. +-inverses 38.4%

       \[\leadsto \left|\frac{\color{blue}{0}}{\frac{2}{ew}} - \sin t \cdot eh\right| \]

15. Simplified 38.4%

    \[\leadsto \left|\frac{\color{blue}{0}}{\frac{2}{ew}} - \sin t \cdot eh\right| \]

16. Final simplification 38.4%

    \[\leadsto \left|\frac{0}{\frac{2}{ew}} - eh \cdot \sin t\right| \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))))))