Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.3s

Alternatives: 13

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

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 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 45.6%

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

   4. sqrt-unprod 92.8%

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

   5. sqr-neg 92.8%

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

   6. sqrt-unprod 54.2%

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

   7. add-sqr-sqrt 99.8%

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

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 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[((-eh) / N[(ew / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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]), $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 98.8%

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

   1. mul-1-neg 77.6%

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

   2. associate-/l* 77.6%

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

   3. distribute-neg-frac 77.6%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 75.8%

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

   2. expm1-udef 56.5%

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

6. Applied egg-rr 58.0%

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

   1. expm1-def 77.4%

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

   2. expm1-log1p 99.8%

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

   3. associate-/l* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 75.8%

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

   2. expm1-udef 56.5%

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

6. Applied egg-rr 58.0%

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

   1. expm1-def 77.4%

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

   2. expm1-log1p 99.8%

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

   3. associate-/l* 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in t around 0 98.7%

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

    1. mul-1-neg 77.6%

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

    2. associate-/l* 77.6%

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

    3. distribute-neg-frac 77.6%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

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

Alternative 5: 98.2% 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 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 45.6%

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

   4. sqrt-unprod 92.8%

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

   5. sqr-neg 92.8%

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

   6. sqrt-unprod 54.2%

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

   7. add-sqr-sqrt 99.8%

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

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 98.1%

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

8. Taylor expanded in ew around 0 98.1%

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

9. Final simplification 98.1%

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

Alternative 6: 98.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - 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)) - (sin(t) * (eh * sin(atan((-eh / (ew / t))))))));
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[((-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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. add-sqr-sqrt 45.6%

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

   4. sqrt-unprod 92.8%

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

   5. sqr-neg 92.8%

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

   6. sqrt-unprod 54.2%

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

   7. add-sqr-sqrt 99.8%

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

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 98.1%

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

8. Taylor expanded in t around 0 97.7%

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

   1. mul-1-neg 77.6%

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

   2. associate-/l* 77.6%

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

   3. distribute-neg-frac 77.6%

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

10. Simplified 97.7%

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

11. Final simplification 97.7%

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

Alternative 7: 79.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in t around 0 77.6%

   \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
8. Step-by-step derivation

   1. pow-base-1 77.6%

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

   2. *-lft-identity 77.6%

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

9. Simplified 77.6%

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

10. Final simplification 77.6%

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

Alternative 8: 79.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in t around 0 77.6%

   \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
8. Step-by-step derivation

   1. pow-base-1 77.6%

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

   2. *-lft-identity 77.6%

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

9. Simplified 77.6%

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

10. Taylor expanded in t around 0 77.6%

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

    1. mul-1-neg 77.6%

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

    2. associate-/l* 77.6%

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

    3. distribute-neg-frac 77.6%

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

12. Simplified 77.6%

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

13. Final simplification 77.6%

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

Alternative 9: 79.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in t around 0 77.6%

   \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
8. Step-by-step derivation

   1. pow-base-1 77.6%

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

   2. *-lft-identity 77.6%

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

9. Simplified 77.6%

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

10. Taylor expanded in t around 0 77.6%

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

    1. mul-1-neg 77.6%

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

    2. associate-/l* 77.6%

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

    3. distribute-neg-frac 77.6%

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

12. Simplified 77.6%

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

    1. clear-num 77.6%

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

    2. associate-/r/ 77.6%

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

    3. clear-num 77.6%

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

    4. add-sqr-sqrt 38.1%

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

    5. sqrt-unprod 73.1%

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

    6. sqr-neg 73.1%

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

    7. sqrt-unprod 39.4%

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

    8. add-sqr-sqrt 77.5%

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

14. Applied egg-rr 77.5%

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

15. Final simplification 77.5%

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

Alternative 10: 63.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-6} \lor \neg \left(t \leq 0.0033\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \left(t \cdot \frac{\frac{eh}{\mathsf{hypot}\left(1, t \cdot \frac{eh}{ew}\right)}}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -5.5e-6) (not (<= t 0.0033)))
   (fabs
    (-
     ew
     (* (* eh (sin t)) (* t (/ (/ eh (hypot 1.0 (* t (/ eh ew)))) (- ew))))))
   (fabs (- ew (* (sin (atan (/ (- eh) (/ ew t)))) (* t eh))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -5.5e-6) || !(t <= 0.0033)) {
		tmp = fabs((ew - ((eh * sin(t)) * (t * ((eh / hypot(1.0, (t * (eh / ew)))) / -ew)))));
	} else {
		tmp = fabs((ew - (sin(atan((-eh / (ew / t)))) * (t * eh))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -5.5e-6) || !(t <= 0.0033)) {
		tmp = Math.abs((ew - ((eh * Math.sin(t)) * (t * ((eh / Math.hypot(1.0, (t * (eh / ew)))) / -ew)))));
	} else {
		tmp = Math.abs((ew - (Math.sin(Math.atan((-eh / (ew / t)))) * (t * eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -5.5e-6) or not (t <= 0.0033):
		tmp = math.fabs((ew - ((eh * math.sin(t)) * (t * ((eh / math.hypot(1.0, (t * (eh / ew)))) / -ew)))))
	else:
		tmp = math.fabs((ew - (math.sin(math.atan((-eh / (ew / t)))) * (t * eh))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -5.5e-6) || !(t <= 0.0033))
		tmp = abs(Float64(ew - Float64(Float64(eh * sin(t)) * Float64(t * Float64(Float64(eh / hypot(1.0, Float64(t * Float64(eh / ew)))) / Float64(-ew))))));
	else
		tmp = abs(Float64(ew - Float64(sin(atan(Float64(Float64(-eh) / Float64(ew / t)))) * Float64(t * eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -5.5e-6) || ~((t <= 0.0033)))
		tmp = abs((ew - ((eh * sin(t)) * (t * ((eh / hypot(1.0, (t * (eh / ew)))) / -ew)))));
	else
		tmp = abs((ew - (sin(atan((-eh / (ew / t)))) * (t * eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -5.5e-6], N[Not[LessEqual[t, 0.0033]], $MachinePrecision]], N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(eh / N[Sqrt[1.0 ^ 2 + N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[Sin[N[ArcTan[N[((-eh) / N[(ew / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.4999999999999999e-6 or 0.0033 < t

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. cancel-sign-sub 99.6%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t around 0 59.3%

      \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
   8. Step-by-step derivation

      1. pow-base-1 59.3%

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

      2. *-lft-identity 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in t around 0 59.3%

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

       1. mul-1-neg 59.3%

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

       2. associate-/l* 59.3%

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

       3. distribute-neg-frac 59.3%

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

   12. Simplified 59.3%

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

       1. sin-atan 14.5%

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

       2. frac-2neg 14.5%

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

       3. remove-double-neg 14.5%

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

       4. associate-/l/ 15.0%

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

       5. hypot-1-def 24.4%

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

       6. div-inv 23.5%

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

       7. add-sqr-sqrt 10.0%

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

       8. sqrt-unprod 19.6%

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

       9. sqr-neg 19.6%

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

       10. sqrt-unprod 13.5%

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

       11. add-sqr-sqrt 23.5%

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

       12. clear-num 23.4%

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

       13. distribute-neg-frac 23.4%

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

   14. Applied egg-rr 23.4%

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

       1. associate-/r* 23.4%

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

       2. associate-/r/ 25.4%

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

       3. *-commutative 25.4%

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

       4. associate-*l/ 23.6%

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

       5. associate-*r/ 26.0%

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

   16. Simplified 26.0%

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

   if -5.4999999999999999e-6 < t < 0.0033

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. cancel-sign-sub 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in t around 0 98.2%

      \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
   8. Step-by-step derivation

      1. pow-base-1 98.2%

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

      2. *-lft-identity 98.2%

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

   9. Simplified 98.2%

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

   10. Taylor expanded in t around 0 98.2%

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

       1. mul-1-neg 98.2%

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

       2. associate-/l* 98.2%

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

       3. distribute-neg-frac 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in t around 0 97.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-6} \lor \neg \left(t \leq 0.0033\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \left(t \cdot \frac{\frac{eh}{\mathsf{hypot}\left(1, t \cdot \frac{eh}{ew}\right)}}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-6} \lor \neg \left(t \leq 0.0033\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \frac{\frac{eh}{ew}}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{t}{ew}\right)}{t}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -7.5e-6) (not (<= t 0.0033)))
   (fabs
    (- ew (* (* eh (sin t)) (/ (/ eh ew) (/ (hypot 1.0 (* eh (/ t ew))) t)))))
   (fabs (- ew (* (sin (atan (/ (- eh) (/ ew t)))) (* t eh))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7.5e-6) || !(t <= 0.0033)) {
		tmp = fabs((ew - ((eh * sin(t)) * ((eh / ew) / (hypot(1.0, (eh * (t / ew))) / t)))));
	} else {
		tmp = fabs((ew - (sin(atan((-eh / (ew / t)))) * (t * eh))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7.5e-6) || !(t <= 0.0033)) {
		tmp = Math.abs((ew - ((eh * Math.sin(t)) * ((eh / ew) / (Math.hypot(1.0, (eh * (t / ew))) / t)))));
	} else {
		tmp = Math.abs((ew - (Math.sin(Math.atan((-eh / (ew / t)))) * (t * eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -7.5e-6) or not (t <= 0.0033):
		tmp = math.fabs((ew - ((eh * math.sin(t)) * ((eh / ew) / (math.hypot(1.0, (eh * (t / ew))) / t)))))
	else:
		tmp = math.fabs((ew - (math.sin(math.atan((-eh / (ew / t)))) * (t * eh))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -7.5e-6) || !(t <= 0.0033))
		tmp = abs(Float64(ew - Float64(Float64(eh * sin(t)) * Float64(Float64(eh / ew) / Float64(hypot(1.0, Float64(eh * Float64(t / ew))) / t)))));
	else
		tmp = abs(Float64(ew - Float64(sin(atan(Float64(Float64(-eh) / Float64(ew / t)))) * Float64(t * eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -7.5e-6) || ~((t <= 0.0033)))
		tmp = abs((ew - ((eh * sin(t)) * ((eh / ew) / (hypot(1.0, (eh * (t / ew))) / t)))));
	else
		tmp = abs((ew - (sin(atan((-eh / (ew / t)))) * (t * eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -7.5e-6], N[Not[LessEqual[t, 0.0033]], $MachinePrecision]], N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[Sin[N[ArcTan[N[((-eh) / N[(ew / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000019e-6 or 0.0033 < t

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. cancel-sign-sub 99.6%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t around 0 59.3%

      \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
   8. Step-by-step derivation

      1. pow-base-1 59.3%

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

      2. *-lft-identity 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in t around 0 59.3%

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

       1. mul-1-neg 59.3%

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

       2. associate-/l* 59.3%

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

       3. distribute-neg-frac 59.3%

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

   12. Simplified 59.3%

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

       1. sin-atan 14.5%

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

       2. associate-/r/ 15.9%

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

       3. associate-/l* 16.6%

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

       4. add-sqr-sqrt 7.4%

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

       5. sqrt-unprod 15.2%

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

       6. sqr-neg 15.2%

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

       7. sqrt-unprod 9.2%

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

       8. add-sqr-sqrt 16.6%

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

       9. hypot-1-def 25.7%

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

       10. div-inv 25.1%

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

       11. add-sqr-sqrt 10.8%

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

       12. sqrt-unprod 20.1%

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

       13. sqr-neg 20.1%

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

       14. sqrt-unprod 14.2%

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

       15. add-sqr-sqrt 25.1%

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

       16. clear-num 25.0%

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

   14. Applied egg-rr 25.0%

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

   if -7.50000000000000019e-6 < t < 0.0033

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. cancel-sign-sub 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in t around 0 98.2%

      \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
   8. Step-by-step derivation

      1. pow-base-1 98.2%

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

      2. *-lft-identity 98.2%

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

   9. Simplified 98.2%

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

   10. Taylor expanded in t around 0 98.2%

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

       1. mul-1-neg 98.2%

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

       2. associate-/l* 98.2%

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

       3. distribute-neg-frac 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in t around 0 97.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-6} \lor \neg \left(t \leq 0.0033\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \frac{\frac{eh}{ew}}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{t}{ew}\right)}{t}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in t around 0 77.6%

   \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
8. Step-by-step derivation

   1. pow-base-1 77.6%

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

   2. *-lft-identity 77.6%

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

9. Simplified 77.6%

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

10. Taylor expanded in t around 0 77.6%

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

    1. mul-1-neg 77.6%

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

    2. associate-/l* 77.6%

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

    3. distribute-neg-frac 77.6%

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

12. Simplified 77.6%

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

    1. sin-atan 48.0%

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

    2. div-inv 47.8%

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

    3. clear-num 47.8%

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

    4. associate-/l* 48.1%

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

    5. add-sqr-sqrt 23.1%

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

    6. sqrt-unprod 40.7%

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

    7. sqr-neg 40.7%

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

    8. sqrt-unprod 24.9%

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

    9. add-sqr-sqrt 48.1%

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

    10. hypot-1-def 56.0%

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

    11. div-inv 56.0%

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

    12. add-sqr-sqrt 28.3%

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

    13. sqrt-unprod 46.2%

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

    14. sqr-neg 46.2%

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

    15. sqrt-unprod 27.6%

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

    16. add-sqr-sqrt 56.0%

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

    17. clear-num 56.0%

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

14. Applied egg-rr 56.0%

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

15. Final simplification 56.0%

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

Alternative 13: 54.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in t around 0 77.6%

   \[\leadsto \left|\color{blue}{{1}^{0.3333333333333333} \cdot ew} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right| \]
8. Step-by-step derivation

   1. pow-base-1 77.6%

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

   2. *-lft-identity 77.6%

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

9. Simplified 77.6%

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

10. Taylor expanded in t around 0 77.6%

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

    1. mul-1-neg 77.6%

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

    2. associate-/l* 77.6%

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

    3. distribute-neg-frac 77.6%

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

12. Simplified 77.6%

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

13. Taylor expanded in t around 0 51.4%

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

14. Final simplification 51.4%

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

Reproduce

herbie shell --seed 2024019 
(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)))))))