Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 25.9s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. frac-2neg 99.8%

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

   3. metadata-eval 99.8%

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

   4. hypot-1-def 99.8%

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

   5. add-sqr-sqrt 54.5%

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

   6. sqrt-unprod 95.8%

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

   7. sqr-neg 95.8%

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

   8. sqrt-unprod 45.2%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. neg-mul-1 99.8%

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

   2. associate-/r* 99.8%

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

   3. metadata-eval 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.1%

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

   1. associate-/l* 99.1%

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

   2. associate-*r/ 99.1%

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

   3. neg-mul-1 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 4: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.7 \cdot 10^{+82} \lor \neg \left(ew \leq 1.35 \cdot 10^{+98}\right):\\ \;\;\;\;\left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{-ew} \cdot \tan t\right)\right) + eh \cdot 0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.7e+82) (not (<= ew 1.35e+98)))
   (fabs
    (+ (* (cos t) (* ew (cos (atan (* (/ eh (- ew)) (tan t)))))) (* eh 0.0)))
   (fabs (+ (* eh (sin t)) (/ ew (hypot 1.0 (/ eh (/ ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.7e+82) || !(ew <= 1.35e+98)) {
		tmp = fabs(((cos(t) * (ew * cos(atan(((eh / -ew) * tan(t)))))) + (eh * 0.0)));
	} else {
		tmp = fabs(((eh * sin(t)) + (ew / hypot(1.0, (eh / (ew / tan(t)))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.7e+82) || !(ew <= 1.35e+98)) {
		tmp = Math.abs(((Math.cos(t) * (ew * Math.cos(Math.atan(((eh / -ew) * Math.tan(t)))))) + (eh * 0.0)));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) + (ew / Math.hypot(1.0, (eh / (ew / Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.7e+82) or not (ew <= 1.35e+98):
		tmp = math.fabs(((math.cos(t) * (ew * math.cos(math.atan(((eh / -ew) * math.tan(t)))))) + (eh * 0.0)))
	else:
		tmp = math.fabs(((eh * math.sin(t)) + (ew / math.hypot(1.0, (eh / (ew / math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.7e+82) || !(ew <= 1.35e+98))
		tmp = abs(Float64(Float64(cos(t) * Float64(ew * cos(atan(Float64(Float64(eh / Float64(-ew)) * tan(t)))))) + Float64(eh * 0.0)));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) + Float64(ew / hypot(1.0, Float64(eh / Float64(ew / tan(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.7e+82) || ~((ew <= 1.35e+98)))
		tmp = abs(((cos(t) * (ew * cos(atan(((eh / -ew) * tan(t)))))) + (eh * 0.0)));
	else
		tmp = abs(((eh * sin(t)) + (ew / hypot(1.0, (eh / (ew / tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.7e+82], N[Not[LessEqual[ew, 1.35e+98]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * N[(ew * N[Cos[N[ArcTan[N[(N[(eh / (-ew)), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(ew / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.69999999999999997e82 or 1.35e98 < ew

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 98.8%

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

      1. associate-/l* 98.8%

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

      2. associate-*r/ 98.8%

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

      3. neg-mul-1 98.8%

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

   6. Simplified 98.8%

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

   8. Applied egg-rr 94.5%

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

      1. *-commutative 94.5%

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

      2. associate-/l* 94.5%

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

      3. +-inverses 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \frac{\color{blue}{0}}{\frac{2}{eh}}\right| \]

      4. associate-/r/ 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{\frac{0}{2} \cdot eh}\right| \]

      5. metadata-eval 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{0} \cdot eh\right| \]

      6. +-inverses 94.5%

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

      7. *-commutative 94.5%

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

      8. +-inverses 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - eh \cdot \color{blue}{0}\right| \]

   10. Simplified 94.5%

       \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{eh \cdot 0}\right| \]

   if -1.69999999999999997e82 < ew < 1.35e98

   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%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. sin-atan 72.5%

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

      3. associate-*l/ 72.5%

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

      4. associate-/l* 72.5%

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

      5. add-sqr-sqrt 37.4%

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

      6. sqrt-unprod 56.3%

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

      7. sqr-neg 56.3%

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

      8. sqrt-unprod 34.2%

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

      9. add-sqr-sqrt 71.4%

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

   5. Applied egg-rr 80.6%

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

      1. associate-/l* 80.7%

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

      2. associate-*l* 80.6%

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

      3. associate-/r/ 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in eh around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in t around 0 86.3%

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

       1. *-commutative 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

       2. associate-*r/ 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

       3. mul-1-neg 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

       4. distribute-lft-neg-in 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

       5. *-commutative 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

   13. Simplified 86.3%

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

       1. expm1-log1p-u 28.5%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)\right)} - eh \cdot 0\right| \]

       2. expm1-udef 10.4%

          \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)} - 1\right)} - eh \cdot 0\right| \]

   15. Applied egg-rr 62.6%

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

       1. expm1-def 79.2%

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

       2. expm1-log1p 86.3%

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

       3. *-commutative 86.3%

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

       4. associate-/r/ 86.3%

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

   17. Simplified 86.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.7 \cdot 10^{+82} \lor \neg \left(ew \leq 1.35 \cdot 10^{+98}\right):\\ \;\;\;\;\left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{-ew} \cdot \tan t\right)\right) + eh \cdot 0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 82.1%

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

   3. associate-*l/ 82.1%

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

   4. associate-/l* 82.1%

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

   5. add-sqr-sqrt 44.9%

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

   6. sqrt-unprod 71.1%

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

   7. sqr-neg 71.1%

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

   8. sqrt-unprod 36.5%

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

   9. add-sqr-sqrt 81.3%

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

5. Applied egg-rr 87.3%

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

   1. associate-/l* 87.3%

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

   2. associate-*l* 87.2%

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

   3. associate-/r/ 87.4%

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

7. Simplified 87.4%

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

8. Taylor expanded in eh around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

10. Simplified 98.2%

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

    1. associate-*r* 63.6%

       \[\leadsto \left|\color{blue}{\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)} - eh \cdot 0\right| \]

    2. cos-atan 63.3%

       \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \color{blue}{\frac{1}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}}} - eh \cdot 0\right| \]

    3. un-div-inv 63.3%

       \[\leadsto \left|\color{blue}{\frac{\cos t \cdot ew}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}}} - eh \cdot 0\right| \]

    4. metadata-eval 63.3%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{\color{blue}{1 \cdot 1} + \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}} - eh \cdot 0\right| \]

    5. associate-*r/ 63.3%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\frac{\tan t \cdot eh}{-ew}} \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}} - eh \cdot 0\right| \]

    6. *-commutative 63.3%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{\color{blue}{eh \cdot \tan t}}{-ew} \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}} - eh \cdot 0\right| \]

    7. associate-*r/ 63.3%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{eh \cdot \tan t}{-ew} \cdot \color{blue}{\frac{\tan t \cdot eh}{-ew}}}} - eh \cdot 0\right| \]

    8. *-commutative 63.3%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{eh \cdot \tan t}{-ew} \cdot \frac{\color{blue}{eh \cdot \tan t}}{-ew}}} - eh \cdot 0\right| \]

    9. frac-times 55.3%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\frac{\left(eh \cdot \tan t\right) \cdot \left(eh \cdot \tan t\right)}{\left(-ew\right) \cdot \left(-ew\right)}}}} - eh \cdot 0\right| \]

    10. sqr-neg 55.3%

        \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{\left(eh \cdot \tan t\right) \cdot \left(eh \cdot \tan t\right)}{\color{blue}{ew \cdot ew}}}} - eh \cdot 0\right| \]

    11. frac-times 63.3%

        \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\frac{eh \cdot \tan t}{ew} \cdot \frac{eh \cdot \tan t}{ew}}}} - eh \cdot 0\right| \]

    12. *-commutative 63.3%

        \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{\color{blue}{\tan t \cdot eh}}{ew} \cdot \frac{eh \cdot \tan t}{ew}}} - eh \cdot 0\right| \]

    13. associate-*r/ 63.3%

        \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)} \cdot \frac{eh \cdot \tan t}{ew}}} - eh \cdot 0\right| \]

12. Applied egg-rr 98.2%

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

13. Final simplification 98.2%

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

Alternative 6: 87.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{+82} \lor \neg \left(ew \leq 1.6 \cdot 10^{+100}\right):\\ \;\;\;\;\left|eh \cdot 0 + \frac{\cos t \cdot ew}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.65e+82) (not (<= ew 1.6e+100)))
   (fabs (+ (* eh 0.0) (/ (* (cos t) ew) (hypot 1.0 (* (/ eh ew) (tan t))))))
   (fabs (+ (* eh (sin t)) (/ ew (hypot 1.0 (/ eh (/ ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.65e+82) || !(ew <= 1.6e+100)) {
		tmp = fabs(((eh * 0.0) + ((cos(t) * ew) / hypot(1.0, ((eh / ew) * tan(t))))));
	} else {
		tmp = fabs(((eh * sin(t)) + (ew / hypot(1.0, (eh / (ew / tan(t)))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.65e+82) || !(ew <= 1.6e+100)) {
		tmp = Math.abs(((eh * 0.0) + ((Math.cos(t) * ew) / Math.hypot(1.0, ((eh / ew) * Math.tan(t))))));
	} else {
		tmp = Math.abs(((eh * Math.sin(t)) + (ew / Math.hypot(1.0, (eh / (ew / Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.65e+82) or not (ew <= 1.6e+100):
		tmp = math.fabs(((eh * 0.0) + ((math.cos(t) * ew) / math.hypot(1.0, ((eh / ew) * math.tan(t))))))
	else:
		tmp = math.fabs(((eh * math.sin(t)) + (ew / math.hypot(1.0, (eh / (ew / math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.65e+82) || !(ew <= 1.6e+100))
		tmp = abs(Float64(Float64(eh * 0.0) + Float64(Float64(cos(t) * ew) / hypot(1.0, Float64(Float64(eh / ew) * tan(t))))));
	else
		tmp = abs(Float64(Float64(eh * sin(t)) + Float64(ew / hypot(1.0, Float64(eh / Float64(ew / tan(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.65e+82) || ~((ew <= 1.6e+100)))
		tmp = abs(((eh * 0.0) + ((cos(t) * ew) / hypot(1.0, ((eh / ew) * tan(t))))));
	else
		tmp = abs(((eh * sin(t)) + (ew / hypot(1.0, (eh / (ew / tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.65e+82], N[Not[LessEqual[ew, 1.6e+100]], $MachinePrecision]], N[Abs[N[(N[(eh * 0.0), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(ew / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.6499999999999999e82 or 1.5999999999999999e100 < ew

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 98.8%

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

      1. associate-/l* 98.8%

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

      2. associate-*r/ 98.8%

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

      3. neg-mul-1 98.8%

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

   6. Simplified 98.8%

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

   8. Applied egg-rr 94.5%

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

      1. *-commutative 94.5%

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

      2. associate-/l* 94.5%

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

      3. +-inverses 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \frac{\color{blue}{0}}{\frac{2}{eh}}\right| \]

      4. associate-/r/ 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{\frac{0}{2} \cdot eh}\right| \]

      5. metadata-eval 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{0} \cdot eh\right| \]

      6. +-inverses 94.5%

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

      7. *-commutative 94.5%

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

      8. +-inverses 94.5%

         \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - eh \cdot \color{blue}{0}\right| \]

   10. Simplified 94.5%

       \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{eh \cdot 0}\right| \]
   11. Step-by-step derivation

       1. associate-*r* 94.5%

          \[\leadsto \left|\color{blue}{\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)} - eh \cdot 0\right| \]

       2. cos-atan 94.4%

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \color{blue}{\frac{1}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}}} - eh \cdot 0\right| \]

       3. un-div-inv 94.4%

          \[\leadsto \left|\color{blue}{\frac{\cos t \cdot ew}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}}} - eh \cdot 0\right| \]

       4. metadata-eval 94.4%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{\color{blue}{1 \cdot 1} + \left(\tan t \cdot \frac{eh}{-ew}\right) \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}} - eh \cdot 0\right| \]

       5. associate-*r/ 94.4%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\frac{\tan t \cdot eh}{-ew}} \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}} - eh \cdot 0\right| \]

       6. *-commutative 94.4%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{\color{blue}{eh \cdot \tan t}}{-ew} \cdot \left(\tan t \cdot \frac{eh}{-ew}\right)}} - eh \cdot 0\right| \]

       7. associate-*r/ 94.4%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{eh \cdot \tan t}{-ew} \cdot \color{blue}{\frac{\tan t \cdot eh}{-ew}}}} - eh \cdot 0\right| \]

       8. *-commutative 94.4%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{eh \cdot \tan t}{-ew} \cdot \frac{\color{blue}{eh \cdot \tan t}}{-ew}}} - eh \cdot 0\right| \]

       9. frac-times 83.4%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\frac{\left(eh \cdot \tan t\right) \cdot \left(eh \cdot \tan t\right)}{\left(-ew\right) \cdot \left(-ew\right)}}}} - eh \cdot 0\right| \]

       10. sqr-neg 83.4%

           \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{\left(eh \cdot \tan t\right) \cdot \left(eh \cdot \tan t\right)}{\color{blue}{ew \cdot ew}}}} - eh \cdot 0\right| \]

       11. frac-times 94.4%

           \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\frac{eh \cdot \tan t}{ew} \cdot \frac{eh \cdot \tan t}{ew}}}} - eh \cdot 0\right| \]

       12. *-commutative 94.4%

           \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \frac{\color{blue}{\tan t \cdot eh}}{ew} \cdot \frac{eh \cdot \tan t}{ew}}} - eh \cdot 0\right| \]

       13. associate-*r/ 94.4%

           \[\leadsto \left|\frac{\cos t \cdot ew}{\sqrt{1 \cdot 1 + \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)} \cdot \frac{eh \cdot \tan t}{ew}}} - eh \cdot 0\right| \]

   12. Applied egg-rr 94.4%

       \[\leadsto \left|\color{blue}{\frac{\cos t \cdot ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}} - eh \cdot 0\right| \]

   if -1.6499999999999999e82 < ew < 1.5999999999999999e100

   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%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. sin-atan 72.5%

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

      3. associate-*l/ 72.5%

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

      4. associate-/l* 72.5%

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

      5. add-sqr-sqrt 37.4%

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

      6. sqrt-unprod 56.3%

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

      7. sqr-neg 56.3%

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

      8. sqrt-unprod 34.2%

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

      9. add-sqr-sqrt 71.4%

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

   5. Applied egg-rr 80.6%

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

      1. associate-/l* 80.7%

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

      2. associate-*l* 80.6%

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

      3. associate-/r/ 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in eh around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in t around 0 86.3%

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

       1. *-commutative 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

       2. associate-*r/ 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

       3. mul-1-neg 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

       4. distribute-lft-neg-in 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

       5. *-commutative 34.8%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

   13. Simplified 86.3%

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

       1. expm1-log1p-u 28.5%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)\right)} - eh \cdot 0\right| \]

       2. expm1-udef 10.4%

          \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)} - 1\right)} - eh \cdot 0\right| \]

   15. Applied egg-rr 62.6%

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

       1. expm1-def 79.2%

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

       2. expm1-log1p 86.3%

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

       3. *-commutative 86.3%

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

       4. associate-/r/ 86.3%

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

   17. Simplified 86.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{+82} \lor \neg \left(ew \leq 1.6 \cdot 10^{+100}\right):\\ \;\;\;\;\left|eh \cdot 0 + \frac{\cos t \cdot ew}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}\right|\\ \end{array} \]

Alternative 7: 42.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot 0 + ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* eh 0.0) (* ew (cos (atan (* (/ eh ew) (- (tan t)))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((eh * 0.0) + (ew * cos(atan(((eh / ew) * -tan(t)))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((eh * 0.0d0) + (ew * cos(atan(((eh / ew) * -tan(t)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((eh * 0.0) + (ew * Math.cos(Math.atan(((eh / ew) * -Math.tan(t)))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((eh * 0.0) + (ew * math.cos(math.atan(((eh / ew) * -math.tan(t)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(eh * 0.0) + Float64(ew * cos(atan(Float64(Float64(eh / ew) * Float64(-tan(t))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((eh * 0.0) + (ew * cos(atan(((eh / ew) * -tan(t)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * 0.0), $MachinePrecision] + N[(ew * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * (-N[Tan[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. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.1%

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

   1. associate-/l* 99.1%

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

   2. associate-*r/ 99.1%

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

   3. neg-mul-1 99.1%

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

6. Simplified 99.1%

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

8. Applied egg-rr 63.6%

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

   1. *-commutative 63.6%

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

   2. associate-/l* 63.6%

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

   3. +-inverses 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \frac{\color{blue}{0}}{\frac{2}{eh}}\right| \]

   4. associate-/r/ 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{\frac{0}{2} \cdot eh}\right| \]

   5. metadata-eval 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{0} \cdot eh\right| \]

   6. +-inverses 63.6%

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

   7. *-commutative 63.6%

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

   8. +-inverses 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - eh \cdot \color{blue}{0}\right| \]

10. Simplified 63.6%

    \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{eh \cdot 0}\right| \]

11. Taylor expanded in t around 0 43.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)} - eh \cdot 0\right| \]
12. Step-by-step derivation

    1. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

    2. associate-*r/ 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

    3. mul-1-neg 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    4. distribute-lft-neg-in 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    5. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

13. Simplified 43.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

14. Final simplification 43.7%

    \[\leadsto \left|eh \cdot 0 + ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right| \]

Alternative 8: 54.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 82.1%

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

   3. associate-*l/ 82.1%

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

   4. associate-/l* 82.1%

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

   5. add-sqr-sqrt 44.9%

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

   6. sqrt-unprod 71.1%

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

   7. sqr-neg 71.1%

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

   8. sqrt-unprod 36.5%

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

   9. add-sqr-sqrt 81.3%

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

5. Applied egg-rr 87.3%

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

   1. associate-/l* 87.3%

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

   2. associate-*l* 87.2%

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

   3. associate-/r/ 87.4%

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

7. Simplified 87.4%

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

8. Taylor expanded in eh around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in t around 0 78.8%

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

    1. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

    2. associate-*r/ 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

    3. mul-1-neg 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    4. distribute-lft-neg-in 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    5. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

13. Simplified 78.8%

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

14. Taylor expanded in t around 0 56.8%

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

    1. *-commutative 56.8%

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

    2. mul-1-neg 56.8%

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

    3. *-commutative 56.8%

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

16. Simplified 56.8%

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

17. Final simplification 56.8%

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

Alternative 9: 78.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 82.1%

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

   3. associate-*l/ 82.1%

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

   4. associate-/l* 82.1%

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

   5. add-sqr-sqrt 44.9%

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

   6. sqrt-unprod 71.1%

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

   7. sqr-neg 71.1%

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

   8. sqrt-unprod 36.5%

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

   9. add-sqr-sqrt 81.3%

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

5. Applied egg-rr 87.3%

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

   1. associate-/l* 87.3%

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

   2. associate-*l* 87.2%

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

   3. associate-/r/ 87.4%

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

7. Simplified 87.4%

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

8. Taylor expanded in eh around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in t around 0 78.8%

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

    1. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

    2. associate-*r/ 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

    3. mul-1-neg 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    4. distribute-lft-neg-in 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    5. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

13. Simplified 78.8%

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

14. Taylor expanded in t around 0 77.3%

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

    1. associate-/l* 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\frac{eh}{\frac{ew}{t}}}\right) - eh \cdot 0\right| \]

    2. mul-1-neg 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\frac{eh}{\frac{ew}{t}}\right)} - eh \cdot 0\right| \]

    3. distribute-frac-neg 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{-eh}{\frac{ew}{t}}\right)} - eh \cdot 0\right| \]

    4. associate-/r/ 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot t\right)} - eh \cdot 0\right| \]

16. Simplified 77.3%

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

17. Final simplification 77.3%

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

Alternative 10: 79.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 82.1%

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

   3. associate-*l/ 82.1%

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

   4. associate-/l* 82.1%

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

   5. add-sqr-sqrt 44.9%

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

   6. sqrt-unprod 71.1%

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

   7. sqr-neg 71.1%

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

   8. sqrt-unprod 36.5%

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

   9. add-sqr-sqrt 81.3%

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

5. Applied egg-rr 87.3%

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

   1. associate-/l* 87.3%

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

   2. associate-*l* 87.2%

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

   3. associate-/r/ 87.4%

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

7. Simplified 87.4%

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

8. Taylor expanded in eh around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in t around 0 78.8%

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

    1. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

    2. associate-*r/ 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

    3. mul-1-neg 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    4. distribute-lft-neg-in 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    5. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

13. Simplified 78.8%

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

    1. expm1-log1p-u 26.8%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)\right)} - eh \cdot 0\right| \]

    2. expm1-udef 15.1%

       \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)} - 1\right)} - eh \cdot 0\right| \]

15. Applied egg-rr 49.6%

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

    1. expm1-def 60.4%

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

    2. expm1-log1p 78.8%

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

    3. *-commutative 78.8%

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

    4. associate-/r/ 78.8%

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

17. Simplified 78.8%

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

18. Final simplification 78.8%

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

Alternative 11: 41.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot 0 + ew \cdot \cos \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* eh 0.0) (* ew (cos (atan (* t (/ (- eh) ew))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((eh * 0.0) + (ew * cos(atan((t * (-eh / ew)))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((eh * 0.0d0) + (ew * cos(atan((t * (-eh / ew)))))))
end function

Java

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

Python

def code(eh, ew, t):
	return math.fabs(((eh * 0.0) + (ew * math.cos(math.atan((t * (-eh / ew)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(eh * 0.0) + Float64(ew * cos(atan(Float64(t * Float64(Float64(-eh) / ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((eh * 0.0) + (ew * cos(atan((t * (-eh / ew)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * 0.0), $MachinePrecision] + N[(ew * N[Cos[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.1%

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

   1. associate-/l* 99.1%

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

   2. associate-*r/ 99.1%

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

   3. neg-mul-1 99.1%

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

6. Simplified 99.1%

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

8. Applied egg-rr 63.6%

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

   1. *-commutative 63.6%

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

   2. associate-/l* 63.6%

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

   3. +-inverses 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \frac{\color{blue}{0}}{\frac{2}{eh}}\right| \]

   4. associate-/r/ 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{\frac{0}{2} \cdot eh}\right| \]

   5. metadata-eval 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{0} \cdot eh\right| \]

   6. +-inverses 63.6%

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

   7. *-commutative 63.6%

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

   8. +-inverses 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - eh \cdot \color{blue}{0}\right| \]

10. Simplified 63.6%

    \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{eh \cdot 0}\right| \]

11. Taylor expanded in t around 0 43.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)} - eh \cdot 0\right| \]
12. Step-by-step derivation

    1. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

    2. associate-*r/ 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

    3. mul-1-neg 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    4. distribute-lft-neg-in 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    5. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

13. Simplified 43.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

14. Taylor expanded in t around 0 42.1%

    \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)} - eh \cdot 0\right| \]
15. Step-by-step derivation

    1. associate-/l* 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\frac{eh}{\frac{ew}{t}}}\right) - eh \cdot 0\right| \]

    2. mul-1-neg 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\frac{eh}{\frac{ew}{t}}\right)} - eh \cdot 0\right| \]

    3. distribute-frac-neg 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{-eh}{\frac{ew}{t}}\right)} - eh \cdot 0\right| \]

    4. associate-/r/ 42.1%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot t\right)} - eh \cdot 0\right| \]

16. Simplified 42.1%

    \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot t\right)} - eh \cdot 0\right| \]

17. Final simplification 42.1%

    \[\leadsto \left|eh \cdot 0 + ew \cdot \cos \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right| \]

Alternative 12: 42.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot 0 + \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* eh 0.0) (/ ew (hypot 1.0 (* (/ eh ew) (tan t)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((eh * 0.0) + (ew / hypot(1.0, ((eh / ew) * tan(t))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((eh * 0.0) + (ew / Math.hypot(1.0, ((eh / ew) * Math.tan(t))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((eh * 0.0) + (ew / math.hypot(1.0, ((eh / ew) * math.tan(t))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(eh * 0.0) + Float64(ew / hypot(1.0, Float64(Float64(eh / ew) * tan(t))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((eh * 0.0) + (ew / hypot(1.0, ((eh / ew) * tan(t))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * 0.0), $MachinePrecision] + N[(ew / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.1%

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

   1. associate-/l* 99.1%

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

   2. associate-*r/ 99.1%

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

   3. neg-mul-1 99.1%

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

6. Simplified 99.1%

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

8. Applied egg-rr 63.6%

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

   1. *-commutative 63.6%

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

   2. associate-/l* 63.6%

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

   3. +-inverses 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \frac{\color{blue}{0}}{\frac{2}{eh}}\right| \]

   4. associate-/r/ 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{\frac{0}{2} \cdot eh}\right| \]

   5. metadata-eval 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{0} \cdot eh\right| \]

   6. +-inverses 63.6%

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

   7. *-commutative 63.6%

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

   8. +-inverses 63.6%

      \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - eh \cdot \color{blue}{0}\right| \]

10. Simplified 63.6%

    \[\leadsto \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right) - \color{blue}{eh \cdot 0}\right| \]

11. Taylor expanded in t around 0 43.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)} - eh \cdot 0\right| \]
12. Step-by-step derivation

    1. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{\color{blue}{\tan t \cdot eh}}{ew}\right) - eh \cdot 0\right| \]

    2. associate-*r/ 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(-1 \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot 0\right| \]

    3. mul-1-neg 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(-\tan t \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    4. distribute-lft-neg-in 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)} - eh \cdot 0\right| \]

    5. *-commutative 43.7%

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]

13. Simplified 43.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)} - eh \cdot 0\right| \]
14. Step-by-step derivation

    1. expm1-log1p-u 26.8%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)\right)} - eh \cdot 0\right| \]

    2. expm1-udef 15.1%

       \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)} - 1\right)} - eh \cdot 0\right| \]

15. Applied egg-rr 15.1%

    \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} - 1\right)} - eh \cdot 0\right| \]
16. Step-by-step derivation

    1. expm1-def 26.6%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)} - eh \cdot 0\right| \]

    2. expm1-log1p 43.4%

       \[\leadsto \left|\color{blue}{\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}} - eh \cdot 0\right| \]

17. Simplified 43.4%

    \[\leadsto \left|\color{blue}{\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}} - eh \cdot 0\right| \]

18. Final simplification 43.4%

    \[\leadsto \left|eh \cdot 0 + \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew} \cdot \tan t\right)}\right| \]

Reproduce

herbie shell --seed 2023305 
(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)))))))