Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.9s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l/ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l/ 99.8%

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

6. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 53.4%

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

   6. sqrt-unprod 94.5%

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

   7. sqr-neg 94.5%

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

   8. sqrt-unprod 46.4%

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

   9. add-sqr-sqrt 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 99.0%

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

   1. mul-1-neg 87.6%

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

   2. associate-/l* 87.6%

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

   3. distribute-neg-frac 87.6%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 53.4%

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

   6. sqrt-unprod 94.5%

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

   7. sqr-neg 94.5%

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

   8. sqrt-unprod 46.4%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.3%

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

9. Taylor expanded in ew around 0 98.3%

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

10. Final simplification 98.3%

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

Alternative 5: 89.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] - N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[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. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 76.0%

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

   3. associate-*r/ 74.3%

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

   4. *-commutative 74.3%

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

   5. div-inv 74.2%

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

   6. div-inv 74.3%

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

   7. add-sqr-sqrt 37.1%

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

   8. sqrt-unprod 61.3%

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

   9. sqr-neg 61.3%

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

   10. sqrt-unprod 36.6%

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

   11. add-sqr-sqrt 73.2%

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

   12. hypot-1-def 76.6%

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

   13. div-inv 76.5%

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

5. Applied egg-rr 76.6%

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

   1. associate-/l* 84.3%

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

   2. associate-/l* 84.2%

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

   3. associate-*r/ 84.2%

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

   4. associate-*l/ 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-*r/ 88.3%

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

   7. associate-*l/ 88.4%

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

   8. *-commutative 88.4%

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

7. Simplified 88.4%

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

8. Taylor expanded in eh around inf 98.2%

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

9. Taylor expanded in t around 0 87.6%

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

    1. mul-1-neg 87.6%

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

    2. associate-/l* 87.6%

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

    3. distribute-neg-frac 87.6%

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

11. Simplified 87.6%

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

12. Final simplification 87.6%

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

Alternative 6: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. 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. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 76.0%

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

   3. associate-*r/ 74.3%

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

   4. *-commutative 74.3%

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

   5. div-inv 74.2%

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

   6. div-inv 74.3%

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

   7. add-sqr-sqrt 37.1%

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

   8. sqrt-unprod 61.3%

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

   9. sqr-neg 61.3%

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

   10. sqrt-unprod 36.6%

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

   11. add-sqr-sqrt 73.2%

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

   12. hypot-1-def 76.6%

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

   13. div-inv 76.5%

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

5. Applied egg-rr 76.6%

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

   1. associate-/l* 84.3%

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

   2. associate-/l* 84.2%

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

   3. associate-*r/ 84.2%

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

   4. associate-*l/ 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-*r/ 88.3%

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

   7. associate-*l/ 88.4%

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

   8. *-commutative 88.4%

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

7. Simplified 88.4%

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

8. Taylor expanded in eh around inf 98.2%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

10. Applied egg-rr 97.1%

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

    1. rem-cube-cbrt 98.2%

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

    2. expm1-log1p-u 74.5%

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

    3. expm1-udef 59.2%

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

    4. associate-*l* 59.2%

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

    5. cos-atan 61.6%

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

    6. un-div-inv 61.6%

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

    7. hypot-1-def 61.6%

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

    8. clear-num 61.6%

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

    9. un-div-inv 61.6%

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

12. Applied egg-rr 61.6%

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

    1. expm1-def 76.9%

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

    2. expm1-log1p 98.2%

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

    3. associate-*r/ 98.2%

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

    4. associate-/l* 98.1%

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

    5. associate-/r/ 98.2%

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

    6. associate-/l* 98.2%

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

    7. associate-*r/ 98.2%

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

14. Simplified 98.2%

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

15. Final simplification 98.2%

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

Alternative 7: 79.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 53.4%

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

   6. sqrt-unprod 94.5%

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

   7. sqr-neg 94.5%

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

   8. sqrt-unprod 46.4%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 98.3%

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

9. Taylor expanded in t around 0 74.3%

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

10. Final simplification 74.3%

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

Alternative 8: 79.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 76.0%

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

   3. associate-*r/ 74.3%

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

   4. *-commutative 74.3%

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

   5. div-inv 74.2%

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

   6. div-inv 74.3%

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

   7. add-sqr-sqrt 37.1%

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

   8. sqrt-unprod 61.3%

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

   9. sqr-neg 61.3%

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

   10. sqrt-unprod 36.6%

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

   11. add-sqr-sqrt 73.2%

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

   12. hypot-1-def 76.6%

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

   13. div-inv 76.5%

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

5. Applied egg-rr 76.6%

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

   1. associate-/l* 84.3%

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

   2. associate-/l* 84.2%

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

   3. associate-*r/ 84.2%

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

   4. associate-*l/ 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-*r/ 88.3%

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

   7. associate-*l/ 88.4%

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

   8. *-commutative 88.4%

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

7. Simplified 88.4%

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

8. Taylor expanded in eh around inf 98.2%

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

9. Taylor expanded in t around 0 74.3%

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

    1. mul-1-neg 74.3%

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

    2. distribute-frac-neg 74.3%

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

    3. *-commutative 74.3%

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

    4. distribute-rgt-neg-in 74.3%

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

11. Simplified 74.3%

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

    1. expm1-log1p-u 57.4%

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

    2. expm1-udef 57.3%

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

    3. add-sqr-sqrt 31.9%

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

    4. sqrt-unprod 55.5%

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

    5. sqr-neg 55.5%

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

    6. sqrt-unprod 28.2%

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

    7. add-sqr-sqrt 59.2%

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

13. Applied egg-rr 59.2%

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

    1. expm1-def 59.4%

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

    2. expm1-log1p 74.3%

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

    3. *-commutative 74.3%

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

15. Simplified 74.3%

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

16. Final simplification 74.3%

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

Alternative 9: 79.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 76.0%

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

   3. associate-*r/ 74.3%

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

   4. *-commutative 74.3%

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

   5. div-inv 74.2%

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

   6. div-inv 74.3%

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

   7. add-sqr-sqrt 37.1%

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

   8. sqrt-unprod 61.3%

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

   9. sqr-neg 61.3%

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

   10. sqrt-unprod 36.6%

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

   11. add-sqr-sqrt 73.2%

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

   12. hypot-1-def 76.6%

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

   13. div-inv 76.5%

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

5. Applied egg-rr 76.6%

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

   1. associate-/l* 84.3%

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

   2. associate-/l* 84.2%

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

   3. associate-*r/ 84.2%

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

   4. associate-*l/ 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-*r/ 88.3%

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

   7. associate-*l/ 88.4%

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

   8. *-commutative 88.4%

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

7. Simplified 88.4%

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

8. Taylor expanded in eh around inf 98.2%

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

9. Taylor expanded in t around 0 74.3%

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

    1. mul-1-neg 74.3%

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

    2. distribute-frac-neg 74.3%

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

    3. *-commutative 74.3%

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

    4. distribute-rgt-neg-in 74.3%

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

11. Simplified 74.3%

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

    1. cos-atan 74.3%

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

    2. hypot-1-def 74.3%

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

    3. add-cube-cbrt 74.3%

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

    4. times-frac 74.3%

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

    5. add-sqr-sqrt 40.1%

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

    6. sqrt-unprod 70.0%

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

    7. sqr-neg 70.0%

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

    8. sqrt-unprod 34.2%

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

    9. add-sqr-sqrt 74.3%

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

    10. times-frac 74.3%

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

    11. add-cube-cbrt 74.3%

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

    12. associate-*r/ 74.3%

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

    13. clear-num 74.3%

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

    14. un-div-inv 74.3%

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

13. Applied egg-rr 74.3%

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

14. Final simplification 74.3%

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

Alternative 10: 78.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 76.0%

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

   3. associate-*r/ 74.3%

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

   4. *-commutative 74.3%

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

   5. div-inv 74.2%

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

   6. div-inv 74.3%

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

   7. add-sqr-sqrt 37.1%

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

   8. sqrt-unprod 61.3%

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

   9. sqr-neg 61.3%

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

   10. sqrt-unprod 36.6%

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

   11. add-sqr-sqrt 73.2%

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

   12. hypot-1-def 76.6%

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

   13. div-inv 76.5%

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

5. Applied egg-rr 76.6%

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

   1. associate-/l* 84.3%

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

   2. associate-/l* 84.2%

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

   3. associate-*r/ 84.2%

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

   4. associate-*l/ 84.2%

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

   5. *-commutative 84.2%

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

   6. associate-*r/ 88.3%

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

   7. associate-*l/ 88.4%

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

   8. *-commutative 88.4%

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

7. Simplified 88.4%

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

8. Taylor expanded in eh around inf 98.2%

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

9. Taylor expanded in t around 0 74.3%

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

    1. mul-1-neg 74.3%

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

    2. distribute-frac-neg 74.3%

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

    3. *-commutative 74.3%

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

    4. distribute-rgt-neg-in 74.3%

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

11. Simplified 74.3%

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

12. Taylor expanded in t around 0 73.0%

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

    1. associate-*r/ 73.0%

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

    2. mul-1-neg 73.0%

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

    3. *-commutative 73.0%

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

    4. distribute-rgt-neg-in 73.0%

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

14. Simplified 73.0%

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

15. Final simplification 73.0%

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

Reproduce

herbie shell --seed 2023319 
(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)))))))