Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.1s

Alternatives: 15

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.7%

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

   3. clear-num 99.7%

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

   4. add-sqr-sqrt 47.9%

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

   5. sqrt-unprod 93.4%

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

   6. sqr-neg 93.4%

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

   7. sqrt-unprod 51.8%

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

   8. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. cos-atan 64.4%

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

   2. hypot-1-def 64.4%

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

   3. div-inv 64.4%

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

   4. add-sqr-sqrt 30.8%

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

   5. sqrt-unprod 57.6%

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

   6. sqr-neg 57.6%

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

   7. sqrt-unprod 33.6%

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

   8. add-sqr-sqrt 64.4%

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

   9. clear-num 64.4%

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

5. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * (cos(t) * (1.0 / hypot(1.0, (tan(t) * (eh / ew)))))) - (sin(t) * (eh * sin(atan((-eh * (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[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in eh around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. mul-1-neg 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

   5. distribute-rgt-neg-in 99.7%

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

6. Simplified 99.7%

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

   1. cos-atan 64.4%

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

   2. hypot-1-def 64.4%

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

   3. div-inv 64.4%

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

   4. add-sqr-sqrt 30.8%

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

   5. sqrt-unprod 57.6%

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

   6. sqr-neg 57.6%

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

   7. sqrt-unprod 33.6%

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

   8. add-sqr-sqrt 64.4%

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

   9. clear-num 64.4%

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

8. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in t around 0 98.9%

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

12. Final simplification 98.9%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in eh around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. mul-1-neg 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

   5. distribute-rgt-neg-in 99.7%

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

6. Simplified 99.7%

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

   1. cos-atan 64.4%

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

   2. hypot-1-def 64.4%

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

   3. div-inv 64.4%

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

   4. add-sqr-sqrt 30.8%

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

   5. sqrt-unprod 57.6%

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

   6. sqr-neg 57.6%

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

   7. sqrt-unprod 33.6%

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

   8. add-sqr-sqrt 64.4%

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

   9. clear-num 64.4%

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

8. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in t around 0 97.9%

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

12. Final simplification 97.9%

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

Alternative 5: 90.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in t around 0 88.8%

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

   1. mul-1-neg 88.8%

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

   2. associate-/l* 88.8%

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

   3. distribute-neg-frac 88.8%

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

6. Simplified 88.8%

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

   1. associate-*r* 88.8%

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

   2. sin-atan 72.7%

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

   3. associate-*r/ 70.1%

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

   4. div-inv 70.0%

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

   5. add-sqr-sqrt 34.1%

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

   6. sqrt-unprod 59.9%

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

   7. sqr-neg 59.9%

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

   8. sqrt-unprod 35.9%

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

   9. add-sqr-sqrt 69.6%

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

   10. clear-num 69.5%

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

   11. hypot-1-def 73.5%

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

   12. div-inv 73.6%

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

8. Applied egg-rr 73.6%

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

   1. associate-/l* 80.5%

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

   2. associate-/r/ 80.5%

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

   3. *-commutative 80.5%

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

   4. associate-*l/ 80.4%

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

   5. associate-*r/ 77.1%

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

   6. *-commutative 77.1%

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

   7. associate-*l/ 77.1%

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

   8. associate-*r/ 77.0%

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

10. Simplified 77.0%

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

11. Taylor expanded in eh around inf 88.2%

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

12. Final simplification 88.2%

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

Alternative 6: 90.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in t around 0 88.8%

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

   1. mul-1-neg 88.8%

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

   2. associate-/l* 88.8%

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

   3. distribute-neg-frac 88.8%

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

6. Simplified 88.8%

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

   1. associate-*r* 88.8%

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

   2. sin-atan 72.7%

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

   3. associate-*r/ 70.1%

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

   4. div-inv 70.0%

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

   5. add-sqr-sqrt 34.1%

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

   6. sqrt-unprod 59.9%

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

   7. sqr-neg 59.9%

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

   8. sqrt-unprod 35.9%

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

   9. add-sqr-sqrt 69.6%

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

   10. clear-num 69.5%

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

   11. hypot-1-def 73.5%

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

   12. div-inv 73.6%

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

8. Applied egg-rr 73.6%

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

   1. associate-/l* 80.5%

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

   2. associate-/r/ 80.5%

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

   3. *-commutative 80.5%

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

   4. associate-*l/ 80.4%

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

   5. associate-*r/ 77.1%

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

   6. *-commutative 77.1%

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

   7. associate-*l/ 77.1%

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

   8. associate-*r/ 77.0%

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

10. Simplified 77.0%

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

11. Taylor expanded in eh around -inf 88.2%

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

    1. mul-1-neg 88.2%

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

    2. *-commutative 88.2%

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

    3. distribute-lft-neg-in 88.2%

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

13. Simplified 88.2%

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

14. Final simplification 88.2%

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

Alternative 7: 61.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   ew
   (*
    (cos t)
    (cos (atan (/ (- eh) (+ (* -0.3333333333333333 (* ew t)) (/ ew t)))))))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * (cos(t) * cos(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (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((-eh / (((-0.3333333333333333d0) * (ew * t)) + (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((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))))))));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * (math.cos(t) * math.cos(math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))))))))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * Float64(cos(t) * cos(atan(Float64(Float64(-eh) / Float64(Float64(-0.3333333333333333 * Float64(ew * t)) + Float64(ew / t))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * (cos(t) * cos(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[((-eh) / N[(N[(-0.3333333333333333 * N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(ew / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in t around 0 64.7%

   \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-eh}{\color{blue}{-0.3333333333333333 \cdot \left(t \cdot ew\right) + \frac{ew}{t}}}\right)\right) - 0\right| \]

9. Final simplification 64.7%

   \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right| \]

Alternative 8: 46.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.7 \cdot 10^{-32} \lor \neg \left(eh \leq 4.5 \cdot 10^{+55}\right):\\ \;\;\;\;\left|ew\right| \cdot \left|\frac{1}{\mathsf{hypot}\left(1, \frac{\tan t}{ew} \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t \cdot t}{\frac{ew}{eh \cdot eh}} + ew \cdot \left(\cos t \cdot \frac{-1}{\mathsf{hypot}\left(1, \frac{-t}{\frac{ew}{eh}}\right)}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.7e-32) (not (<= eh 4.5e+55)))
   (* (fabs ew) (fabs (/ 1.0 (hypot 1.0 (* (/ (tan t) ew) eh)))))
   (fabs
    (+
     (/ (* t t) (/ ew (* eh eh)))
     (* ew (* (cos t) (/ -1.0 (hypot 1.0 (/ (- t) (/ ew eh))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.7e-32) || !(eh <= 4.5e+55)) {
		tmp = fabs(ew) * fabs((1.0 / hypot(1.0, ((tan(t) / ew) * eh))));
	} else {
		tmp = fabs((((t * t) / (ew / (eh * eh))) + (ew * (cos(t) * (-1.0 / hypot(1.0, (-t / (ew / eh))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.7e-32) || !(eh <= 4.5e+55)) {
		tmp = Math.abs(ew) * Math.abs((1.0 / Math.hypot(1.0, ((Math.tan(t) / ew) * eh))));
	} else {
		tmp = Math.abs((((t * t) / (ew / (eh * eh))) + (ew * (Math.cos(t) * (-1.0 / Math.hypot(1.0, (-t / (ew / eh))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2.7e-32) or not (eh <= 4.5e+55):
		tmp = math.fabs(ew) * math.fabs((1.0 / math.hypot(1.0, ((math.tan(t) / ew) * eh))))
	else:
		tmp = math.fabs((((t * t) / (ew / (eh * eh))) + (ew * (math.cos(t) * (-1.0 / math.hypot(1.0, (-t / (ew / eh))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.7e-32) || !(eh <= 4.5e+55))
		tmp = Float64(abs(ew) * abs(Float64(1.0 / hypot(1.0, Float64(Float64(tan(t) / ew) * eh)))));
	else
		tmp = abs(Float64(Float64(Float64(t * t) / Float64(ew / Float64(eh * eh))) + Float64(ew * Float64(cos(t) * Float64(-1.0 / hypot(1.0, Float64(Float64(-t) / Float64(ew / eh))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2.7e-32) || ~((eh <= 4.5e+55)))
		tmp = abs(ew) * abs((1.0 / hypot(1.0, ((tan(t) / ew) * eh))));
	else
		tmp = abs((((t * t) / (ew / (eh * eh))) + (ew * (cos(t) * (-1.0 / hypot(1.0, (-t / (ew / eh))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.7e-32], N[Not[LessEqual[eh, 4.5e+55]], $MachinePrecision]], N[(N[Abs[ew], $MachinePrecision] * N[Abs[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(t * t), $MachinePrecision] / N[(ew / N[(eh * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(-1.0 / N[Sqrt[1.0 ^ 2 + N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.69999999999999981e-32 or 4.49999999999999998e55 < eh

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. sub0-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--r+ 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 45.4%

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

      2. associate-*r/ 45.4%

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

   5. Applied egg-rr 42.3%

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

      1. +-inverses 42.3%

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

      2. *-commutative 42.3%

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

      3. associate-/l* 42.3%

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

      4. div0 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in t around 0 32.1%

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

      1. *-commutative 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-*r/ 32.1%

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

      4. associate-*l* 32.1%

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

      5. neg-mul-1 32.1%

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

      6. *-commutative 32.1%

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

      7. associate-*l/ 32.1%

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

      8. distribute-rgt-neg-in 32.1%

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

   10. Simplified 32.1%

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

       1. distribute-frac-neg 32.1%

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

       2. *-commutative 32.1%

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

       3. associate-*r/ 32.1%

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

       4. distribute-lft-neg-out 32.1%

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

       5. cos-atan 31.5%

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

       6. hypot-1-def 31.6%

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

       7. add-sqr-sqrt 20.9%

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

       8. sqrt-unprod 20.3%

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

       9. sqr-neg 20.3%

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

       10. sqrt-unprod 10.7%

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

       11. add-sqr-sqrt 31.6%

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

       12. associate-*r/ 31.6%

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

       13. *-commutative 31.6%

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

       14. associate-*r/ 31.6%

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

       15. frac-2neg 31.6%

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

       16. metadata-eval 31.6%

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

   12. Applied egg-rr 31.6%

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

       1. /-rgt-identity 31.6%

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

       2. /-rgt-identity 31.6%

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

       3. *-commutative 31.6%

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

       4. associate-*l/ 31.6%

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

       5. associate-*r/ 31.6%

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

   14. Simplified 31.6%

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

   if -2.69999999999999981e-32 < eh < 4.49999999999999998e55

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. sub0-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. associate--r+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 88.5%

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

      1. mul-1-neg 88.5%

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

      2. associate-/l* 88.5%

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

      3. distribute-neg-frac 88.5%

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

   6. Simplified 88.5%

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

      1. associate-*r* 88.4%

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

      2. sin-atan 85.6%

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

      3. associate-*r/ 85.6%

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

      4. div-inv 85.5%

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

      5. add-sqr-sqrt 38.4%

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

      6. sqrt-unprod 82.8%

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

      7. sqr-neg 82.8%

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

      8. sqrt-unprod 47.2%

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

      9. add-sqr-sqrt 84.9%

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

      10. clear-num 85.0%

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

      11. hypot-1-def 87.8%

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

      12. div-inv 87.8%

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

   8. Applied egg-rr 87.8%

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

      1. associate-/l* 87.8%

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

      2. associate-/r/ 87.8%

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

      3. *-commutative 87.8%

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

      4. associate-*l/ 87.8%

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

      5. associate-*r/ 87.8%

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

      6. *-commutative 87.8%

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

      7. associate-*l/ 87.8%

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

      8. associate-*r/ 87.8%

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

   10. Simplified 87.8%

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

   11. Taylor expanded in t around 0 66.3%

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

       1. associate-/l* 66.1%

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

       2. unpow2 66.1%

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

       3. unpow2 66.1%

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

   13. Simplified 66.1%

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

       1. cos-atan 66.0%

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

       2. hypot-1-def 66.0%

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

   15. Applied egg-rr 66.0%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.7 \cdot 10^{-32} \lor \neg \left(eh \leq 4.5 \cdot 10^{+55}\right):\\ \;\;\;\;\left|ew\right| \cdot \left|\frac{1}{\mathsf{hypot}\left(1, \frac{\tan t}{ew} \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t \cdot t}{\frac{ew}{eh \cdot eh}} + ew \cdot \left(\cos t \cdot \frac{-1}{\mathsf{hypot}\left(1, \frac{-t}{\frac{ew}{eh}}\right)}\right)\right|\\ \end{array} \]

Alternative 9: 61.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

   1. cos-atan 64.4%

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

   2. hypot-1-def 64.4%

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

   3. div-inv 64.4%

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

   4. add-sqr-sqrt 30.8%

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

   5. sqrt-unprod 57.6%

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

   6. sqr-neg 57.6%

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

   7. sqrt-unprod 33.6%

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

   8. add-sqr-sqrt 64.4%

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

   9. clear-num 64.4%

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

9. Applied egg-rr 64.4%

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

10. Final simplification 64.4%

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

Alternative 10: 46.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3 \cdot 10^{-32} \lor \neg \left(eh \leq 2.65 \cdot 10^{+54}\right):\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{\tan t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t \cdot t}{\frac{ew}{eh \cdot eh}} + ew \cdot \left(\cos t \cdot \frac{-1}{\mathsf{hypot}\left(1, \frac{-t}{\frac{ew}{eh}}\right)}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3e-32) (not (<= eh 2.65e+54)))
   (fabs (* ew (cos (atan (/ (* (tan t) eh) ew)))))
   (fabs
    (+
     (/ (* t t) (/ ew (* eh eh)))
     (* ew (* (cos t) (/ -1.0 (hypot 1.0 (/ (- t) (/ ew eh))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3e-32) || !(eh <= 2.65e+54)) {
		tmp = fabs((ew * cos(atan(((tan(t) * eh) / ew)))));
	} else {
		tmp = fabs((((t * t) / (ew / (eh * eh))) + (ew * (cos(t) * (-1.0 / hypot(1.0, (-t / (ew / eh))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3e-32) || !(eh <= 2.65e+54)) {
		tmp = Math.abs((ew * Math.cos(Math.atan(((Math.tan(t) * eh) / ew)))));
	} else {
		tmp = Math.abs((((t * t) / (ew / (eh * eh))) + (ew * (Math.cos(t) * (-1.0 / Math.hypot(1.0, (-t / (ew / eh))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -3e-32) or not (eh <= 2.65e+54):
		tmp = math.fabs((ew * math.cos(math.atan(((math.tan(t) * eh) / ew)))))
	else:
		tmp = math.fabs((((t * t) / (ew / (eh * eh))) + (ew * (math.cos(t) * (-1.0 / math.hypot(1.0, (-t / (ew / eh))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3e-32) || !(eh <= 2.65e+54))
		tmp = abs(Float64(ew * cos(atan(Float64(Float64(tan(t) * eh) / ew)))));
	else
		tmp = abs(Float64(Float64(Float64(t * t) / Float64(ew / Float64(eh * eh))) + Float64(ew * Float64(cos(t) * Float64(-1.0 / hypot(1.0, Float64(Float64(-t) / Float64(ew / eh))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -3e-32) || ~((eh <= 2.65e+54)))
		tmp = abs((ew * cos(atan(((tan(t) * eh) / ew)))));
	else
		tmp = abs((((t * t) / (ew / (eh * eh))) + (ew * (cos(t) * (-1.0 / hypot(1.0, (-t / (ew / eh))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3e-32], N[Not[LessEqual[eh, 2.65e+54]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t * t), $MachinePrecision] / N[(ew / N[(eh * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(-1.0 / N[Sqrt[1.0 ^ 2 + N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3e-32 or 2.65000000000000009e54 < eh

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. sub0-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--r+ 99.8%

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

   3. Simplified 99.8%

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

      1. sin-mult 45.1%

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

      2. associate-*r/ 45.1%

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

   5. Applied egg-rr 41.9%

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

      1. +-inverses 41.9%

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

      2. *-commutative 41.9%

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

      3. associate-/l* 41.9%

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

      4. div0 41.9%

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

   7. Simplified 41.9%

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

      1. expm1-log1p-u 41.9%

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

      2. add-sqr-sqrt 27.2%

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

      3. sqrt-unprod 28.8%

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

      4. sqr-neg 28.8%

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

      5. sqrt-unprod 14.8%

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

      6. add-sqr-sqrt 41.9%

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

      7. div-inv 41.9%

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

      8. clear-num 41.9%

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

   9. Applied egg-rr 41.9%

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

   10. Taylor expanded in t around 0 31.8%

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

   if -3e-32 < eh < 2.65000000000000009e54

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. sub0-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. associate--r+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. associate-/l* 88.4%

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

      3. distribute-neg-frac 88.4%

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

   6. Simplified 88.4%

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

      1. associate-*r* 88.4%

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

      2. sin-atan 86.2%

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

      3. associate-*r/ 86.2%

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

      4. div-inv 86.1%

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

      5. add-sqr-sqrt 38.6%

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

      6. sqrt-unprod 83.4%

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

      7. sqr-neg 83.4%

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

      8. sqrt-unprod 47.6%

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

      9. add-sqr-sqrt 85.5%

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

      10. clear-num 85.6%

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

      11. hypot-1-def 87.7%

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

      12. div-inv 87.7%

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

   8. Applied egg-rr 87.7%

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

      1. associate-/l* 87.7%

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

      2. associate-/r/ 87.7%

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

      3. *-commutative 87.7%

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

      4. associate-*l/ 87.7%

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

      5. associate-*r/ 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-*l/ 87.7%

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

      8. associate-*r/ 87.7%

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

   10. Simplified 87.7%

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

   11. Taylor expanded in t around 0 66.8%

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

       1. associate-/l* 66.5%

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

       2. unpow2 66.5%

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

       3. unpow2 66.5%

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

   13. Simplified 66.5%

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

       1. cos-atan 66.5%

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

       2. hypot-1-def 66.5%

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

   15. Applied egg-rr 66.5%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3 \cdot 10^{-32} \lor \neg \left(eh \leq 2.65 \cdot 10^{+54}\right):\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{\tan t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t \cdot t}{\frac{ew}{eh \cdot eh}} + ew \cdot \left(\cos t \cdot \frac{-1}{\mathsf{hypot}\left(1, \frac{-t}{\frac{ew}{eh}}\right)}\right)\right|\\ \end{array} \]

Alternative 11: 42.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in t around 0 45.0%

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

   1. *-commutative 45.0%

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

   2. *-commutative 45.0%

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

   3. associate-*r/ 45.0%

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

   4. associate-*l* 45.0%

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

   5. neg-mul-1 45.0%

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

   6. *-commutative 45.0%

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

   7. associate-*l/ 45.0%

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

   8. distribute-rgt-neg-in 45.0%

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

10. Simplified 45.0%

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

    1. distribute-frac-neg 45.0%

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

    2. *-commutative 45.0%

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

    3. associate-*r/ 45.0%

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

    4. distribute-lft-neg-out 45.0%

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

    5. cos-atan 44.8%

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

    6. hypot-1-def 44.8%

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

    7. add-sqr-sqrt 23.3%

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

    8. sqrt-unprod 39.5%

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

    9. sqr-neg 39.5%

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

    10. sqrt-unprod 21.5%

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

    11. add-sqr-sqrt 44.8%

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

    12. associate-*r/ 44.8%

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

    13. *-commutative 44.8%

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

    14. associate-*r/ 44.8%

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

    15. frac-2neg 44.8%

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

    16. metadata-eval 44.8%

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

12. Applied egg-rr 44.8%

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

    1. /-rgt-identity 44.8%

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

    2. /-rgt-identity 44.8%

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

    3. *-commutative 44.8%

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

    4. associate-*l/ 44.8%

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

    5. associate-*r/ 44.8%

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

14. Simplified 44.8%

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

15. Final simplification 44.8%

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

Alternative 12: 52.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in t around 0 54.3%

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

   1. mul-1-neg 88.8%

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

   2. associate-/l* 88.8%

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

   3. distribute-neg-frac 88.8%

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

10. Simplified 54.3%

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

11. Final simplification 54.3%

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

Alternative 13: 41.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in t around 0 45.0%

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

   1. *-commutative 45.0%

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

   2. *-commutative 45.0%

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

   3. associate-*r/ 45.0%

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

   4. associate-*l* 45.0%

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

   5. neg-mul-1 45.0%

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

   6. *-commutative 45.0%

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

   7. associate-*l/ 45.0%

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

   8. distribute-rgt-neg-in 45.0%

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

10. Simplified 45.0%

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

11. Taylor expanded in t around 0 43.7%

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

12. Final simplification 43.7%

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

Alternative 14: 40.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in t around 0 45.0%

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

   1. *-commutative 45.0%

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

   2. *-commutative 45.0%

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

   3. associate-*r/ 45.0%

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

   4. associate-*l* 45.0%

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

   5. neg-mul-1 45.0%

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

   6. *-commutative 45.0%

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

   7. associate-*l/ 45.0%

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

   8. distribute-rgt-neg-in 45.0%

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

10. Simplified 45.0%

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

11. Taylor expanded in t around 0 43.7%

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

    1. cos-atan 42.4%

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

    2. *-commutative 42.4%

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

    3. *-commutative 42.4%

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

13. Applied egg-rr 42.4%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{-eh \cdot t}{ew} \cdot \frac{-eh \cdot t}{ew}}}} - 0\right| \]
14. Step-by-step derivation

    1. hypot-1-def 42.5%

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

    2. distribute-lft-neg-in 42.5%

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

    3. associate-/l* 42.6%

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

15. Simplified 42.6%

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

16. Final simplification 42.6%

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

Alternative 15: 40.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. sub0-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. associate--r+ 99.7%

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

3. Simplified 99.7%

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

   1. sin-mult 66.1%

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

   2. associate-*r/ 66.1%

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

5. Applied egg-rr 64.6%

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

   1. +-inverses 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-/l* 64.6%

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

   4. div0 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in t around 0 45.0%

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

   1. *-commutative 45.0%

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

   2. *-commutative 45.0%

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

   3. associate-*r/ 45.0%

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

   4. associate-*l* 45.0%

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

   5. neg-mul-1 45.0%

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

   6. *-commutative 45.0%

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

   7. associate-*l/ 45.0%

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

   8. distribute-rgt-neg-in 45.0%

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

10. Simplified 45.0%

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

11. Taylor expanded in t around 0 43.7%

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

    1. cos-atan 42.4%

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

    2. *-commutative 42.4%

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

    3. *-commutative 42.4%

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

13. Applied egg-rr 42.4%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{-eh \cdot t}{ew} \cdot \frac{-eh \cdot t}{ew}}}} - 0\right| \]
14. Step-by-step derivation

    1. distribute-frac-neg 42.4%

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

    2. associate-*r/ 42.4%

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

    3. distribute-frac-neg 42.4%

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

    4. associate-*r/ 42.6%

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

    5. sqr-neg 42.6%

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

    6. hypot-1-def 42.6%

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

    7. associate-*r/ 42.5%

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

    8. *-commutative 42.5%

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

    9. *-lft-identity 42.5%

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

    10. times-frac 42.6%

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

    11. /-rgt-identity 42.6%

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

15. Simplified 42.6%

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

16. Final simplification 42.6%

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

Reproduce

herbie shell --seed 2023187 
(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)))))))