Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.1s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.7%

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

   4. associate-*r/ 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 97.8%

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

   1. mul-1-neg 77.9%

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

   2. associate-/l* 77.9%

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

   3. distribute-neg-frac 77.9%

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

10. Simplified 97.8%

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

11. Final simplification 97.8%

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

Alternative 3: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.7%

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

   4. associate-*r/ 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 97.8%

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

   1. mul-1-neg 77.9%

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

   2. associate-/l* 77.9%

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

   3. distribute-neg-frac 77.9%

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

10. Simplified 97.8%

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

    1. associate-/r/ 97.8%

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

    2. add-sqr-sqrt 50.2%

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

    3. sqrt-unprod 96.4%

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

    4. sqr-neg 96.4%

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

    5. sqrt-unprod 47.5%

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

    6. add-sqr-sqrt 97.6%

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

12. Applied egg-rr 97.6%

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

13. Final simplification 97.6%

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

Alternative 4: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in ew around inf 97.5%

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

9. Final simplification 97.5%

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

Alternative 5: 78.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 77.9%

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

   1. associate-*r/ 77.9%

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

   2. add-sqr-sqrt 39.9%

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

   3. sqrt-unprod 75.9%

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

   4. sqr-neg 75.9%

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

   5. sqrt-unprod 38.0%

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

   6. add-sqr-sqrt 77.9%

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

10. Applied egg-rr 77.9%

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

11. Final simplification 77.9%

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

Alternative 6: 78.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 77.9%

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

   1. add-sqr-sqrt 53.9%

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

   2. pow2 53.9%

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

10. Applied egg-rr 51.5%

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

    1. unpow2 51.5%

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

    2. add-sqr-sqrt 77.9%

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

    3. *-commutative 77.9%

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

12. Applied egg-rr 77.9%

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

13. Final simplification 77.9%

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

Alternative 7: 78.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 77.9%

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

9. Taylor expanded in t around 0 77.9%

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

    1. mul-1-neg 77.9%

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

    2. associate-/l* 77.9%

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

    3. distribute-neg-frac 77.9%

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

11. Simplified 77.9%

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

12. Final simplification 77.9%

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

Alternative 8: 78.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|ew + eh \cdot \sin t\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (+ ew (* eh (sin t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(ew + Float64(eh * sin(t))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-neg-in 99.8%

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

   3. cancel-sign-sub 99.8%

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

3. Simplified 99.8%

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

   1. add-log-exp 40.0%

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

   2. *-un-lft-identity 40.0%

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

   3. log-prod 40.0%

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

   4. metadata-eval 40.0%

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

   5. add-log-exp 99.8%

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

   6. cos-atan 99.8%

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

   7. un-div-inv 99.8%

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

   8. hypot-1-def 99.8%

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

   9. add-sqr-sqrt 47.6%

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

   10. sqrt-unprod 92.2%

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

   11. sqr-neg 92.2%

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

   12. sqrt-unprod 52.2%

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

   13. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 77.9%

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

   1. sin-atan 53.9%

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

   2. div-inv 53.9%

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

   3. associate-*r/ 54.0%

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

   4. add-sqr-sqrt 28.1%

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

   5. sqrt-unprod 43.2%

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

   6. sqr-neg 43.2%

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

   7. sqrt-unprod 25.8%

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

   8. add-sqr-sqrt 54.0%

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

   9. associate-*r/ 53.9%

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

   10. hypot-1-def 67.0%

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

   11. associate-*r/ 67.1%

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

   12. add-sqr-sqrt 34.0%

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

   13. sqrt-unprod 55.6%

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

   14. sqr-neg 55.6%

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

   15. sqrt-unprod 32.9%

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

   16. add-sqr-sqrt 67.1%

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

   17. associate-*r/ 67.0%

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

10. Applied egg-rr 67.0%

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

11. Taylor expanded in eh around -inf 77.8%

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

12. Final simplification 77.8%

    \[\leadsto \left|ew + eh \cdot \sin t\right| \]

Reproduce

herbie shell --seed 2023318 
(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)))))))