Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.2s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.2%

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

   1. associate-*r/ 97.7%

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

   2. remove-double-neg 97.7%

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

   3. neg-mul-1 97.7%

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

   4. times-frac 97.7%

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

   5. metadata-eval 97.7%

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

   6. *-lft-identity 97.7%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 3: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.2%

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

9. Final simplification 98.2%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.2%

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

   1. add-sqr-sqrt 49.7%

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

   2. sqrt-unprod 94.5%

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

   3. sqr-neg 94.5%

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

   4. sqrt-unprod 48.4%

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

   5. add-sqr-sqrt 98.1%

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

   6. associate-*r/ 98.1%

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

10. Applied egg-rr 98.1%

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

11. Final simplification 98.1%

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

Alternative 5: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.2%

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

   1. expm1-log1p-u 81.1%

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

   2. expm1-udef 72.1%

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

   3. associate-*l* 72.1%

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

   4. distribute-lft-neg-out 72.1%

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

   5. atan-neg 72.1%

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

10. Applied egg-rr 72.1%

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

    1. expm1-def 81.1%

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

    2. expm1-log1p 98.2%

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

    3. *-commutative 98.2%

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

    4. sin-neg 98.2%

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

    5. distribute-lft-neg-out 98.2%

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

12. Simplified 98.2%

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

13. Final simplification 98.2%

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

Alternative 6: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.2%

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

9. Taylor expanded in t around 0 97.7%

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

    1. associate-*r/ 97.7%

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

    2. remove-double-neg 97.7%

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

    3. neg-mul-1 97.7%

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

    4. times-frac 97.7%

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

    5. metadata-eval 97.7%

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

    6. *-lft-identity 97.7%

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

11. Simplified 97.7%

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

12. Taylor expanded in ew around 0 97.7%

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

13. Final simplification 97.7%

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

Alternative 7: 85.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.65 \cdot 10^{-35} \lor \neg \left(eh \leq 1.75 \cdot 10^{+15}\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\log \left(e^{eh \cdot 0}\right) - ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.65e-35) (not (<= eh 1.75e+15)))
   (fabs (- ew (* (* eh (sin t)) (sin (atan (/ (* t eh) (- ew)))))))
   (fabs (- (log (exp (* eh 0.0))) (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.65e-35) || !(eh <= 1.75e+15)) {
		tmp = fabs((ew - ((eh * sin(t)) * sin(atan(((t * eh) / -ew))))));
	} else {
		tmp = fabs((log(exp((eh * 0.0))) - (ew * cos(t))));
	}
	return tmp;
}

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) :: tmp
    if ((eh <= (-1.65d-35)) .or. (.not. (eh <= 1.75d+15))) then
        tmp = abs((ew - ((eh * sin(t)) * sin(atan(((t * eh) / -ew))))))
    else
        tmp = abs((log(exp((eh * 0.0d0))) - (ew * cos(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.65e-35) || !(eh <= 1.75e+15)) {
		tmp = Math.abs((ew - ((eh * Math.sin(t)) * Math.sin(Math.atan(((t * eh) / -ew))))));
	} else {
		tmp = Math.abs((Math.log(Math.exp((eh * 0.0))) - (ew * Math.cos(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.65e-35) or not (eh <= 1.75e+15):
		tmp = math.fabs((ew - ((eh * math.sin(t)) * math.sin(math.atan(((t * eh) / -ew))))))
	else:
		tmp = math.fabs((math.log(math.exp((eh * 0.0))) - (ew * math.cos(t))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.65e-35) || !(eh <= 1.75e+15))
		tmp = abs(Float64(ew - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * eh) / Float64(-ew)))))));
	else
		tmp = abs(Float64(log(exp(Float64(eh * 0.0))) - Float64(ew * cos(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.65e-35) || ~((eh <= 1.75e+15)))
		tmp = abs((ew - ((eh * sin(t)) * sin(atan(((t * eh) / -ew))))));
	else
		tmp = abs((log(exp((eh * 0.0))) - (ew * cos(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.65e-35], N[Not[LessEqual[eh, 1.75e+15]], $MachinePrecision]], N[Abs[N[(ew - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Log[N[Exp[N[(eh * 0.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.65e-35 or 1.75e15 < eh

   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%

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

      1. cos-atan 99.7%

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

      2. hypot-1-def 99.7%

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

      3. add-sqr-sqrt 55.3%

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

      4. sqrt-unprod 89.9%

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

      5. sqr-neg 89.9%

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

      6. sqrt-unprod 44.4%

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

      7. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 98.2%

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

   9. Taylor expanded in t around 0 97.9%

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

       1. associate-*r/ 97.9%

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

       2. remove-double-neg 97.9%

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

       3. neg-mul-1 97.9%

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

       4. times-frac 97.9%

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

       5. metadata-eval 97.9%

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

       6. *-lft-identity 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in t around 0 93.5%

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

   if -1.65e-35 < eh < 1.75e15

   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%

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

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. add-sqr-sqrt 44.5%

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

      4. sqrt-unprod 98.7%

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

      5. sqr-neg 98.7%

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

      6. sqrt-unprod 55.3%

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

      7. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 98.1%

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

      1. add-log-exp 86.9%

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

      2. associate-*l* 86.9%

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

      3. distribute-lft-neg-out 86.9%

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

      4. atan-neg 86.9%

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

   10. Applied egg-rr 86.9%

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

       1. sin-mult 86.9%

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

       2. div-sub 86.9%

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

   12. Applied egg-rr 86.9%

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

       1. +-inverses 86.9%

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

   14. Simplified 86.9%

       \[\leadsto \left|ew \cdot \left(\cos t \cdot 1\right) - \log \left(e^{eh \cdot \color{blue}{0}}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.65 \cdot 10^{-35} \lor \neg \left(eh \leq 1.75 \cdot 10^{+15}\right):\\ \;\;\;\;\left|ew - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\log \left(e^{eh \cdot 0}\right) - ew \cdot \cos t\right|\\ \end{array} \]

Alternative 8: 60.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left|\log \left(e^{eh \cdot 0}\right) - ew \cdot \cos t\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (- (log (exp (* eh 0.0))) (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	return fabs((log(exp((eh * 0.0))) - (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((log(exp((eh * 0.0d0))) - (ew * cos(t))))
end function

Java

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

Python

def code(eh, ew, t):
	return math.fabs((math.log(math.exp((eh * 0.0))) - (ew * math.cos(t))))

Julia

function code(eh, ew, t)
	return abs(Float64(log(exp(Float64(eh * 0.0))) - Float64(ew * cos(t))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((log(exp((eh * 0.0))) - (ew * cos(t))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Log[N[Exp[N[(eh * 0.0), $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%

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 94.3%

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

   5. sqr-neg 94.3%

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

   6. sqrt-unprod 49.8%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.2%

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

   1. add-log-exp 60.3%

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

   2. associate-*l* 60.3%

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

   3. distribute-lft-neg-out 60.3%

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

   4. atan-neg 60.3%

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

10. Applied egg-rr 60.3%

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

    1. sin-mult 61.2%

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

    2. div-sub 61.2%

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

12. Applied egg-rr 61.2%

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

    1. +-inverses 61.2%

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

14. Simplified 61.2%

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

15. Final simplification 61.2%

    \[\leadsto \left|\log \left(e^{eh \cdot 0}\right) - ew \cdot \cos t\right| \]

Reproduce

herbie shell --seed 2023229 
(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)))))))