Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.9s

Alternatives: 10

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \left|\mathsf{fma}\left(ew, \sin t \cdot \cos t\_1, eh \cdot \left(\cos t \cdot \sin t\_1\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.9%

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

   2. fma-define 99.9%

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

   3. associate-/r* 99.9%

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

   4. associate-*l* 99.9%

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

   5. associate-/r* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \sin t \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right|} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ \left|\left(ew \cdot \sin t\right) \cdot {\left(1 + {t\_1}^{2}\right)}^{-0.5} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (+
     (* (* ew (sin t)) (pow (+ 1.0 (pow t_1 2.0)) -0.5))
     (* (* eh (cos t)) (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs((((ew * sin(t)) * pow((1.0 + pow(t_1, 2.0)), -0.5)) + ((eh * cos(t)) * sin(atan(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 = (eh / ew) / tan(t)
    code = abs((((ew * sin(t)) * ((1.0d0 + (t_1 ** 2.0d0)) ** (-0.5d0))) + ((eh * cos(t)) * sin(atan(t_1)))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / Math.tan(t);
	return Math.abs((((ew * Math.sin(t)) * Math.pow((1.0 + Math.pow(t_1, 2.0)), -0.5)) + ((eh * Math.cos(t)) * Math.sin(Math.atan(t_1)))));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / math.tan(t)
	return math.fabs((((ew * math.sin(t)) * math.pow((1.0 + math.pow(t_1, 2.0)), -0.5)) + ((eh * math.cos(t)) * math.sin(math.atan(t_1)))))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(Float64(Float64(Float64(ew * sin(t)) * (Float64(1.0 + (t_1 ^ 2.0)) ^ -0.5)) + Float64(Float64(eh * cos(t)) * sin(atan(t_1)))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs((((ew * sin(t)) * ((1.0 + (t_1 ^ 2.0)) ^ -0.5)) + ((eh * cos(t)) * sin(atan(t_1)))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. clear-num 99.8%

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

   3. associate-/r/ 99.8%

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

   4. associate-/r* 99.8%

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

4. Applied egg-rr 99.8%

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

   1. cos-atan 99.8%

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

   2. inv-pow 99.8%

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

   3. hypot-1-def 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*r/ 99.8%

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

   6. div-inv 99.8%

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

   7. hypot-undefine 99.8%

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

   8. sqrt-pow2 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \color{blue}{{\left(1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)}^{-0.5}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
7. Step-by-step derivation

   1. associate-/r* 99.9%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot {\left(1 + {\color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}}^{2}\right)}^{-0.5} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

8. Simplified 99.9%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \color{blue}{{\left(1 + {\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}\right)}^{-0.5}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
9. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in eh around 0 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in eh around 0 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 89.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in eh around 0 97.8%

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

8. Taylor expanded in t around 0 92.7%

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

9. Final simplification 92.7%

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

Alternative 7: 79.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in eh around 0 97.8%

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

8. Taylor expanded in t around 0 85.7%

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

9. Taylor expanded in ew around 0 85.7%

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

10. Final simplification 85.7%

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

Alternative 8: 74.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.092 \lor \neg \left(t \leq 6.2 \cdot 10^{+33}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.092) (not (<= t 6.2e+33)))
   (fabs (* ew (sin t)))
   (fabs (+ (* ew t) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.092) || !(t <= 6.2e+33)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(((ew * t) + (eh * sin(atan((eh / (ew * tan(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 ((t <= (-0.092d0)) .or. (.not. (t <= 6.2d+33))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.092) || !(t <= 6.2e+33)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(((ew * t) + (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.092) or not (t <= 6.2e+33):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(((ew * t) + (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.092) || !(t <= 6.2e+33))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(Float64(ew * t) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.092) || ~((t <= 6.2e+33)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(((ew * t) + (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.092], N[Not[LessEqual[t, 6.2e+33]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * t), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.091999999999999998 or 6.2e33 < t

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

      2. cos-atan 99.7%

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

      3. hypot-1-def 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in eh around 0 97.8%

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

   8. Taylor expanded in t around 0 74.5%

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

   9. Taylor expanded in ew around inf 69.5%

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]

   if -0.091999999999999998 < t < 6.2e33

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

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

      2. cos-atan 100.0%

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

      3. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eh around 0 97.7%

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

   8. Taylor expanded in t around 0 95.0%

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

   9. Taylor expanded in t around 0 94.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.092 \lor \neg \left(t \leq 6.2 \cdot 10^{+33}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-19} \lor \neg \left(t \leq 3 \cdot 10^{-149}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -8e-19) (not (<= t 3e-149)))
   (fabs (* ew (sin t)))
   (fabs (* eh (sin (atan (/ (/ eh ew) (tan t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -8e-19) || !(t <= 3e-149)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((eh * sin(atan(((eh / ew) / tan(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 ((t <= (-8d-19)) .or. (.not. (t <= 3d-149))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((eh * sin(atan(((eh / ew) / tan(t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -8e-19) || !(t <= 3e-149)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -8e-19) or not (t <= 3e-149):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((eh * math.sin(math.atan(((eh / ew) / math.tan(t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -8e-19) || !(t <= 3e-149))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -8e-19) || ~((t <= 3e-149)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((eh * sin(atan(((eh / ew) / tan(t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -8e-19], N[Not[LessEqual[t, 3e-149]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.9999999999999998e-19 or 3.0000000000000002e-149 < t

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

      2. cos-atan 99.7%

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

      3. hypot-1-def 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-/r* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eh around 0 97.6%

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

   8. Taylor expanded in t around 0 77.9%

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

   9. Taylor expanded in ew around inf 64.7%

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]

   if -7.9999999999999998e-19 < t < 3.0000000000000002e-149

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

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

      2. cos-atan 100.0%

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

      3. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eh around 0 97.9%

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

   8. Taylor expanded in t around 0 97.9%

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

   9. Taylor expanded in ew around 0 81.3%

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

       1. associate-/r* 81.3%

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

   11. Simplified 81.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-19} \lor \neg \left(t \leq 3 \cdot 10^{-149}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * 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 * sin(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in eh around 0 97.8%

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

8. Taylor expanded in t around 0 85.7%

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

9. Taylor expanded in ew around inf 48.1%

   \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))