Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 24.3s

Alternatives: 7

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 99.8%

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

   1. *-commutative 99.8%

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

7. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.8%

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

   3. cos-atan 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

   6. associate-/r/ 99.7%

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

   7. un-div-inv 99.7%

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

   8. associate-/l/ 99.7%

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

   9. associate-/r* 99.7%

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

9. Applied egg-rr 99.7%

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

    1. associate-/r/ 99.8%

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

    2. associate-/r* 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 99.8%

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

   1. *-commutative 99.8%

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

7. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.8%

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

   3. cos-atan 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

   6. associate-/r/ 99.7%

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

   7. un-div-inv 99.7%

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

   8. associate-/l/ 99.7%

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

   9. associate-/r* 99.7%

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

9. Applied egg-rr 99.7%

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

    1. associate-/r/ 99.8%

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

    2. associate-/r* 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in t around 0 98.8%

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

    1. *-commutative 98.8%

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

14. Simplified 98.8%

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

15. Final simplification 98.8%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.0%

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

   2. pow3 99.0%

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

   3. *-commutative 99.0%

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

   4. cos-atan 99.0%

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

   5. un-div-inv 99.0%

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

   6. hypot-1-def 99.0%

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

   7. associate-/l/ 99.0%

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

   8. associate-/r* 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in ew around inf 97.6%

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

   1. rem-cube-cbrt 98.3%

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

   2. *-commutative 98.3%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in t around inf 98.3%

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

11. Final simplification 98.3%

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

Alternative 5: 93.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((sin(t) * ew) + (cos(t) * (eh * sin(atan(((eh / (t * ew)) + (eh * ((t / ew) * (-0.3333333333333333d0))))))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.sin(t) * ew) + (Math.cos(t) * (eh * Math.sin(Math.atan(((eh / (t * ew)) + (eh * ((t / ew) * -0.3333333333333333)))))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.0%

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

   2. pow3 99.0%

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

   3. *-commutative 99.0%

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

   4. cos-atan 99.0%

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

   5. un-div-inv 99.0%

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

   6. hypot-1-def 99.0%

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

   7. associate-/l/ 99.0%

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

   8. associate-/r* 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in ew around inf 97.6%

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

   1. rem-cube-cbrt 98.3%

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

   2. *-commutative 98.3%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in t around 0 94.8%

    \[\leadsto \left|\sin t \cdot ew + \cos t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew} + \frac{eh}{ew \cdot t}\right)}\right)\right| \]
11. Step-by-step derivation

    1. +-commutative 94.8%

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

    2. *-commutative 94.8%

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

    3. associate-/l* 94.8%

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

    4. associate-*l* 94.8%

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

12. Simplified 94.8%

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

13. Final simplification 94.8%

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

Alternative 6: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.0%

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

   2. pow3 99.0%

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

   3. *-commutative 99.0%

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

   4. cos-atan 99.0%

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

   5. un-div-inv 99.0%

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

   6. hypot-1-def 99.0%

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

   7. associate-/l/ 99.0%

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

   8. associate-/r* 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in ew around inf 97.6%

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

   1. rem-cube-cbrt 98.3%

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

   2. *-commutative 98.3%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in t around 0 88.3%

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

11. Final simplification 88.3%

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

Alternative 7: 79.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $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 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.0%

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

   2. pow3 99.0%

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

   3. *-commutative 99.0%

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

   4. cos-atan 99.0%

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

   5. un-div-inv 99.0%

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

   6. hypot-1-def 99.0%

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

   7. associate-/l/ 99.0%

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

   8. associate-/r* 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in ew around inf 97.6%

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

   1. rem-cube-cbrt 98.3%

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

   2. *-commutative 98.3%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in t around 0 76.6%

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

11. Final simplification 76.6%

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

Reproduce

herbie shell --seed 2024048 
(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))))))))