Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 35.3s

Alternatives: 12

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

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

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(Float64(ew * sin(t)), cos(t_1), Float64(Float64(eh * 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[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $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. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 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. Taylor expanded in t around 0 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) + (cos(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)) + (cos(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)) + (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.cos(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(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t)))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * sin(t)) + (cos(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[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.3%

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

7. Taylor expanded in eh around 0 98.3%

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

8. Final simplification 98.3%

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

Alternative 5: 94.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan(((eh / (ew * t)) + (-0.3333333333333333 * ((t * eh) / ew))))))));
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 / N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(N[(t * eh), $MachinePrecision] / ew), $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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.3%

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

7. Taylor expanded in t around 0 94.8%

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

8. Final simplification 94.8%

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

Alternative 6: 89.2% 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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.3%

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

7. Taylor expanded in t around 0 90.1%

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

8. Final simplification 90.1%

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

Alternative 7: 79.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* ew (sin t)) (* eh (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(Float64(eh / 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[(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. Taylor expanded in t around 0 82.2%

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

   1. add-cbrt-cube 64.3%

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

   2. pow3 64.3%

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

   3. cos-atan 64.9%

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

   4. hypot-1-def 64.9%

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

   5. associate-/l/ 64.9%

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

   6. un-div-inv 64.9%

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

4. Applied egg-rr 64.9%

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

5. Taylor expanded in ew around inf 81.0%

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

6. Final simplification 81.0%

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

Alternative 8: 36.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t * N[(t * N[(N[(ew * ew), $MachinePrecision] / eh), $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. Taylor expanded in t around 0 82.2%

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

   1. add-cbrt-cube 64.3%

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

   2. pow3 64.3%

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

   3. cos-atan 64.9%

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

   4. hypot-1-def 64.9%

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

   5. associate-/l/ 64.9%

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

   6. un-div-inv 64.9%

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

4. Applied egg-rr 64.9%

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

5. Taylor expanded in t around 0 40.5%

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

   1. associate-/l* 40.0%

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

   2. unpow2 40.0%

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

   3. unpow2 40.0%

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

7. Simplified 40.0%

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

8. Taylor expanded in t around 0 40.5%

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

   1. unpow2 40.5%

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

   2. associate-*r/ 40.0%

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

   3. unpow2 40.0%

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

   4. associate-*l/ 42.7%

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

   5. associate-*r* 44.0%

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

   6. associate-*l/ 41.6%

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

10. Simplified 41.6%

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

11. Final simplification 41.6%

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

Alternative 9: 37.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[(ew / eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $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. Taylor expanded in t around 0 82.2%

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

   1. add-cbrt-cube 64.3%

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

   2. pow3 64.3%

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

   3. cos-atan 64.9%

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

   4. hypot-1-def 64.9%

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

   5. associate-/l/ 64.9%

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

   6. un-div-inv 64.9%

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

4. Applied egg-rr 64.9%

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

5. Taylor expanded in t around 0 40.5%

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

   1. associate-/l* 40.0%

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

   2. unpow2 40.0%

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

   3. unpow2 40.0%

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

7. Simplified 40.0%

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

   1. clear-num 40.0%

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

   2. associate-/r/ 40.0%

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

   3. associate-/r* 42.7%

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

   4. associate-/r/ 42.7%

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

   5. clear-num 42.7%

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

9. Applied egg-rr 42.7%

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

10. Final simplification 42.7%

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

Alternative 10: 40.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(ew * N[(ew * N[(t / N[(eh / t), $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. Taylor expanded in t around 0 82.2%

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

   1. add-cbrt-cube 64.3%

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

   2. pow3 64.3%

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

   3. cos-atan 64.9%

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

   4. hypot-1-def 64.9%

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

   5. associate-/l/ 64.9%

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

   6. un-div-inv 64.9%

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

4. Applied egg-rr 64.9%

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

5. Taylor expanded in t around 0 40.5%

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

   1. associate-/l* 40.0%

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

   2. unpow2 40.0%

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

   3. unpow2 40.0%

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

7. Simplified 40.0%

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

   1. associate-/r/ 40.0%

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

   2. associate-*r* 45.4%

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

   3. associate-/l* 45.7%

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

9. Applied egg-rr 45.7%

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

10. Final simplification 45.7%

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

Alternative 11: 40.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(ew * t), $MachinePrecision] * N[(ew * t), $MachinePrecision]), $MachinePrecision] / eh), $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. Taylor expanded in t around 0 82.2%

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

   1. add-cbrt-cube 64.3%

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

   2. pow3 64.3%

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

   3. cos-atan 64.9%

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

   4. hypot-1-def 64.9%

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

   5. associate-/l/ 64.9%

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

   6. un-div-inv 64.9%

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

4. Applied egg-rr 64.9%

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

5. Taylor expanded in t around 0 40.5%

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

   1. unpow2 40.5%

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

   2. unpow2 40.5%

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

   3. unswap-sqr 46.2%

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

7. Simplified 46.2%

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

8. Final simplification 46.2%

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

Alternative 12: 34.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(t * t), $MachinePrecision] / N[(eh / N[(ew * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * 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. Taylor expanded in t around 0 82.2%

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

   1. add-cbrt-cube 64.3%

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

   2. pow3 64.3%

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

   3. cos-atan 64.9%

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

   4. hypot-1-def 64.9%

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

   5. associate-/l/ 64.9%

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

   6. un-div-inv 64.9%

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

4. Applied egg-rr 64.9%

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

5. Taylor expanded in t around 0 40.5%

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

   1. associate-/l* 40.0%

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

   2. unpow2 40.0%

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

   3. unpow2 40.0%

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

7. Simplified 40.0%

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

8. Taylor expanded in t around 0 39.7%

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

9. Final simplification 39.7%

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

Reproduce

herbie shell --seed 2023274 
(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))))))))